Zero The Biography of a Dangerous Idea Summary


    Zero The Biography of a Dangerous Idea Summary

    On September 21st, 1997, the USS Yorktown was hit by a Zero. The billion-dollar missile cruiser shuttered to a halt, and engineers spent two days getting rid of the Zero, repairing the engines and putting the Yorktown back into fighting trim.

    Computer failures are just a faint shadow of the power of Zero, which has been feared and hated throughout history.

    Zero is the story of a mysterious number, its birth in ancient times, its growth and nourishment in the East, its struggle for acceptance in Europe, and its ascendance in the West.

    Zero and Infinity were at the heart of the battle between east and West, religion and science, and the dark core of a black hole and the brilliant flash of the Big Bang are struggles to defeat Zero.

    The story of Zero is an ancient one. It was a foreign and frightening idea for ancient peoples, born in the Fertile Crescent a few centuries before the birth of Christ.

    Within Xero, there is the power to shatter the framework of logic. Civilizations functioned perfectly well for millennia before Zero was discovered.

    Life without Zero is difficult to imagine, but there was a time before the beginning of history when there was no zero, just as there was no Seven or Thirty One. Researchers discovered that stone age mathematicians used wolves instead of blackboards.

    Early humans could count up to two with wolf bones, but couldn't express quantities other than one and many. Some languages still don't have words for anything larger than three, but clever tribesmen began stringing number words in a row to yield more numbers.

    The Kyrie and Baroro peoples of Brazil use a binary system, which is similar to the Bachari and Baroro.

    Gogg's Wolf Bone had 55 little notches, arranged into groups of five. It looks suspiciously like he was counting by fives and tallied groups in bunches of five.

    Gog the Wolf Carver used a five-based or quinary counting system, but why five? Deep down it’s an arbitrary decision, and people all over the world shared Gog’s preference for counting in groups of five rather than four.

    In the Germanic proto-languages that English came from, 10 was the basic unit, and thus, those people used a base 10 number system. In French, 80 is Katravan four twenties, and ninety is Catra Vandis for twenties and Ten.

    People got along without zero for so long because it simply didn't occur to anybody to assign a symbol to the absence of objects.

    In prehistoric times, being able to count was considered a mystical and arcane talent. A dead soul had to recite a counting rhyme to satisfy the fairy men who carried departed souls across a river in the netherworld.

    Though counting abilities were rare in the ancient world, numbers and the fundamentals of counting always developed before writing and reading. Scribes simply had to figure out a coding method to set the numbers down in a more permanent form.

    The Ancient Egyptians used a system for transcribing their decimal system, where pictures stood for numbers. To write down a number, a scribe had to record groups of these symbols, and they did not have or need a zero yet.

    Most ancient peoples used a lunar calendar, but the length of the month was between 29 and 30 days, and the solar year determined the time for harvest and planting, not the lunar year.

    The Egyptians came up with a better system than the Lunar calendar, which used the sun to keep track of the passage of the days. This system was adopted by Greece and Rome, where it was modified by adding leap years, and then became the standard calendar of the Western world.

    The Egyptians invented the solar calendar and the art of geometry, and the Nile River overflowed its banks every year, erasing boundary markers and making the Nile Delta the richest farmland in the Ancient world.

    The Egyptians took property rights very seriously and had surveyors who measured land and reset boundary markers. These surveyors learned how to measure objects like pyramids and determined the areas of plots of land by dividing them into rectangles and triangles.

    The Egyptians were of a practical bent, and their best mathematicians were unable to use the principles of geometry for anything unrelated to real-world problems. The Greeks were different, and embraced the abstract and the philosophical, but they were not inclined to put math into their philosophy.

    The Greek system of numbers was quite similar to the Egyptian system, but the Greeks outgrew this primitive way of writing numbers and developed a more sophisticated system that had distinct letters for 2, 3, 300, and many other numbers.

    Though the Greek number system was more sophisticated than the Egyptian system, the Babylonian style of Counting was more advanced.

    The Babylonian system was based on the number sixty, and used only two marks to represent their numbers. These marks could be used to represent a multitude of different numbers, making the system seem perverse.

    The Babylonians invented a machine called the abacus that relied upon sliding stones to keep track of amounts. The words calculate and calcium all come from the Latin word for pebble calculus.

    A skilled user can add large numbers on an abacus by moving the stones up and down.

    The Babylonian system of numbering was like an abacus, with each grouping representing a different number of stones.

    The symbols in the number 111 stand for 1, 10, and 100, respectively. The symbol wedge stands for 1 60 or 3600.

    The Babylonians had no way to denote which column a written symbol was in, so they started using two slanted wedges to represent an empty space an empty column on the abacus. This made it easy to tell which number was represented.

    The zero mark was born out of the need to give any given sequence of Babylonian digits a unique, permanent meaning. However, it didn't really have a numerical value of its own, and it wasn't a digit, not a number.

    Using the digit zero as a placeholder without connecting it to the number Zero is unnatural because zero has a definite numerical value of its own and must sit in its rightful place on the number line before one and after negative one.

    The Mayan people of Mexico and Central America had a number system and a calendar that made more sense than ours does. They had a vigesimal base 20 system and a zero to keep track of what each digit meant.

    The Mayans had a solar calendar that had 18 months of 20 days each, and a special period of five days at the end called Uaeb. They started numbering days with the number zero, and each month had 20 days numbered 0 through 19.

    The Mayan calendar was wonderfully complicated, but the western calendar was created at a time when there was no zero. This omission caused a great deal of trouble, and we are stuck with a troublesome, zero free calendar.

    The Egyptians had a cumbersome way of handling fractions, which made ratios extremely difficult to handle. Zero makes this cumbersome system obsolete.

    The Babylonian system with zero in it makes it easy to write fractions, even better than our modern day base 10 system.

    The Greeks and Romans hated Zero so much that they refused to admit it into their writings, even though they saw how useful it was. They were afraid of nothing, and zero was inextricably linked with the void and chaos.

    Most ancient peoples believed that emptiness and chaos were present before the universe came to be. The Hebrew creation myths say that the earth Was chaotic and void before God showedered it with light and formed its features.

    The ancients feared zero because it behaved differently from the other numbers, and because adding zero to itself violated a basic principle of numbers called the Axiom of Archimedes. Zero also refused to make any other number bigger.

    This number threatens to undermine the simplest operations in mathematics like multiplication and division. When you multiply by zero, the whole number line collapses, and there is no way to get around this unpleasant fact.

    A toy store sells balls and blocks in groups of two and three, and a neighboring toy store sells a combination pack with two balls and three blocks.

    The distributive property says that multiplying by zero is the same as multiplying by open parentheses, but when you add two times zero to itself, it stays the same, so two times zero equals two times zero plus two times zero.

    Multiplying by two stretches the number line by a factor of two, dividing by two relaxes the rubber band by a factor of two, and dividing by zero destroys the number line. However, division by zero doesn't undo the destruction of the number line.

    Close parentheses divided by zero should equal four, but two times zero, three times zero, and four times zero each equal zero, so zero divided by zero equals two, three, and four.

    Multiplying by zero undoes division by 0, but dividing by zero destroys the entire framework of mathematics. Dividing by zero allows you to prove anything in the universe, but multiplying by zero collapses the number line.

    The West could not accept zero for nearly two millennia because it clashed with one of the central tenets of Western philosophy, a dictum whose roots were in the number philosophy of Pythagoras, and whose importance came from the paradoxes of Xeno.

    The Egyptians thought little about mathematics, but the Greeks took them very seriously and went overboard when it came to numbers. Hepassus of Metapontum was sentenced to death by drowning for revealing a secret that threatened to undermine the entire Greek way of thinking.

    Pythagoras was an ancient radical who believed he was the reincarnated soul of Euphorbus, a trojan hero. He was a strict vegetarian and prohibited beans as they generated flatulence and were like the genitalia.

    Pythagoras was an ancient New Age thinker, renowned scholar, and charismatic teacher. He attracted a retinue of followers.

    The Pythagoreans believed that all is number, and that the most beautiful number shapes were sacred. They believed that all disease is caused by indigestion, and that one should eat raw food and drink only water.

    The Pentagram is a five-pointed star that contains a pentagon that contains a smaller pentagon that contains a tinier star with its tiny pentagon and so forth. The golden ratio is hidden within the lines of the pentagon.

    Pythagoras discovered that strings have a peculiar yet predictable behavior. By moving a sliding bridge up and down a monocord, Pythagoras changed the notes that the device played, until he discovered that each half of the string played the same note.

    Shifting the bridge slightly might create two notes that form a perfect fifth.

    Different ratios gave different tones that could soothe or disturb. The discordant tritone was dubbed the devil in music and was rejected by medieval musicians. To Pythagoras, music was a mathematical act like squares and triangles, and the harmony of the monochord was the harmony of mathematics and the harmony of the universe. Ratios governed not only music, but also all other types of beauty, and understanding nature was as simple as understanding the mathematics of proportions.

    The Pythagoreans and later Greek mathematicians spent much of their energy investigating the properties of proportions, and one of their means yielded the most beautiful number in the world: the golden ratio.

    The pentagram was the most sacred symbol of the Pythagorean brotherhood, and the golden ratio was favored by artists and nature alike. The link between aesthetics, ratios, and the universe became one of the central tenets of Western civilization.

    Zero had no place within the Pythagorean framework because it was impossible to imagine something with width 0 and height 0 being a square.

    In Ancient Greece, the great unsolved problems in mathematics were geometric with only a straight edge and compasses. Zero was a number that didn't seem to make any geometric sense, so the Greeks chose not to include it.

    The Pythagoreans tried to squelch the irrational, a concept that would punch a hole in the neat Pythagorean order of the universe. When the secret leaked out, the cult turned to violence.

    To the Greeks, counting was tantamount to measuring a line. To make a ratio, you divide a line into tiny pieces and compare them to a standard common yardstick.

    The Pythagoreans hoped that the universe would be governed by ratios, but the universe is not really that orderly. Some numbers cannot be expressed as a simple ratio of A divided by B.

    If you draw a diagonal line from one corner to the opposite corner, the irrational appears as a concrete example. If you use a ruler 1 6 of an inch long, the diagonal comes out to more than 101, but fewer than 102 segments.

    When measuring in bits a millionth of an inch each, it is impossible to choose a common yardstick that will measure both the side and the diagonal perfectly. This means that the diagonal of a square is irrational and cannot be expressed as A divided by B.

    Pythagoras believed that the square's diagonal irrationality threatened the basis of his ratio universe, and he soon discovered that the Golden Ratio was an irrational number.

    To keep the irrationals from ruining the Pythagorean doctrine, everyone in the Pythagorean brotherhood was already tight-lipped. However, someone was going to let the secret of the irrationals out, and the fate of Hepassus mathematicians is uncertain.

    Hepassus was reviled by his brothers, but the Pythagoreans kept the irrationals from contaminating their view of the universe by considering the anomalous and irrational.

    The irrationals didn't kill Pythagoras, beans did. The master died in a bizarre way, after his house was set ablaze by his enemies who were angry at not being considered worthy to be admitted into Pythagoras presence.

    The essence of the Pythagorean teachings lived on, and was soon to become the basis of the most influential philosophy in Western history. Zero clashed with the Aristotelian doctrine, but the Greeks could ignore it because of the number shaped duality in Greek numbers.

    Around 500 bc , the placeholder zero began to appear in Babylonian writings. It naturally spread to the Greek astronomical community, but the Greeks didn't like zero at all and used it as infrequently as possible.

    The Greeks rejected zero because it conflicted with their philosophical beliefs. These ideas, the Void and the infinite, would eventually destroy Aristotelian philosophy.

    The Greek philosophers believed that the universe was governed by ratios and shapes, and that there could not be an infinite number of nested spheres. Zeno of Ilia, a philosopher reckoned by his contemporaries to be the most annoying man in the west, would defeat Greek philosophy.

    Zeno had a paradox, which baffled Greek philosophers and mathematicians for nearly 2 000 years. It proved the impossible, that nothing in the universe could move, and that swift Achilles can never catch up with a lumbering tortoise.

    Imagine that Achilles runs at one foot a second, while the tortoise runs at half that speed. Achilles catches up to the tortoise in a flash, but the tortoise scoots ahead each time, and Achilles never catches up.

    The Greeks were stumped by Zeno's Paradox, but they found the source of the trouble: Infinity. Because there are an infinite number of tiny steps, the race would go on forever and ever, but modern mathematicians have learned to handle the infinite.

    The infinite must be approached very carefully, but Xero can help you master it. Zero is the key to solving Zeno's puzzle.

    To get a finite result when adding infinite terms together, you must approach zero. This is the case with Achilles and the Tortoise.

    The Greeks rejected the number zero, so they just saw the terms as getting smaller and smaller, meandering outside the realm of numbers.

    The sum of the steps Achilles takes gets closer and closer to zero, because as we subtract the terms from two, we have nothing left.

    Achilles takes an infinite number of steps to catch up to the tortoise, but he takes only two seconds to do it.

    The Greeks couldn't do this neat little mathematical trick because they didn't believe in Zero, and they couldn't handle the infinite. This is the biggest failure in Greek mathematics, and it's the only thing that kept them from discovering Calculus Infinity Zero.

    Xeno himself didn’t have a proper solution to the paradox, but it suited his philosophy perfectly. He believed that everything is one and changeless, and his puzzles were his chief support for this theory.

    There were other schools of thought, including the atomists, who believed that the universe is made up of little particles called atoms, which are indivisible and eternal. The atomists required that the universe be filled with emptiness.

    The atomists embraced the concept of the infinite vacuum, which was a shocking conclusion, but the indivisible kernels of matter in atomic theory got around the problem of Xenos paradoxes.

    The Aristotelian system, which was later refined by the Alexandrian astronomer Ptolemy, turned the universe into a cozy nutshell and explained away Zeno's paradoxes by stating that mathematicians do not need the infinite or use it.

    The universe was contained in a nutshell, with the outermost sphere being a midnight blue globe encrusted with tiny glowing points of light.

    The heavenly spheres are slowly spinning in their places, making a music that suffuses the cosmos, but something must be moving them. This something is God, the prime mover.

    Christianity swept through the West and became closely tied to the Aristotelian view of the universe. The Aristotelian system would survive until Elizabethan times.

    With the long-standing acceptance of Aristotle came a rejection of the infinite and the void, for the existence of the void implied the existence of the infinite, and the void itself implied the existence of the infinite.

    Zero destroys Aristotle's neat argument, and the Greeks were forced to reject Void, the infinite, and infinity. However, it is not so easy to reject both infinity and zero, because the universe must have always existed.

    Aristotle rejected Zeno's infinities, but his view of physics was so influential that it eclipsed all opposing views for more than a millennium.

    Zeno got himself into serious trouble around 435 bc when he conspired to overthrow the tyrant of Elia Nirkus. Nirkus had Zeno tortured, but Zeno refused to let go and was eventually stabbed to death.

    Archimedes, the eccentric mathematician of Syracuse, was the only thinker of his day to glimpse the infinite. He was asked by the king of Syracuse to determine whether his crown was pure gold or had been mixed with lead, and he solved the problem by measuring how much water it displaced.

    Archimedes' talents were useful to the Syracusan military as well. He built stone throwers, cranes, and mirrors that set Roman ships of fire at a great distance by reflecting sunlight.

    Archimedes' war mirrors focused light from the sun onto a small area, and this was how he first glimpsed the infinite.

    Archimedes studied the properties of the parabola, and he first started toying with the infinite. He figured out a way to measure the area of a section of a parabola by resorting to the infinite, which was expressly disallowed by his mathematical colleagues.

    Archimedes included a proof based upon the accepted mathematics of the time that relied upon the so-called axiom of Archimedes, but rejected zero, which is the bridge between the realms of the finite and the infinite.

    Archimedes calculated how many grains of sand would fit in the universe using the length of a poppy seed, a fingers breadth, and a stadium. He concluded that 10 to the 51st grains of sand would fill the entire universe.

    Archimedes invented a new method of denoting really huge numbers by hitting the reset button and starting over at a myriad myriads, setting 100 million equal to one, and counting again, calling these new numbers numbers of the second order.

    Archimedes' way of doing business was cumbersome, but it got the job done and went far beyond what he needed to solve his thought experiment. The infinite was not needed in the Greek universe.

    The Romans were too powerful for the Syracusans, and poured through the city, but Archimedes was deaf to the panic around him and sat on the ground drawing circles in the sand trying to prove a theorem.

    Archimedes, the greatest mind in the Ancient world, slaughtered needlessly by the Romans. In all that time, there were no significant mathematical developments, and it would be another seven centuries before zero reappeared in the west.

    The monks couldn't be faulted for their ignorance, as they were the only Westerners who studied math. They needed math for two things: prayer and money, and they needed to know the time and the date to pray.

    Dionysius Xiglius, a monk in the sixth century, calculated the dates when Easter would fall for the next few hundred years. While translating and recalculating the Easter Tables, he realized that he could figure out just when Jesus Christ was born.

    Dionysius decided that the year of Christ's birth should thenceforth be the year One on Odomani, but he got the date of Christ's birth wrong. Today, most scholars believe that Christ was born in 4 Bc.

    Dionysius's calendar zero had a more serious problem: there was no year zero. He didn't even have a choice, because he was brought up after the decline of the Roman Empire, and the Romans were not exactly math whizzes.

    Pope John died, and all the philosophers and mathematicians were kicked out of office. Anisius Boethius was a powerful courtier who was among the finest medieval Western mathematicians, but he was clubbed to death soon afterward. The new calendar languished for years, and Bead, a monk from the northern part of England, extended it again, but his book had one significant flaw: it started with the year 60 Bc, 60 years before Dionysius' reference year.

    This style of numbering went negative, and so the proper place for a zero is nowhere to be seen.

    In 1996, a Washington Post article told people that 1996 was the 2000th year since Jesus' birth, but it was actually 1,997 years.

    Imagine a child born on January 1st in the year 4 Bc. In the years following, he would be one year old, in the years following, he would be two years old, in the years following, he would be three years old, and so forth.

    Century is only 99 years old in the year 100, and celebrates its 100th birthday in the year 101. The 21st century begins in the year 2001.

    Hotels and restaurants around the world were completely booked well in advance for December 31st, 1999, but not so for December 31st, 2000. Even the Royal Greenwich Observatory planned to be swamped by the revelers on the wrong date. Astronomers measure time in days since January 1st, 4713 Bc, a date that avoids all the weirdness caused by calendars that were constantly under construction.

    The modified Julian date system was introduced in 1858, and puts the zero hour at midnight on November 17, 1858. On December 31, 1999, the great odometer in the sky clicked ahead to the year 2000.

    Watsoff Sierpinski, a great Polish mathematician, lost one piece of his luggage, but his wife said all six pieces were here.

    When you count backward, it seems unnatural to suggest that Dionysius Embiid made a mistake when they forgot to include Zero in their calendar. However, when you look carefully enough, you will see that people usually do start counting with Zero.

    When zero came into the fold, the neat relationship between a number's cardinality and its ordinality was ruined. Now, the first hour of the day starts at zero seconds past midnight, the second hour starts at 1 A.M. and the third hour starts at 2 A.M.

    After a baby finishes his twelfth month, we say that the child is one year old. If the baby turns one before that time, we say that the child is zero years old.

    Dionysius started the calendar with year one, because people of those times thought in terms of cardinality and ordinality.

    The gaping void was not absolute nothingness, but a kind of formlessness without any definition. Medieval scholars inherited the ancient prejudices, a fear of the infinite, and a horror of the void.

    The Aristotelian system was Greek, but the Judeo-christian story of creation was Semitic and didn't have a fear of the void. Christian scholars tried to fix the Bible to match Aristotle.

    Zero was an outcast in Europe, but flourished in India and later in the Arab lands. In India, it became a number, and had no value on its own.

    Indian mathematicians first learned about the Babylonian system of numbers and about zero through Alexander the Great's invasion of India in the 4th century bc. India was also insulated from the rise of Christianity and the fall of Rome in the fourth and fifth century A.d.

    Hinduism was steeped in the symbolism of duality, and the god Shiva was both creator and destroyer of the world. However, the Hindu cosmos was infinite in extent, and to achieve nothingness again, became the ultimate goal of mankind.

    Death tells a disciple that the atman, the spirit, is concealed in the heart of all beings and is immortal. The goal of the Hindu is to free the atman entirely from the cycle of rebirth.

    The body is under the power of pleasure and pain, and if you can separate yourself from the whims of the flesh, you will be liberated.

    India accepted zero as a number, and changed its role from placeholder to number.

    Indian mathematicians used rope stretchers to survey fields and layout temples, and had a sophisticated system of astronomy. They changed their style of numbering from a Greek-like system to a Babylonian-style one around the 5th century A.d.

    Nobody knows when the Indians switched to a Babylonian-style place value number system, but the Hindu numerals were around before the Bishop wrote about them, and the placeholder in the base 10 numbering system was certainly in use by the 9th century.

    Indian numerals allowed people to add, subtract, multiply, and divide numbers without using an abacus, and they could multiply large numbers faster than an abacist could tally.

    The Indian number system was useful for everyday tasks like addition and multiplication, but its true impact was considerably deeper. It was the birth of what we now know as algebra, and it freed the Indians from the shortcomings of the Greek system of thought.

    Negative numbers made perfect sense in India and China, and Brahmagupta gave rules for dividing numbers by each other that we recognize today.

    If the numbers signs are the same as two, Minus three, then Two Minus two is now a number, and it is placed between one, two minus one, and negative one, two, minus three. However, even the Indians thought that zero was a pretty bizarre number.

    Brahmagupta tried to figure out what zero divided by zero and one divided by zero were and failed. He wrote positive or negative, Divided by Cipher is not, and he didn't make a whole lot of sense.

    Brahmagupta's mistake did not last for very long in time, because the Indians realized that one divided by Zero was infinite, and that God was found in infinity and in Zero.

    Islam began in the East with Muhammad falling into a trance on Mount Hira. By the seventh century, the Muslims had captured Egypt, Syria, Mesopotamia, and Persia, and had advanced as far as France in the east.

    Muslims absorbed the wisdom of the peoples that they conquered, and founded a great library in Baghdad. One of its first scholars was the mathematician Muhammad ibn Musa al Kawarismi, who wrote several important books, including one on the Hindu numeral system.

    The Arabs adopted Hindu Arabic numerals and called zero seifer. The West was still far from adopting zero, and even the Muslim world was heavily contaminated by the teachings of Aristotle thanks to the conquests of Alexander the Great.

    Moses Maimonides, a 12th century Jewish scholar, described the Kalam the beliefs of Islamic theologians with horror. The Muslims turned to the atomists, who believed that matter was composed of individual particles called atoms, and that there had to be a vacuum between them.

    As Islam spread throughout the Muslim-controlled world, Zero caused discord. A Muslim philosopher declared that clinging to Aristotelian doctrine should be punishable by death.

    In the 10th century, Jews were welcomed in Islamic Spain. They were wed to Aristotle's doctrines, which conflicted with Jewish theology, and Maimonides, the 12th century Rabbi, wrote a tome to reconcile the Semitic Eastern Bible with the Greek Western philosophy.

    God Maimonides argument was a proof of God's existence, but it also contradicted Aristotle's proof that the universe had always existed. Maimonides made the void a holiness for Jews, and the years after his death became the era of nothing.

    In the 13th century, a new doctrine spread called Kabbalism, which used gematria to interpret the hidden meaning of words. The Hebrews used letters from their alphabet to represent numbers, so every word had a numerical value, and words with the same numerical value were mystically linked.

    The Kabbalah was much more than number crunching. It seized upon the idea of the dual nature of God, and used the term Einsof, which meant infinite, to represent the creator aspect of God.

    As the Jews pitted their Western sensibilities against their Eastern Bible, warrior monks, scholars, and traitors began to bring Islamic ideas back to the west. The new numbers didn't catch on, even though Pope Sylvester Ii was an admirer of them.

    In the 13th century, Aristotle still had a firm grip on the church, and its finest thinkers still rejected the infinitely large, the infinitely small, and the void.

    Etienne Tompier abolished many Aristotelian doctrines that contradicted God's Omnipotence, and signaled that the foundations of Aristotelian philosophy were crumbling. The church would cling to Aristotle for a few more centuries, but the fall of Aristotle was clearly beginning.

    Zero arrived in the West in the mid 12th century, just as the Church was breaking the shackles of Aristotelianism.

    Imagine that a farmer has a pair of baby rabbits that take two months to reach maturity. The first pair reproduces, the second pair reproduces, the third pair reproduces, the fourth pair reproduces, and so on.

    Mathematicians instantly realize the importance of this series, and how it approaches the golden ratio. Fibonacci's Liber Abachi introduced Europe to Arabic numerals including zero, and the Italian merchants and bankers quickly seized upon the new system.

    Before Arabic numerals came around, money counters had to make do with an abacus or accounting board. They used tally sticks to record loans and kept the biggest piece, the stock.

    Italian merchants loved the Arabic numbers, but the local governments hated them. The governments relented in the face of commercial pressure, and the Arabic notation was allowed into Italy and soon spread throughout Europe.

    Zero had arrived, as had the Void, and the Aristotelian wall was crumbling thanks to the influence of the Muslims and the Hindus. The proof of God was no longer valid, and a new proof was needed, which was found in Zero Chapter 4.

    The Papacy was blind to the danger of the void and the infinite, but when the church was threatened, it retreated into its old philosophy, but it was too late. The void and the infinite destroyed the Aristotelian foundation of the church.

    At the beginning of the Renaissance, artists did not know how to use an infinite Zero, but an Italian architect did and created a realistic painting by using a vanishing point.

    Imagine that a gigantic hand comes down and squashes a book flat. The book is now a flat, floppy rectangle, and when turned sideways, the book is no longer a rectangle, but a line.

    Brunelleschi placed a point in the center of a drawing of a famous Florentine building, The Baptistry, and used a mirror to compare the drawing to the real thing. The reflected image matched the building's geometry exactly.

    The vanishing point, a concept that became very important later in the history of science, was caused by zero, and artists were amateur mathematicians. They perfected the technique of perspective and soon could depict arbitrary objects in three dimensions.

    The Church dabbled with Zero and the infinite, but didn't yet realize how dangerous that idea was.

    The medieval Aristotelian doctrine held that earth was the only world capable of containing life, and that any people in the heavens would naturally fall to earth. God could create life on other worlds if he so desired.

    The sky was littered with stars, planets and moons, and maybe earth glows brightly in their heavens.

    Copernicus learned mathematics so he could cast astrological tables, but his dabblings with the planets and stars showed him how complicated the old Greek system of tracking the planets was. His idea was to place the sun at the center of the universe and the planets move in simple circles.

    The Heliocentric idea was correct, but it was much simpler than the Ptolemaic system. Nicholas of Cusa and Nicolas Copernicus cracked open the nutshell universe of Aristotle and Ptolemy, and showed that there were other Popes on other planets.

    The Catholic church was under attack, and in 1517 a constipated German monk nailed a list of complaints to the door of the Church in Wittenberg. This was the beginning of the Reformation, and the Catholic church had to strike back.

    The church turned orthodox when threatened with schism, and fell back upon its Orthodox teachings. It also had other tools to fight heresy, like the Spanish Inquisition.

    The Counter-reformation was the church's attempt to rebuild the old order by crushing the new ideas, and Giordano Bruno was burned at the stake for suggesting the earth was not the center of the universe.

    Despite the Church's counter-reformation, the new Philosophy wasn't easily destroyed. Johannes Kepler refined Copernicus's theory, making it even more accurate than the Ptolemaic system, and it would prevail eventually because Kepler was right and Aristotle was wrong.

    Rene Descartes discourse on Method Zero and the Infinite was at the center of the philosophical war taking place during the 16th and 17th centuries. He rejected the void and sought a proof of God in the Void and the Infinite.

    Descartes was a mathematician and philosopher who invented what we now call cartesian coordinates. He realized that he could not start his two reference lines or axes with the number one, so he started counting with zero.

    Descartes notion of coordinates was slightly different from what we use today, because he didn't extend his coordinate system to the negative numbers.

    Descartes unified numbers and shapes no longer, and assumed that nothing can be created out of nothing. He also assumed that all ideas, all philosophies, all notions, all future discoveries already exist in people's brains when they are born.

    Descartes argued that God must exist, because all other beings are less than divine, and lie somewhere between infinity and zero.

    Descartes, a child of the Counter-reformation, learned about Aristotle at the very moment the church was relying upon his principles the most. As a result, Descartes denied the existence of the vacuum, but later in his life, he wrote about atoms and the vacuum.

    Galileo's Secretary Evangelista Toricelli proved that this wasn't True by creating the first vacuum in Italy. He used a kind of pump that could lift water up to 33 feet, but after that the water level stayed the same.

    In 1643, Toricilli placed a tube filled with mercury in a dish filled with mercury. The tube sank downward a bit, leaving a space at the top, and the mercury sank down until it could only rise 30 inches to combat the vacuum above it.

    Pascal's father was an accomplished scientist and mathematician, and the young Blaze was a genius equal to his father. However, Pascal's religion was not a comfortable fit for the young scientist, and he would use science to unravel the secret of the vacuum.

    Blaise Pascal performed experiments on the vacuum, which left the main question unanswered. The theories of the time tried to save a fragment of Aristotle's philosophy by declaring that Nature could only destroy a finite amount of vacuum.

    Pascal sent his brother-in-law up a mountain with a mercury filled tube and found that the mercury rose considerably. Pascal's experiment demolished Aristotle's assertion that nature abhors a vacuum, because the atmosphere pushes the mercury up the tube, and at the top of the mountain, there is less atmosphere pushing down so the air can't push the mercury as high as 30 inches.

    Pascal sought to prove God's existence in Zero and the Infinite by wagering on the mean between nothing and everything.

    Pascal was a mathematician and scientist who invented a new branch of mathematics called probability theory. His theory helped rich aristocrats win more money with their gambling.

    Pascal had an intense spiritual experience and abandoned mathematics and science altogether, except for a brief time when he was unable to sleep owing to illness. He became a theologian, but could not escape his profane past.

    Pascal argued that it was best to believe in God because it was a good bet literally. He analyzed the value of accepting Christ as savior using the mathematics of zero and infinity and concluded that one should assume that God exists.

    Pascal's Wager is exactly like this game, except that it uses a different set of envelopes. If you are offered a choice between the two envelopes, the smart thing to do is to choose B.

    Pascal analyzed the Christian case, but the Atheist case is the logical extension. If you choose the Christian envelope, you go to Heaven if there is a God, or you fade into nothingness.

    The expected value of being a Christian is infinity, because half of infinity is still infinity.

    The value of being a Christian is as bad as you can possibly get, since there is only a 1000th chance that God exists.

    Pascal's Wager, which is based on the idea that there is no chance that God exists, makes no sense, because the expected value of being a Christian would be zero times infinity, and that was gibberish.

    The Scientific Revolution was fueled by the introduction of the infinitely small and infinitely large mathematics usually so strictly ethical fell from grace, and the realm of controversy was inaugurated. However, there was a problem with the scientific world's powerful new tool, Calculus.

    After a thousand-year stupor, European thought revived and the problem of infinity was one of the first to be revived. The Greeks were stumped by Achilles infinite steps, but the West embraced zero and ended Achilles race.

    The infinite is not quite that simple, because the numbers in the sequence get closer and closer to zero.

    A French mathematician tried to sum the terms in an infinite sequence of numbers, but realized that the sum went off to infinity, even though the individual terms went to zero.

    The strangest aspect of infinite sums is that zero itself is not immune to the bizarre nature of infinity. The following series sums to zero after all.

    Open parentheses plus one minus one close parenthesis plus open parentheses plus one minus one close parentheses plus and so forth sum to zero, but beware of grouping the series in a different way, which clearly sums to one.

    The infinite sum of zeros can equal anything. A priest used it to prove that God could create the universe.

    Johannes Kepler, the man who figured out that planets move in ellipses, spent a year gazing into wine barrels and added infinite things to each other to yield their volumes. This may seem like a backward way of going about measuring barrels, but it was a brilliant idea.

    To estimate the area of a triangle, inscribe little rectangles inside the triangle. The area of the triangle is half the base times the height, so if we inscribe three rectangles, we get a value of 24 closer to the actual value of 32.

    Modern notation replaces Sigma with integral and Delta X with derivative, turning the equation into integral function X Derivative, which is the integral in one of Kepler's lesser known works: volume measurement of barrels.

    Kepler and Galileo sliced objects infinitely thin, and Cavalieri added up an infinite number of zeros to figure out the area and volume of geometric objects. For geometers, this statement was troublesome, because adding infinite, zero area lines could not yield a two-dimensional triangle.

    A tangent is a line that just kisses a curve, and it is extremely important in studying motion. For instance, the slope of a tangent line tells you how fast a bicycle is going when it reaches a certain point.

    Mathematicians created different methods for calculating the tangent line to any given point on a curve, but they all came up against the infinitesimal. The most important property of a line is its slope, and to measure this, mathematicians look at how high a line rises in a certain amount of distance.

    When calculating the slope of a tangent line, zero wrecks your approximation process as your approximations of the tangent lines get closer together, and the difference in height between the points goes to zero.

    Mathematicians encountered trouble when dealing with the infinite and zero, but these infinites and zeros are the key to understanding Nature zero and the mystical calculus.

    The tangent problem and the area problem are both aspects of calculus, a scientific tool far more powerful than anything ever seen before, yet their very fabric was infused with zeros and infinities that threatened to destroy the new tool.

    Isaac Newton, the first discoverer of calculus, nearly died before he ever took a breath. He developed a systematic method of solving the tangent problem, but his method of differentiation didn't look very much like the one we use today.

    The amount of time that had passed was represented by the letter O. The equation became open parentheses Y plus a Y dot close parentheses equals open parentheses X plus o X dot close parentheses squared plus open parentheses X plus o X dot close parentheses plus one. Since Y equals X squared plus X plus one, we can subtract Y from the equation and add X squared plus X plus one to leave the equation unchanged.

    Newton declared that O X dot was really really small, so he could ignore it. This gives us o Y dot equals 2x plus 1 and o Y dot divided by O X dot equals 2x plus one.

    The method gave the right answer, but Newton's vanishing act was troubling because it implied that O X dot itself must be equal to zero. Division by zero is forbidden by the Logic of Mathematics Newton's.

    The Method of Fluxions solved the tangent problem and the area problem by relying upon an illegal mathematical operation. The formula for the area under a curve is simply Y equals X squared plus X plus one, where C is any constant you choose.

    Nature speaks in equations. The rules of mathematics were built around counting sheep and surveying property, yet these very rules govern the way the universe works.

    In 1662, Robert Boyle showed that squishing a sealed container with a gas inside would increase the pressure inside, and in 1676, Robert Hook showed that the force exerted by a spring is negative kx.

    These early equation laws were extremely good at expressing simple relationships, but they had limitations that prevented them from being universal laws. For example, the famous equation Rate times Time equals distance is wrong for objects that move at a constant rate.

    Calculus allowed Newton to combine all these equations into one grand set of laws, which applied in all cases under all conditions. Even though mathematicians knew that calculus was deeply flawed thanks to the mathematics of Zero and Infinity, they quickly embraced the new mathematical tools.

    Newton's method of flexions tied together concepts like position, velocity, and acceleration by creating a simple differential equation that describes the motion of all objects in the universe.

    F equals Mx2 dot is a universal law, but it only holds when the mass of an object is constant.

    If you know how an object moves, such as a ball in free fall or a spring, you can use the differential equation to find out what kind of force is being applied.

    Newton's triumph was figuring out the shapes of the planet's orbits using the equation that described the force of gravity. Leibniz's work had the same flaw, and the two theories came up with the same answers, but their notations and their philosophies were very different.

    Newton disliked infinitesimals, and swept them under the rug. Leibniz reveled in the infinitesimal, and his derivatives could be manipulated just like ordinary numbers, but underneath it all, his differentials still had the forbidden zero divided by zero nature that plagued Newton's fluxians.

    Leibniz tried to convert the Chinese to Christianity using his new mathematics, which he had derived from God's division by one and void's division by zero.

    Mathematicians would have to wait many years before they could free calculus from its mystical underpinnings, because they were busy fighting over who invented calculus. In the meantime, Leibniz developed his own calculus, and the English stuck to Newton's fluxian notation, rather than adopting Leibniz's superior differential notation.

    Guillaume Francois, Antoine de L'hopital was a marquis who bought himself the best teacher that money could buy. He persuaded Johann Bernoulli to send him all his new mathematical discoveries in return for cash, and the result was a textbook.

    L'hopital explained the fundamentals of calculus in his textbook, and included some exciting new results. The most famous is known as L'hopital's Rule, which provided a way to figure out the true value of a mathematical function that goes to zero divided by zero at a point.

    Clever manipulations could also resolve other odd expressions, such as Infinity divided by Infinity and Zero to the zeroth power. This is why Zero divided by Zero is dubbed indeterminate.

    Bernoulli claimed that L'hopital had stolen his work, but L'hopital proved himself an able mathematician and his correspondence with Bernoulli backs up his story. L'hopital's Rule was extremely important for resolving some of the difficulties with Zero Divided by Zero, but the underlying problem remained.

    Seven years after Newton's death, George Berkeley wrote a book entitled The Analyst, in which he pounced on Newton's and Leibniz's dirty tricks with zeros, and showed how making infinite decimals disappear with impunity can lead to a contradiction.

    In those days, calculus was very different from other realms of mathematics. It was based on faith, and everybody accepted the fact that making infinitesimals disappear at the right time gave the correct answer.

    A quantity is either something or nothing. The supposition that there is an intermediate state between these two is a chimera.

    Mathematicians all over Europe were having stunning successes with the new tool, calculus. Colin Mclaren and Brooke Taylor discovered how to use calculus to rewrite functions in a totally different form, and Leonard Euler proved that the sum of 1 divided by x cubed plus 1 divided by X squared equals zero.

    Euler was an excellent mathematician, but his careless manipulation of zero and infinity led him astray.

    A bear named Jean Lauren was found on the steps of the Church of Saint Jean-baptist Laurent in 1717. He eventually took the surname Dalamber and was best known for his collaboration on the famed Encyclopedia of Human Knowledge.

    The Achilles Problem vanishes if you consider the limit of the race, where Achilles passes the tortoise at the two foot mark. But how do you prove that two feet is actually the limit of the race?

    If you challenge me to a distance of one thousandth of a foot, I will meet your challenge with 23 millionths of a foot to spare.

    Achilles is getting arbitrarily close to the two-foot mark, even if you challenge him with a distance of one billionth of a foot.

    Achilles runs one foot, one half foot, one foot, one half foot, one foot, one quarter, one point, seven, five feet.

    Dollambear rewrote the infinite sum to be one plus one half plus one quarter plus one eighth, plus and so forth.

    When you have an infinity in an expression, or when you divide by zero, all the mathematical operations go out the window. Even the plus sign doesn't seem so straightforward.

    Achilles sub-races are each finite. You can add, divide, square them, and do whatever you want with them, but the limit doesn't exist for the infinite sum of plus one and negative one.

    Modern mathematicians divide by a number that approaches zero and then take the limit, instead of dividing by zero as Newton and Leibniz did. This logic satisfies the mathematician's strict requirement of logical rigor and puts calculus on a solid foundation.

    Zero and Infinity are equal and opposite Yin and yang, and it is possible to understand the infinite by studying Zero. To learn this, mathematicians had to venture into the world of the imaginary.

    Algebra was a new way of looking at numbers, divorced from Greek geometric ideas. It allowed us to find solutions to equations that encode relationships between different numbers, such as four X minus twelve equals zero.

    When stringing symbols together to get equations, you can wind up with something unexpected, such as a negative number. The East embraced negative numbers, while the West tried to ignore them.

    Algebraists used negative numbers to solve equations such as 4x minus 12 equals 0. But they soon turned to more difficult problems.

    Quadratic equations are more complicated than regular equations, and can have two different roots. If X is 1 or negative 1, the expression goes to 0.

    Quadratic equations are more complicated than linear equations, but there is a simple way to figure out what the roots are. The quadratic formula is the crowning achievement of high school algebra class.

    Al Khawarizmi knew how to solve quadratic equations, but didn't consider negative numbers as roots. Later, algebraists accepted negative numbers as valid solutions to equations.

    No number seems to solve the equation X Squared plus one equals zero. The quadratic equation gives two silly sounding answers plus negative one and minus negative one.

    The Indian mathematician Bhaskara wrote that a negative number is not a square, but when you square a positive number, you get a positive number back.

    Negative two times negative equals four. Positive numbers, negative numbers, and zero all give you non-negative squares.

    Descartes thought that square roots of negative numbers were even worse than negative numbers, and gave them the scornful name imaginary numbers.

    When you allow I into the realm of numbers, every polynomial becomes solvable. Cubic expressions split three ways, Quartic expressions split four ways, and Quintix expressions split five ways.

    Mathematicians used complex numbers to solve polynomials as early as the 16th century, but it took two important developments in mathematics before the true relationship between zero and infinity was uncovered: Point and counterpoint.

    Projective geometry was born in the turmoil of war in the 1700s. France, England, Austria, Prussia, Spain, the Netherlands, and other countries fought each other for nine years before France capitulated and the Seven Years War was over.

    The French Revolution began four years after the American Revolution, and a mathematician signed the record of the king's execution. His work was so important to the military that it was made into a state secret.

    Jean Victor Poncelle learned about three-dimensional geometry while training to become an engineer for Napoleon's army. He was captured by the Russians and left for dead on the battlefield.

    Porcelain founded a new discipline, Projective Geometry, which was the culmination of the work begun by artists and architects in the 15th century. Johannes Kepler took this idea and discovered that planets travel in ellipses.

    A parabola is an ellipse with one focus at infinity. When you stretch out an ellipse, it opens up and becomes a parabola, and all the lines that converge to a point become parallel lines.

    Kepler's Point at Infinity proved that parabolas and ellipses are actually the same thing, and this was the beginning of projective geometry.

    Gerard Desarga, a 17th century French architect, used the Point at Infinity to prove several important theorems, but his work was forgotten.

    Jean-victor Poncelle, a prisoner of war, reinvented the concept of a point at infinity and raised the discipline to a high art, however possibly had no idea that projective geometry would reveal the mysterious nature of zero.

    Carl Friedrich Gauss, born in 1777, was a German prodigy who began his mathematical career with an investigation of imaginary numbers. His work on curvature would become a key component of Einstein's General Theory of Relativity, but it was his way of graphing complex numbers that revealed a whole new structure in mathematics.

    A simple construction called the complex plane revealed a lot about the way numbers work. For example, the angle between I and the X-axis is 90 degrees, and I squared equals negative one.

    A student of Gauss's Georg Friedrich Bernhard Riemann combined projective geometry with the complex numbers and all of a sudden lines became circles, zero and infinity became the poles on a globe full of numbers.

    Riemann imagined a translucent ball with a south pole touching zero, and a north pole like the point at infinity that Kepler and Poncelet imagined. Every point on the ball has a shadow on the complex plane.

    Mathematicians could see multiplication, division, and other difficult operations by analyzing the way the sphere deformed and rotated. For example, multiplying by the number I was equivalent to rotating the whole globe by 90 degrees.

    If you flip a sphere upside down and reflect it in a mirror, the north pole becomes the south pole and the south pole becomes the north pole.

    Multiplying by two is like stretching a rubber cover away from the south pole and toward the north pole, multiplying by one half is like stretching a rubber cover toward the north pole and away from the south pole.

    The numbers in the complex plane are drawn inexorably towards zero or toward infinity. The only numbers that escape are the ones that are equally distant from the two.

    Infinity was no longer mystical, it became an ordinary number, and mathematicians were quick to analyze it. But in the deepest infinity, nestled within the vast continuum of numbers, Zero kept appearing, and mathematicians started analyzing and classifying the points where a function blows up singularities.

    A curve divided by x has a singularity at the point x equals zero, and a curve sine divided by x has an essential singularity at x equals zero. These curves go absolutely berserk near a singularity of this sort.

    Gerg Cantor, the master anatomist of the infinite, lived in Germany, the land of Gauss and Riemann, and the land of Leopold Chronicler, the mathematician who would hound Cantor into a mental institution. Cantor had a vision of the Infinite, which he described with a simple puzzle.

    Cantor generalized this trick and said that two sets of numbers are the same size when one set of numbers can sit on top of another set of numbers one to a customer with none left over.

    When we start removing numbers from the set of whole numbers, the size of the set doesn't change at all. We can ensure that everybody has a seat and that every seat is filled by rearranging the seating pattern slightly.

    The even numbers, odd numbers, whole numbers, and integers are all the same size, and are called countable.

    Even the rational numbers are an olive not sized set, because the real numbers are much bigger than the rationals.

    Imagine that we have a perfect seating plan for the real numbers. We can make a list of seats showing a seat's number along with the real number that is sitting in it.

    Cantor created a real number that was not on the list, and we can easily prevent that from happening.

    Since the first number on the list starts with a 3, we know that the two numbers are different. However, there is a minor point that can be easily overcome.

    We can make sure that our new number is different from the second number on the list by changing the second digit of the second number to a 7. We can do the same thing on down the list.

    Seat One Real Number Point Three, One, Two Five, One Two Three Our new numbers. The first digit is two different from three, the second digit is seven different from eight, the third digit is eight different from nine, etcetera.

    We can create a new number by taking all the numbers on a seating list and ensuring that their first digits don't match. This ensures that the new number is different from all the other numbers on the list.

    The real numbers are a bigger infinity than the rational numbers. Mathematicians struggled to determine whether the Continuum infinity was indeed Aleph one, but Paul Cohen proved that this puzzle was neither provable nor disprovable.

    Cantor's mind was filled with an infinite number of infinities, the trans-finite numbers, each nested in the other, leading to the ultimate infinity, God.

    Leopold Chronicler, one of Contour's teachers, believed that God would never allow such ugliness as the Irrationals, much less an ever increasing set of Russian doll infinities. Chronicler launched vitriolic attacks against Contour's work and made it extremely difficult for him to publish papers.

    Using set theory, mathematicians created numbers that were previously unheard of, and can be added to, multiplied by, subtracted from, and divided by other infinities. Contour was in and out of mental institutions for the remainder of his life.

    Contour's theory would show that the rational numbers are nothing, but an infinite zero, while Cantor's hierarchy of infinities would show that the rational numbers take up very little space on the number line.

    It takes a clever trick to figure out the area of an irregularly shaped object, such as a stain on a wood floor. If a rectangular carpet covers the stain entirely, we can approximate the area of the stain using a one square foot carpet.

    Let's do the same thing with the rational numbers, but this time our carpets are sets of numbers. This has some very odd consequences, as contour soon showed thanks to his seating chart.

    Take the first rational number and imagine it on the number line. Cover it with a carpet of size one, then with a carpet of size one half, and so forth.

    The total size of the carpets is two, and the rational numbers take up less than two units of space. If we start with a carpet of size one-half, the total size of the carpets is one, and the rational numbers take up less than one unit of space.

    All the rational numbers take up less room than one 500th unit, and yet all can fit on a carpet the size of half an atom.

    We can get smaller and smaller. We can cover the rationals with carpets that some together fit in the size of half an atom or a neutron or a quark, but we can still cover the rationals because they take up no space at all.

    Mathematicians had to learn to live with zero and infinity, but physicists had no choice but to encounter zeros in the natural world. A zero became an uncrossable barrier, the coldest temperature possible, a black hole, and a bizarre source of energy.

    When you cannot measure something and express it in numbers, your knowledge is meager and unsatisfactory.

    William Thompson, Lord Kelvin, discovered a law that had been in use for half a century. This law was discovered by Jacques Alexander Charles, a French physicist already famous for being the first to fly aboard a hydrogen balloon.

    Charles' Law describes the relationship of the volume of a gas to its temperature, but William Thompson noticed something odd about Charles' Law.

    The specter of Zero says that a balloon of gas must shrink to zero space, but it can't go on shrinking forever. This is absolute zero, a little more than 273 degrees celsius below the freezing point of water.

    Zero Degrees is the freezing point of water in the Kelvin Scale. Absolute Zero is an unattainable goal because everything in the universe is conspiring to stop you from actually reaching absolute zero.

    You can't stop the wiggling of the atoms in the box that a banana is in, or the radiation they give off, because the banana is constantly absorbing energy from the box and the tweezers you use to manipulate it and the refrigerator coil you use to cool it down.

    Every object is influenced by the environment, so it is impossible to cool anything in the universe to Absolute Zero. This discovery was disappointing news to the physics world, but it was the beginning of a new branch of physics.

    Thermodynamics is the study of the way heat and energy behave. It tells you that it is impossible to create a perpetual motion machine and that even getting a machine to run without wasting energy is impossible.

    Thermodynamics is worse than a casino because you can't win. Statistical mechanics explains Charles's law, which states that gas pushes harder on the walls of its container as the temperature increases.

    Scientists believed that light was composed of little particles that flowed from every bright object over time, but in 1801, a British scientist discovered that light interferes with itself.

    When two stones are dropped into a pond, the ripples cancel out, and you can see lines of still, wave-free water. The same thing happens when light shines through two small slits, and you can see faint dark lines at home.

    The phenomenon of interference seemed to settle the question of light's nature once and for all. Physicists concluded that light was a wave of electric and magnetic fields.

    The hotter an object is, the faster its molecules move, and the more energetic the ripples of light it sends out.

    The Stefan Boltzmann Equation relates the temperature of an object to the total amount of light energy it radiates. This is the law that physicists used to determine that Heaven is more than 500 degrees Kelvin.

    At the turn of the century, two British physicists tried to use the Wiggle Theory to solve a simple problem, and came up with an equation that predicted the amount of large wavelength low energy light that comes off a hot object at high energies.

    According to the Raleigh Genes equation, every object is constantly radiating an infinite amount of energy. However, following the then accepted rules of physics led inexorably to the conclusion that ice cubes don't wipe out civilization with bursts of gamma rays.

    The Ultraviolet Catastrophe led to the Quantum Revolution, which got rid of the zero in the classical theory of light.

    In 1900, German experimenters showed that the Raleigh Gene's formula was failing to predict the true amount of light that comes off objects at various temperatures. Max Planck's formula explained the new measurements and solved the Ultraviolet Catastrophe, but had to change the laws of Physics to accommodate it.

    According to Planck, molecules are forbidden to move in most ways, and can only vibrate with certain acceptable energies called quantum. Nature doesn't move in jumps, and a quantum car would behave in exactly this way.

    Quantum people could not grow very easily, but when they stepped on the gas, they would instantly pop. Be driving 40 miles an hour.

    The quantum hypothesis violates everything our everyday experience tells us, even though it is the correct formula for the frequencies of light that come off an object.

    Albert Einstein, a 26 year old patent clerk, showed the physics world that nature worked in quanta rather than smooth increments. He would later become the chief opponent of the theory he helped create.

    Even a dim beam of ultraviolet high frequency light causes electrons to get knocked out of the metal. However, if you lower the frequency just a little bit beyond a critical threshold, making the light a wee bit too red, the sparking stops all of a sudden.

    Einstein solved the puzzle of the photoelectric effect by proposing that light comes in little packets of energy called photons. This idea contradicted the wave theory of light, but explained the photoelectric effect brilliantly.

    Light acts like a particle sometimes and like a wave at other times. It is neither particle nor wave, but a strange combination of the two, and the force of nothing is caused by the Heisenberg uncertainty principle.

    The concept of uncertainty pertains to scientists ability to describe the properties of a particle. For instance, we can't measure a particle's position and velocity with perfect accuracy at the same time because the very active measuring destroys some of the information we are trying to gather.

    The Heisenberg uncertainty principle shows that you can never measure a particle's position and its velocity with perfect accuracy at the same time, and vice versa.

    Imagine an extremely tiny volume in space, like a really small box. We can make some assumptions about what is going on inside the box.

    The Heisenberg uncertainty Principle implies that we must have some uncertainty about the particle's velocity, their energy. The answer to how the vacuum can have any energy comes from Einstein's Famous E equals M C squared formula.

    The vacuum is never truly empty, instead it is seething with virtual particles. The zero-point energy is limitless according to the equations of quantum mechanics.

    When an equation has an infinity in it, physicists usually assume that something is wrong. However, sometimes the zero point energy does matter.

    Heinrich Bg. Casimir and Dick Polder realized that the zero-point energy can't always be ignored when studying the forces between atoms.

    Pythagoras saw that waves traveled up and down a plucked string, and that certain notes were allowed and others were forbidden. He realized that matter waves were not so different from string waves, and that certain particles were forbidden from being inside a box. Casimir realized that the forbidden particle waves would affect the zero point energy of the vacuum, and that this would cause two metal plates to crush together.

    In 1995, Stephen Lamoreaux measured the casimir force, a mysterious phantom force exerted by nothing at all, and found that it was about the weight of one slice of an ant chopped into thirty thousand pieces.

    The theory of Relativity was born in light, but the speed of light created a paradox. The speed of light should depend on whether you are running toward or running away from the light bulb that's shining on you.

    The American physicists Albert Michelson and Edward Morley were baffled when they tried to measure the speed of light.

    Einstein assumed that if a number of people watch the same phenomenon, the laws of physics are the same for each observer, even if they disagree about some of the details. This is the principle of Relativity in the Special Theory of Relativity.

    Einstein assumed that everyone agrees about the speed of light in a vacuum being 300 million meters per second, a constant denoted by the letter C. This assumption challenged everything physicists had assumed about the motion of objects.

    A person who zooms away at great speed takes more than a second off a stationary observer's stopwatch.

    Time and length change with speed. At nine tenths of the speed of light, an object gets shorter and heavier, but its weight stays the same.

    When a subatomic particle travels very fast, its clock slows down, and a very precise clock slows down when flown in an airplane at great speed.

    When a spaceship approaches the speed of light, time slows down more and more and more. The astronaut aboard the spaceship experiences a multiplier of zero, but it is not so easy to stop time as the spaceship goes ever faster.

    You can accelerate a spaceship to near the speed of light, but it can never reach the speed of light.

    The General Theory of Relativity describes the ultimate zero and the worst infinity of them all, the Black Hole. It changes the way you move through space and through time. Space-time is like a gigantic rubber sheet, and objects sitting on it distort it slightly. This distortion is gravity, and the pull of gravity is just like the tendency of objects to roll into the dimple.

    The curvature of the rubber sheet is an analogy for the curvature of space and time. As space gets distorted close to a massive object, so does time, and so does mass.

    A black hole is a star so dense that nothing can escape its grasp. It begins as a big ball of hot gas, mostly hydrogen, but as it collapses, it gets hotter and denser and hydrogen atoms slam into one another, fusing and creating helium and releasing large quantities of energy.

    During most of its life, a star is in an uneasy equilibrium with its own gravity, but the fusion reaction dims and the equilibrium is upset after a while.

    The sun has about five billion years of fuel left, but after a series of death throws, the star will collapse under its own gravity.

    The Pauli Exclusion Principle states that no two things can be in the same place at the same time, but electrons can't move faster than the Speed of light, so if a star has enough gravity, its electrons will give up their struggle and smash into the protons creating neutrons.

    When a massive star collapses, it winds up being a gigantic ball of neutrons. The pressure of the resulting neutrons can stave off collapse for a little while, but eventually the star disappears due to the gravitational attraction of the surrounding space.

    A black hole is an object so paradoxical that some scientists believe that black holes can be used to travel faster than light and backward in time. Because a black hole takes up zero space, it causes space time to curve, and the curvature goes off to infinity.

    Einstein denied the existence of black holes, but nature has a cosmic sensor that allows spacecraft to escape from black holes. The escape velocity of a spacecraft is faster than the speed of light past the event horizon.

    Even with an impossibly strong spacesuit, you can't tell anybody about what you saw beyond the event horizon. The cosmic sensor can detect black holes in the direction of the Constellation Sagittarius, but it can't see their zeros at their centers.

    If there were no event horizon, a naked singularity might allow you to travel faster than light or backward in time. A wormhole could be formed if a black hole was spinning or had an electric charge.

    Nasa is hoping that wormholes exist, and that they hold the secret to traveling to distant stars. Wormholes are paradoxes caused by a zero in the equations of General Relativity, and might even allow you to track down your mother and kill her before she meets your father.

    Space travel is difficult because there is nothing to push against. You can paddle all you want, but you'll get nowhere.

    Nasa's Breakthrough Propulsion Project is trying to develop an engine that uses zero fuel and could send probes to distant stars in a reasonable amount of time.

    Unfortunately, black hole singularities are unlikely candidates for wormholes in the short term. Even a spinning black hole with a nice ring-shaped singularity will kill an astronaut thanks to mass inflation.

    Millis says that the zero point energy in quantum mechanics might be the ultimate fuel, and that a one-way mirror that reflected virtual particles on one side but let them pass unhindered through the other would be a good way to push a spaceship.

    Millis admits that nobody has any clue how to engineer a quantum sail, because the laws of physics say you can't get something for nothing.

    Harold Putoff believes that a quantum sale would simply alter the properties of the vacuum, and that psychics could view objects remotely.

    The vacuum decays to a slightly lower state, says putoff, and engines could run solely on zero point energy. However, the universe would fall apart slowly.

    The Kasimir Force testifies that the vacuum has energy, but it is possible that the energy of the vacuum is not the lowest possible energy. If it is, tinkering with the energy of the vacuum might cause the universe to self-destruct.

    Putoff believes he can use the casimir effect to extract energy from the void, even at absolute zero. A conducting cylinder-shaped gas would be compressed by the zero-point fluctuations, releasing energy, just as plates are forced together by the zero-point fluctuations.

    Putoff claims to have gotten out 30 times more energy with a bolt of electricity than was put in.

    Putoff's device is one in a long line of free energy machines, none of which have withstood scientific scrutiny.

    Modern physics is a struggle of two titans, General Relativity and Quantum Mechanics. Black holes lie at the juxtaposition of these two theories, and Zero dwells at the center of a black hole, where the two theories clash.

    The Big Bang is the most puzzling event in the history of the universe. To explain it, physicists must marry quantum theory with Relativity, and banish zero.

    When we try to calculate down to zero distance, the equation blows up in our face and gives us meaningless answers.

    General Relativity and Quantum Mechanics are incompatible because zero, the infinite zero of a black hole mass crammed into zero space, punches a hole in the smooth rubber sheet of General Relativity.

    Quantum mechanics has a similar problem, in that scientists don't even know the mass or charge of an electron. The answer lies with zero.

    The electron that scientists see in the laboratory is an imposter. The true electron is hidden in a shroud of particles made up of the zero point fluctuations, and if scientists could invent a tiny device that could get a short distance inside the cloud, they would see more clearly.

    Quantum theory says that as a measuring device gets closer to an electron, it passes more and more virtual particles, and the measured mass and charge go to infinity.

    According to the rules of quantum mechanics, the zero dimensional electron has infinite mass and infinite charge. Scientists learn to ignore this fact and calculate the electron's true mass and charge at an arbitrary distance.

    Quantum mechanics deals with zero dimensional particle points, so when two particles merge at a zero dimensional singularity, it makes no sense, so physicists simply got rid of it.

    String theory was created in the 1970s to explain why black holes and electrons are zero dimensional and why the infinities in General Relativity and quantum mechanics miraculously disappear.

    The infinite mass and charge of an electron vanishes if the electron is a loop of string. This means that the mass and charge don't go off to infinity as you approach the electron, and as two particles merge, they form a smooth, continuous surface in space-time.

    If you imagine a black hole as a string, particles fall through a rip in space-time and merge with the black hole to create bizarre particles such as tachyons.

    String Theory eliminates zero from the universe, smooths out the discontinuous particle-like nature of quantum mechanics, and mends the gashes torn in General Relativity by black Holes.

    Physicists thought that String Theory would unify quantum mechanics with Relativity, but it required ten dimensions, which is one too many for most people. Einstein showed that time is similar to space and that we can change the rate at which we move through time. The other six dimensions are rolled up like little balls and are too tiny to see.

    Six extra dimensions are mathematical constructs that make string theory work like imaginary numbers. We can't see, feel, or smell them, but they are necessary for doing calculations.

    Nowadays, physicists realize that all competing varieties of string theory are actually dual to each other, and that there is a monster theory that lives in 11 dimensions.

    Particle physics uses magnetic fields to move tiny particles very fast, and then they collide with one another, spitting off fragments.

    String Theory is not yet powerful enough to observe strings directly, and no experiment can be designed that will give physicists evidence that black holes and particles are indeed strings.

    Recent theories propose that some dimensions might be larger than 10 to the minus 19 centimeters.

    Newton's Laws of Motion and Gravitation gave physicists an explanation for the way planets and objects move through the universe. Einstein's theories corrected Newton's errors and made testable predictions about the way gravity works.

    String Theory ties together a number of existing theories in a very pretty way, but it is not yet science for the foreseeable future. Zero has not yet been banished from the universe, and it may well be correct, but we may never have the means to find out.

    It took a long time before astrophysicists agreed that the universe was finite. The prejudice against a finite universe is ancient, and Aristotle's idea of an eternal, unchanging universe led Einstein to make the greatest mistake of his career.

    To Einstein, the General Theory of Relativity foretold the end of the universe. The universe would collapse under its own gravity, burn brightly with radiation, destroy all life, and eventually crunch itself into a zero dimensional point like a black hole and disappear forever.

    Einstein proposed a cosmological constant and as yet undetected force that counteracts the force of gravity to stave off the impending destruction of the universe. However, the stars themselves destroyed Einstein's vision of a static, eternal cosmos.

    In 1990, astronomers had little idea that anything lay beyond our own dusty little disk of stars. In the 1920s, an American astronomer named Edwin Hubble used a special type of star called a Cepheid variable to measure the distance to faraway objects.

    Hubble found a Cepheid star blinking in one of these swirly clouds and calculated that the nebula was a million light years away, far beyond the outer reaches of our galaxy.

    Today, astronomers suspect that the universe is about 15 billion light years across and peppered everywhere with clusters of galaxies.

    Hubble's second discovery was that galaxies were fleeing from the Milky Way at high speed, but their velocities were not directly measurable.

    The doppler effect is the same as when a train approaches you, its horn is high pitched, but as it passes you, its pitch drops dramatically. Astronomers can tell how fast a star is moving away from us by looking at its light spectrum.

    Imagine a polka dotted balloon, where the polka dots are like galaxies, and the balloon is the fabric of space-time.

    The universe expands as time runs forward, but if you ran the film backward, the universe would shrink.

    The singularity at the beginning of time and space is the primal zero, the birthplace of the universe, the Big Bang, a furious explosion that created the cosmos. Steady State Theory was an alternative to the Big Bang steady State theory, but was killed by pigeon droppings.

    In 1965, several astrophysicists at Princeton University calculated that the entire universe must have been glowing with bright light right after the Big Bang. This light was the afterglow of the Big Bang.

    The Princeton scientists predicted the existence of cosmic background radiation, which was confirmed at Bell Labs in nearby Murray Hill, New Jersey. The engineers got the Nobel Prize, but the Princeton scientists got nothing.

    The Big Bang had been spotted, and physicists gradually accepted that the universe had a beginning. However, the universe is somewhat lumpy, and all the matter did not wind up in one huge glob.

    If the universe had come from a singularity, the balloon should be evenly shaded or have one giant spot rather than being polka dotted.

    The zero of the vacuum might explain the lumpiness of the universe. The zero point energy in the early universe was greater than it is today, and this extra energy would have pushed objects around, smoothing out the lumpiness of the universe.

    According to the theory of inflation, the universe was created when a pot of water was instantly flash heated to a huge temperature. Little bubbles of true vacuum formed and expanded at the speed of light.

    Zero might hold the secret to the origin of the cosmos, and the existence of an infinite number of other universes. It is at the zero hour of the Big Bang and the Ground Zero of the Black Hole that the mathematical equations that describe our world stop making sense.

    Scientists are trying to eliminate Zero from their equations, but Zero may have the last laugh. If there is enough mass in the universe, the balloon of space-time will not be able to expand forever, and the universe will die a heat death.

    Astronomers peer at distant galaxies and look backward in time to see what light left the galaxy a million years ago.

    If the expansion of the universe is slowing down, the energy from the Big Bang might have given the fabric of space-time enough of a kick to let it expand forever.

    Astronomers have begun to measure the change in the universe's expansion by comparing the doppler shift of galaxies to determine how fast space time is expanding. They found that the expansion of the universe isn't slowing down, but may even be speeding up.

    The force of the vacuum might be the cause of the expansion of the universe, and the fate of our universe may be eternal expansion, cooling, and heat death.

    Astronomers are still cautious about these supernova results, but they are getting more solid with each observation. If they discover a complete theory, it will be understandable in broad principle by everyone, not just a few scientists.

    Zero is behind all of the big puzzles in physics, yet dividing by zero destroys the fabric of mathematics and the framework of logic and threatens to undermine the very basis of science in Pythagoras day.

    Mathematicians and physicists managed to overcome the division by Zero problem in the calculus, but zero returned in the equations of Quantum Mechanics and General Relativity and tainted science once again. Scientists set out to banish zero yet once more and unify the rules that govern the cosmos.

    The arguments of string theorists and cosmologists might be mathematically precise and consistent, but they are also utterly wrong. The universe began and ends with Zero.

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