An Overview of the History of the development of Math The Impact of Zero

The very beginning as we know it Stone age mathematicians were a bit more rugged than modern ones. Instead of whiteboards, they used wolves. In the 1930’s, archaeologist Karl Absolom unearthed a 30,000 year old wolf bone in Czechoslovakia. The interesting thing is that this bone had notches carved into it. An even more interesting thing is that these notches were arranged into groupings of five for a total of 55 notches. This IS a quinary system. Today we use the decinary.

Images digitally enhanced 2003 at the Deutsches Museum in Munich, Germany

Zero – A mysterious concept Zero is powerful because it is infinity’s twin. They are equal yet opposite, yin and yang. Throughout history this number has been feared, hated, and outlawed. The biggest questions in science and religion are about nothingness and eternity, the void and the infinite, zero and infinity. The clashes over zero were the battles that shook the foundations of philosophy, of science, of mathematics, and of religion. Underneath every revolution lay a zero and an infinity

Zero was at the heart of the battle between East and West. Zero was at the center of the struggle between religion and science. Zero became the language of nature and the most important tool in mathematics. And the most prof0und problems in physics – the dark core of a black hole and the brilliant flash of the big bang are struggles to defeat zero. Humanity could never force zero to fit its philosophies. Instead, zero shaped humanity’s view of the universe.

The Power of Zero Sept 21, 1997, the billion dollar missile cruiser, the USS Yorktown, was completely disabled by zero. A new software update controlled the engines. In one part of the program, there was a division by zero; a mistake. When the program attempted to divide by zero the system crashed and 80,000 horsepower instantly became worthless.

The Beginning People first developed a name for one and many. Some languages still have this shortcoming. The Siriono Indians of Bolivia and the Brazilian Yanoama people do not have words for anything larger than three; instead, these two tribes use the words for many and much.

Because of the nature of numbers, they can be added together to create new ones. Clever tribesmen began to string number words together……one, two, two and one, two and two, two and two and one, and so on. This was the first binary system. A south american binary system developed into a quinary system by saying that two and two and one was “this is my hand all together.” The word “hand” was then used in many cultures. It is still used to determine a rough estimate of the height of a horse.

In English, eleven and twelve come from “one over ten” and “two over ten” Meaning one more than ten and two more than ten. Thirteen three and ten Fourteen four and ten Fifteen five and ten I think you get the point. This base of ten was used by the Germanic protolanguages.

The French…those strange people Used a base twenty…. Quatre vingt four twenties eighty Quatrevingt dix four twenties and ten ninety A base twenty system is called a vigesimal number system.

In the past, no one needed to express a symbol for nothing In the past, no one needed to express a symbol for nothing. Zero was not a useful concept. Its representation wasn’t needed. You just didn’t have zero sheep. You had no sheep. As society progressed, scribes needed symbols for larger groupings of numbers. 5,000 years ago, the Egyptians used a system in which pictures stood for groups of numbers. A single vertical mark was one unit. A heel bone represented ten. A swirly snare stood for 100.

The Abacus Instead of writing down 125 tick marks, the Egyptians could write down one snare, two heels, and five vertical marks. In order to perform calculations, colored ropes or an abacus was used. Numbers were primarily used by merchants. A calendar system was developed later. At this time, 5000 years ago, all calendars were lunar. The Egyptians had a lunar calendar 6000 years ago.

Lunacy The length of a month was the time between full moons. This time, however, varied from 29 or 30 days. 12 lunar months added up to 354 days, 11 days short of a solar year.

13 lunar months yielded 19 days too many 13 lunar months yielded 19 days too many. Using the moon to determine when to plant and harvest crops became a problem because the seasons drifted after only a few years.

The ancient egyptians developed a solar calendar 6000 years ago The ancient egyptians developed a solar calendar 6000 years ago. This calendar, based on the position of the sun in the sky and the stars at night, proved to be far more accurate for determining when to plant and harvest. They only performed their calculations up to the year 2012, which is why so many “doom sayers” proclaim that the world will end at this time.

The Egyptians had quickly become masters of mathematics The Egyptians had quickly become masters of mathematics. They had to, thanks to an angry river. Every year the Nile would overflow its banks and flood the delta.

The good news was that the river deposited rich, alluvial silt all over the fields. The bad news was that the river destroyed boundary markers that farmers used to determine what land belong to them. The ancient pharaohs assigned surveyors to assess the damage and reset the boundary markers, and thus geometry was born.

Greek System (2000-86 BC) The Greeks used letters. Greek Timeline H (eta) stood for hekaton 100 M (mu) stood for myriori 10,000 the myriad

Greeks and Romans The Greek system was a step forward from the Egyptian system because it used fewer symbols to represent larger numbers. The Romans took over the Greek empire and supplanted the Greek system for a much less advanced numeral system.

The Babylonians Their style of writing was actually the most sophisticated of the time. With their insight, zero finally appeared in the east, in the Fertile Crescent of present day Iraq.

Present Day Fertile Crescent

Babylonians 2300 – 1550 BC The Babylonian Empire used a base sixty numeral system. ( sexagesimal ) This was very odd since at the time many other societies had already chosen 5, 10, or 20 as base systems.

Babylonians A single symbol could represent 1, 60, or 3600. The user had to know what the symbol basically represented. It was to them a form of code. To help with this code they used an abacus as did many cultures of this time. Japan soroban China suan-pan Russia s’choty Turkey coulba Armenia choreb

The Abacus The abacus relied on sliding stones to keep track of amounts. The words calculate, calculus, and calcium all come from the Latin word for pebble: calculus. Electronic Printer Abacus | 1990    Designer: David Shaw  Award: 1991 ID Magazine Annual Design Review Selection  

Babylonians Zero was used only as a placeholder for the counting system. Zero was represented by two slanted wedges. Zero to them was a digit, a placeholder, not a number. It had no value. The zero didn’t have a spot on the number line. It was just a symbol. It didn’t even have a place in the hierarchy of numbers. Even today, look at a keyboard, the zero comes after the nine, not before the one where it belongs. On an old phone or a cell, it is at the bottom after the nine as well. So why does zero come after nine? We start counting with ONE.

The Mayans 2000 BC to 1450 AD Divided their year into 18 months of 20 days each. This totals 360 days. A special period of five days at the end, called Uayeb, brought the count to 365. This tells us that they definitely used a solar calendar. They did something that no one else did though, they started counting at zero. The first day of the month of Zip Unfortunately, we got our calendar from the Egyptians and Romans.

Mayan Counting System

Mayan Symbol Numbers

Mayan pictographs (glyphs)

The Babylonians and the Mayans were the first societies to use zero The Babylonians and the Mayans were the first societies to use zero. The Greeks feared zero. To the Greeks, zero was linked to the void. Most ancient people believed that emptiness and chaos were present before the universe came to be. The Greeks claimed that Darkness was the mother of all things and from Darkness sprang Chaos. The two spawned the rest of creation. The Hebrew creation myths say that the Earth was chaotic and void before God showered it with light and formed its features.

The Hebrew phrase tohu v’bohu stands for formless and empty The Hebrew phrase tohu v’bohu stands for formless and empty. It is in Genesis 1:2 Tohu, short for Tehomot, was a Semitic dragon that was present at the birth of the universe and whose body became the sky and earth. Bohu, for Behomot, was the famed Behemoth monster of Hebrew legend. The Norse myth tells of an open void that gets covered with ice, and from the chaos caused by the mingling of fire and ice was born the primal Giant. Emptiness and disorder were the primeval, natural state of the cosmos, and there was a nagging fear that at the end of time, disorder and void would reign once again. Zero represented that void.

Why??? Zero misbehaves. Add a number to itself and it changes. One plus one is two. Two plus two is four, three plus three is six and so on. Zero plus zero is zero. It misbehaves. To the Greeks, zero violated the Axiom of Archimedes, which states that if you add something to itself enough times, it will exceed any other number in magnitude.

Zero is just bad It refuses to get bigger. It also refuses to make other numbers bigger. Add one and two and you get three, a larger number. Add zero and two and you get two, the same number you started with. Zero had no substance, and it was a frightening concept in ancient times. Multiplication and division with zero stumped ancient mathematicians and philosophers.

In the realm of numbers, multiplication is a stretch. Literally. Imagine multiplication as if it’s a rubber band number line. Multiply three by two and you get six, which is twice as big as three. You stretched the number line. Divide six by two and you get three, division relaxes the number line. But what happens when you multiply by zero? You just broke the rubber band. The answer is zero and the number line has collapsed.

Think of it like this…. Multiplying two numbers represents an area. 2 times 2 is four. The area of a square with sides equal to two is equal to four units. If you try to determine the area of a square with sides 2 and 0, there is no area. You are left with nothing, zip, nada, zilch.

Now it gets scarier Division is the opposite of multiplication. When you multiply something by zero you get zero. But when you divide by zero you get….what??? 1/0 = infinity. On your calculator you get an error Zero can go into one an infinite number of times and never equal one.

Zero 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 Zeno. The whole Greek universe rested upon this pillar: there is no void. The Greek universe, created by Pythagoras, Aristotle, and Ptolemy, survived long after the collapse of Greek civilization. In that universe there is no such thing as nothing. There is no zero. Because of this, the West could not accept zero for nearly two millennia. The consequences were dire. Zero’s absence would stunt the growth of mathematics, stifle innovation in science, and incidentally, make a mess of the calendar. Before they could accept zero, philosophers in the West would have to destroy their universe.

The Origin of Greek Number Philosophy Hippasus of Metapontum stood on the deck, preparing to die. Around him stood the members of a cult, a secret brotherhood that he had betrayed. Hippasus had revealed a secret that was deadly to the Greek way of thinking, a secret that threatened to undermine the entire philosophy that the brotherhood struggled to build. For revealing that secret, the great Pythagoras himself sentenced Hippasus to death by drowning.

The Greek Mathematical Cult The leader of the cult was Pythagoras, an ancient radical. According to most accounts, he was born in the sixth century BC on Samos, a Greek island off the coast of Turkey famed for a temple to Hera and for really good wine. He was firmly convinced that he was the reincarnated soul of Euphorbus, a Trojan hero. This helped to convince him that all souls transmigrated to other bodies after death. For this, he was a vegetarian.

Pythagoras Pythagoras was a powerful speaker, a renowned scholar, and a charismatic teacher. He was said to have written the constitution for Greeks living in Italy. Students flocked to him. Pythagoras the Samosan On the isle of Samos

The Pythagoreans The Pythagoreans were a cult. They were the followers of the ideas of Pythagoras. They believed that all disease is caused by indigestion, that one should eat raw food and drink only water, and that one must avoid wearing wool. At the center of their philosophy though was an important concept: All is Number.

The Pythagoreans The Greeks inherited their mathematics from the Egyptians. As a result, there was no distinction between shapes and numbers. Every number-shape had a hidden meaning and the most beautiful were sacred. The mystical symbol of this cult was the pentagram. A five pointed star that gave them a glimpse into the infinite.

The Pentagram Nestled within the pentagram are an infinite number of pentagrams with the exact same proportion. Hidden within the lines of the star was, to the Pythagoreans, the ultimate view of the universe: the golden ratio.

Music To Pythagoras, playing music was a mathematical act. The harmony of the monocord was the harmony of the universe. This philosophy led to the earliest model of the planets. Pythagoras argued that the earth was at the center of the universe with the sun, moon, planets, and stars orbiting around it each pinned inside a sphere. The ratios of the sizes of the spheres were nice and orderly. As the spheres moved, they made music. The outermost planets moved the fastest and made the higher pitched notes.

All together, the planets and stars made a “Harmony of the Spheres”. Modern musicians have taken this concept and created an entirely new genre of music. Relaxation music, meditation, sleep therapy, hypnotherapy, massage therapy, massage therapists, teachers, spiritual experiences, cancer clinics, and acupuncture ….to name a few.

Artists and architects today make use of the golden ratio because it is aesthetically pleasing to the eye. This ratio governs the work of many buildings, musical works, and works of art. Some historians and mathematicians argue that the Parthenon, the Athenian temple, was built so that the golden ratio is visible in every aspect of its construction.

The Parthenon

The Nautilus

The Counterclockwise grooves of a Pineapple

More Apparent – The Pinecone

Sunflower Seeds

Modern Art (Comic Books)

As late as Shakespeare’s time (1564-1606), scientists and the general public talked about the revolutions of the orbs of different proportions and discussed the heavenly music that reverberated through the cosmos. Kepler, Galilio, and Newton changed this.