Mathematics in Mesopotamia
By Vickie Chao
Do you like mathematics? No matter what your answer may be, you are not alone.
Mathematics is a challenging subject. Its basic concepts began to emerge when the
world's very first civilization took root in Mesopotamia more than 5,000 years ago.
Back then, the Sumerians developed a unique numeral system, using a base of sixty.
In scientific terms, that system is called a sexagesimal system. Since the Sumerians
counted things with sixty as a unit, they had the same symbol ( ) for 1 and 60. And
they would express 70 ( ) as, literally, the sum of 60 ( ) and
10 ( ). Likewise, they would express 125 (
) as the sum of two units of 60 ( )
and one unit of 5 ( ).
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Today, our decimal numeral system uses ten, not sixty, as a base unit. But that is
not to say that the Sumerians' invention became obsolete. As a matter of fact, it still
plays a critical role in our everyday life. For example, have you ever wondered why
an hour has 60 minutes and a minute has 60 seconds? Have you ever thought about
why a full circle has 360 degrees? As it turns out, that was how the Sumerians kept
track of their time. And that was how they defined a full circle.
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When the Sumerians first came up with their numerals, they did not have a
specific symbol for zero. If they needed to inscribe, say, 506 on a clay tablet, they
would simply put a blank space between the symbols of 5 ( ) and 6 ( ). This way
of denoting zero could be quite confusing and problematic. Neither the Sumerians nor
other people in Mesopotamia (most notably, the Babylonians) were able to come up
with a solution at the time. This issue would remain unsolved until around 500 A.D.
when the Indians developed the Arabic numerals that we are still using today.
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Ahmes and Angles - Egyptian Mathematics
By Colleen Messina
In the 1850s, a man named Mr. Rhind bought an amazing papyrus manuscript.
A scribe named Ahmes, the Moonborn, wrote the manuscript in 1575 B.C., and it
contains most of what we now know about Egyptian mathematics. The manuscript
describes the Egyptian number system, the Egyptian use of fractions to divide rations
of bread and beer among the workers, and geometric calculations. The Rhind Papyrus
hangs in the British Museum in London, and it is one of the oldest mathematical
documents in the world. Although it is hard to pinpoint exact dates for ancient
cultures, the Egyptians' civilization thrived from about 4000 B.C. to 500 B.C., and
they made many strides in the development of mathematics.
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Mathematics had come a long way since the hunters and gatherers first figured out
the lunar cycle. The Egyptians developed a system of writing called hieroglyphics that
used pictures to represent words and numbers, but they still had no zero in their
numerical system. A papyrus leaf represented the number 1; bent-over papyrus leaf
represented 10; a coiled rope represented 100; and the sacred lotus flower represented
1,000 (Egyptians believed that a god who appeared from a lotus created the world).
Animals represented the larger Egyptian numbers; a snake represented 10,000, and a
tadpole was the symbol for 100,000. A figure of a scribe represented the number 1
million, so the scribes were pleased! Repeating the symbols created larger numbers.
For example, three coiled ropes meant 300.
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Scribes like Ahmes learned to read and write, but many Egyptian children did not
attend school. If the future scribes complained about school, they had to listen to a list
of the problems that faced other professions. Metalworkers supposedly choked on
smoke from the furnaces, and weavers had cramped places to work. School was
challenging and the teachers were strict, but the young people had some fun learning
about numbers through games. They learned how to use numbers for practical things,
such as counting household goods, organizing soldiers in the army, and keeping track
of taxes. They also learned calculations to help with farming.
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Farming was one of the most important jobs in ancient Egypt because farmers had
to produce food for everyone. Egyptian farmers needed a more precise calendar. At
first, they still used the lunar calendar to plan their farming, but since this calendar
had only 360 days (12 cycles of 30 days,) they had to add days to remain in harmony
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with the seasons. The Egyptians replaced their lunar calendar with the first solar
calendar in approximately 2772 B.C. This calendar was 365 days long, the actual time
it takes the earth to orbit the sun. Across the globe, in Central America, the Mayan
civilization also developed a solar calendar.
Egyptian farmers had other challenges that led to better methods of measurement.
Each year the Nile River flooded, leaving behind a stretch of fertile land where the
Egyptians grew their crops of barley and emmer wheat. Therefore, each year the
boundaries of the fields had to be accurately redrawn. Egyptian surveyors or "rope
stretchers" used lengths of ropes with equally spaced knots tied in them to measure
land boundaries. When two fields bordered one another, the rope stretchers had to
measure a right angle to form the corners of the fields. The establishment of
boundaries was also important because the area of the land determined the amount of
taxes, and the scribes kept the accounts for taxation.
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Money and Measurement - Babylonian Mathematics
By Colleen Messina
Mesopotamia, which means "The Land between the Two Rivers," was located
one thousand miles east of the delta of the Nile River in between the Tigris and
Euphrates Rivers. The Babylonian civilization flourished at about the same time as the
Egyptian civilization. The Babylonians lived in a large desert, and they had a legal
system, a postal system, and irrigation systems. The environmental differences
between Mesopotamia and Egypt led the Babylonians to develop different areas in
mathematics.
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Mesopotamia engaged in a great deal of foreign trade because they had no wood
or metal in their environment. They were constantly traveling in caravans of donkeys
or camels, or on ships. Merchants traveled far to obtain goods. They went west to
Lebanon for wood, north into Asia for precious metals, and east into India for silks
and spices. These merchants needed a precise way of measuring things because these
goods were rare and expensive.
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The merchants developed scales and standard weights to replace the previous
system of measuring things by donkey load. Heavy items were weighed in talents, and
a talent was approximately 35 pounds. Precious things, like spices, were weighed in
shekels, and a shekel was a little less than one third of an ounce. All of this trade and
commerce also led to the development of money for the first time.
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Proofs and Pythagoras - Greek Mathematics
By Colleen Messina
In the high mountains of Greece in the northeastern part of the Mediterranean,
a civilization was born that influenced the world for centuries. Ancient Greece
made great strides in the areas of art, philosophy, and politics, and its civilization
lasted from approximately 2000 BC to 300 BC. Greece also produced some of the
finest mathematical minds that ever pondered numbers. The Greeks were the first
people of the ancient world who systematically studied geometry, which is the study
of the size and shape of an object. While the first surveyors of Egypt understood
practical elements of geometry, the Greeks asked why these applications worked. The
Greeks wrote down rules for geometry that verified the observations of other ancient
mathematicians.
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The Greek language formed the basis of some of the mathematical words we use
today. The word geometry comes from a Greek word for "earth measuring." Another
modern word that comes from Greek is arithmetic, which comes from arithmos. The
Greeks had fun with numbers, and arithmos, which means number, denoted
discovering secrets and figuring out puzzles about numbers. The Greek sense of
curiosity about numbers probably helped them unravel many problems that earlier
mathematicians could not figure out. The Greeks loved to argue, debate, and figure
out how to prove everything they observed.
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The Greeks had a simple number system, but it was different from the Egyptian
system. While the Egyptians used pictures to represent numbers, the Greeks used the
letters of their alphabet. The Greeks got the idea for an alphabet from the Phoenicians,
a seafaring people who lived around 1500 BC along the coast of Syria. The Greek
number system used units of 5 and 10. The Greek alphabet had twenty-seven letters,
so the first nine letters represented the digits 1 through 9; the second nine letters
represented the tens, and the last nine letters represented the hundreds. The highest
Greek number was 900. The Greeks did not have a zero, and since they rarely needed
numbers higher than hundreds, the system worked fairly well. Even though the Greeks
were logical about numbers, they were surprisingly superstitious too. Some numbers
were evil, while other numbers were friendly or even sacred. Number 10 was the
number of harmony. Number 8 was the symbol of death. Odd numbers were female,
and even numbers were male
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Abacuses and Aqueducts - Roman Mathematics
By Colleen Messina
According to legend, two brothers who were the sons of the god of war
founded Rome in 753 BC. By 146 BC, when the Roman soldiers crushed
Carthage, Rome became the greatest power in the Mediterranean. The Romans were
known as mighty conquerors and had control of southern Europe, Gaul, Britain, North
Africa, and much of Asia. Roman merchants had to develop accounting and
measuring systems that assisted them in keeping track of their trades as they traveled
across the vast empire. Since the Romans also controlled the Greek colonies, they
absorbed a great deal about art, literature, and geometry from them. However, the
Romans didn't copy everything from the Greeks: they devised their own simpler
numerical system, and they also made notable contributions to our modern calendar
and architecture.
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The Roman number system was based on seven symbols: I for 1; V for 5; X for
10; L for 50; C for 100; D for 500; and M for 1,000. Like the Greeks, the Romans had
little need for large numbers. The Romans still did not have a zero in their system, but
the position of the number determined its value. If the number follows a larger
number, the two numbers are added. For example, VI equals 6. When a smaller
number precedes a larger number, the smaller number is subtracted, so IV equals 4.
Today Roman numerals are still sometimes used for dates, or to label volumes in
books, or on the faces of clocks, but calculating with Roman numerals was difficult.
Multiplication or division was practically impossible, so Roman merchants assigned
the task of calculations to slaves, who used a device called an abacus for the task.
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The Roman abacus was a table with columns drawn on its surface. Each column
represented a power of 10. A column on the right was one; the column to the left was
10; the next column to the left was 100, and so on. There were also two columns on
the far right that were used for fractional values. Counters or pebbles, called calculi,
were placed in the columns to represent different numbers, and were moved from
column to column to perform calculations. Calculating anything with an abacus was a
complicated process and required a great deal of training. The calculi were made of
different materials ranging from bronze to gold depending on the wealth of the
merchant.
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Nothing at Last - Indian Mathematics
By Colleen Messina
When you do your algebra, you use Arabic numbers instead of hieroglyphics or
Roman numerals. Arabic numbers were invented in India, but no one knows their
exact origin. Indians loved astronomy, like the Greeks, and Indians had rope
stretchers, like the Egyptians. Babylonian mathematics also influenced Indian
mathematicians. However, in 500 AD, the Indians came up with something
completely original: Arabic numbers. No one knows exactly who designed the
symbols, but one legend said that a glassblower created the shapes of the Arabic
numerals.
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The Arabic system used a base of ten and was more efficient than any prior
mathematical system for several reasons. Each number had a separate symbol and
name. Combinations of these ten symbols and names created larger numbers. Arabic
numbers could go on forever, rather than stopping at 900 like the Greek system, or at
1,000 like the Roman system. Each number stood on its own, unlike the Roman
system that combined more than one symbol to form larger numbers. The Indians
could add, subtract, divide, and multiply numbers without the Roman abacus.
Sometimes there were contests between mathematicians who used the abacus and
ones who used the Arabic numbers. Those who used Arabic numbers won easily!
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One of the greatest advantages of the Arabic numeral system is that at last there
was a symbol for nothing. Ancient peoples did not need a zero because they used
numbers for counting small quantities, but a zero was necessary as a placeholder in
more complex calculations. The word zero comes from an Arabic word, sifr, which
means "empty." When western scholars described the new number to their colleagues,
they turned sifrinto the Latin-sounding word zephirus. This word became "zero." Zero
replaced the blank space that early number systems used for "nothing."
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Mathematics in Western Europe - Intrigue and
Integration
By Colleen Messina
The newly invented Arabic numbers arrived in Europe around 1200 AD.
However, they were not popular right away. Intrigue and opposition accompanied the
change from the old Roman numbers. Sometimes Arabic numbers had to sneak into a
country via a mathematician. One case of this was when a Christian monk named
Adelard of Bath disguised himself as a Muslim and studied in the University of
Cordova in the 12th century. He secretly translated the works of Euclid and smuggled
his translations back to Britain. The difficulties continued into the 14th century as
some insisted on keeping the old system. An Italian University said that price-lists for
books must still be in Roman numerals!
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One mathematician who promoted the use of the new numbers was an Italian
named Leonardo de Pisa. He became most commonly known by his nickname,
Fibonacci. Fibonacci was the son of an Italian diplomat and grew up in North Africa
in the late 12th century. He learned about Arabic numbers as a young boy and later
wrote an influential book about practical geometry. In it, he encouraged the use of the
new Arabic numbers. He also estimated the value of pi as 3.1418. Our value today is
3.14159265.
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With the encouragement of mathematicians like Fibonacci, Europeans finally
adopted the new numbers. By 1400, grateful merchants in Italy, France, Germany, and
Britain used them for accounting. European mathematicians made amazing progress
in many areas of mathematics and science between 1200 and 1700 AD because of the
new number system. Schools taught the new arithmetic throughout Europe. Most
textbooks used the new numbers by the mid-15th century.
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The new textbooks also adopted convenient shortcuts for writing equations for
addition, subtraction, multiplication, and division. These symbols were invented for
practical reasons, and the + and - signs were first used in warehouses. Workers
painted the plus sign on a barrel, for example, to show that it was full. The + and signs first appeared in print in 1526 by Johannes Widmann in a German math book.
The signs for multiplication and division came later, and the equal sign was first used
in England in 1557. These symbols also led to the algebra we recognize today. By
1600, letters were used to represent unknown amounts in equations.
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Logarithms were also invented at this time. Logarithms are intriguing numbers
because if you add two of them together, you can solve complicated multiplication
and division problems! A Scottish mathematician named John Napier first published a
table of these numbers in 1614, and soon books of logarithms became available.
Electronic calculators replaced logarithms by the 1970s, but for centuries, "logs"
simplified complex calculations.
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The Age of Discovery - Gravity and Gauss
By Colleen Messina
By the seventeenth century, mathematics had come a long way from the
tallies and abacuses of the ancient world. Mathematicians had finally adopted the
new Arabic numbers, as well as the symbols for addition, subtraction, multiplication,
and division. Logarithms made difficult problems much easier, and calculus opened
up new possibilities in science. Mathematicians applied these new tools in exciting
ways ranging from world exploration to astronomy. Ships crisscrossed the oceans to
new places, and telescopes scanned the skies and discovered the elliptical orbits of
planets. The understanding of gravity revolutionized military science. It was truly an
age of discovery.
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The discovery of gravity especially changed how people viewed the world. Up
until the 16th century, people thought that heavy objects fell faster. A feisty Italian
named Galileo Galilei had another idea. In 1585, he climbed to the top of the leaning
Tower of Pisa, made sure no one was down below, and dropped two objects. One
object was heavy and the other was light, but both reached the ground at the same
time. Galileo proved that objects fall at the same rate and accelerate as they fall.
Eventually, military engineers understood that a cannonball shoots out in a straight
path, but the force of gravity makes the cannonball fall downward in a curve called a
parabola. The engineers could then fortify their strongholds in the right places, and
artillerymen could shoot their cannons more accurately. Galileo's experiment
revolutionized military science.
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Galileo also did experiments with pendulums that helped clockmakers design
accurate clocks. Seamen needed accurate time-keeping devices to navigate during
long journeys. The weight-driven clocks of the previous centuries were not accurate
enough; now seamen needed to measure minutes and seconds, so the new clocks were
invaluable. Navigators then accurately plotted the daily positions of their ships on
maps that had vertical and horizontal lines of latitude and longitude. When they
connected the dots on these grids, they saw an accurate record of the ship's journey.
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Twentieth Century Mathematics - Riemann to
Relativity
By Colleen Messina
Mathematics moved in new directions by the end of the 19th century as
technology advanced in exciting ways. The Wright brothers flew the first
airplane, and gas lighting became popular. The mathematical giant of that century,
Carl Friedrich Gauss, made many discoveries in mathematics, astronomy, and
physics. He also contemplated unusual problems in geometry, such as how to measure
curved surfaces. His brilliant student, Bernhard Riemann, solved that problem and
eventually laid the mathematical foundations for Einstein's theories of relativity.
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Riemann's father was a Lutheran pastor and wanted his son to follow his example.
Bernhard started to study theology, but eventually his father allowed him to study
mathematics. When he was 28, Bernhard had to give a lecture in order to become an
associate professor. He suggested three possible subjects to his mentor, Carl Gauss.
Gauss selected Riemann's least favorite topic: non-Euclidean geometry. This lecture
was a great success and made Riemann famous. Gauss was so pleased that he made a
comment at end of the presentation that Riemann had solved problems that Gauss had
wondered about for his whole life!
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Riemann's system of "differential geometry" made it possible to measure any
curved surface. The idea of curved space was so revolutionary that it was not fully
understood until the 20th century. Riemann also contributed to the theory of functions,
complex analysis, and number theory, which are important subjects in higher
mathematics. Riemann died at age 39 from tuberculosis. In his short life, he laid the
foundation for Einstein's amazing theories of relativity.
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The Need for Speed - The Age of Computers
By Colleen Messina
Nothing seemed to satisfy the mathematicians' need for speed in calculations.
Pascal's adding machine in 1642 didn't do it; Napier's bones didn't either.
Mathematicians were always trying to figure out a way to go faster. Charles Babbage
designed a machine to print mathematical calculations quickly, a concept far ahead of
most 19th century notions, but that didn't work because the machine couldn't be built.
This need for speed led to an ever-expanding series of ideas and inventions that
spanned more than a century.
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Who would have thought that a silk weaver would start the ball rolling for
computers! A Frenchman named Joseph-Marie Jacquard invented punch cards to
make silk weaving go faster. The cards controlled which needles punched through and
created the patterns in the cloth. Other inventors loved these cards and created
adaptations to use in other places. For example, punched cards decided which music
played on automated pianos. Computers eventually used punched cards to store
programs.
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Herman Hollerith, an American inventor, developed the idea of encoding data in
the punched cards. Hollerith lived in Buffalo, New York, and got the idea for a punch
card tabulation machine by watching a train conductor punch tickets. In 1881, he
started designing a machine that could tabulate census data faster, because traditional
hand counting was painfully slow. The United States Census Bureau had taken eight
years to finish their 1880 census. Everyone was worried that the 1890 census would
take so long that the results wouldn't be finished before the 1900 census had to start.
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Just When Math Got Organized, Chaos Popped Up!
By Colleen Messina
What do you think of when you hear the word chaos? Your bedroom,
perhaps? Mathematicians use the word in a different way. Chaos theory describes
many normal things that seem unorganized, but that have a pattern after all. If this
doesn't make sense to you, don't worry; it took a long time for mathematicians to
understand chaos. Methodical mathematicians spent hundreds of years creating
numbers and inventing calculus. Just when everything seemed perfectly organized,
chaos popped up.
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Where is chaos in real life? There is chaos in the flow of a dripping water faucet,
in weather patterns, and in the ups and downs of the stock market. An excellent
example of chaos is a human heartbeat, which sometimes has a chaotic pattern (and
not just on Valentine's Day) because the time between beats changes depending on
what the person is doing. Under some conditions, the heart beats erratically. The
analysis of a heartbeat can help doctors and researchers find ways to make an
abnormal heartbeat steady again.
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Chaos theory quietly emerged in a meteorologist's office in 1960. Ironically, at
about the same time, computers started organizing the mathematical world. Edward
Lorenz wanted to predict the weather, or maybe he just wanted an extra way to figure
out whether to recommend an umbrella or sunscreen to his friends. In any case, he
programmed his computer to execute 12 equations to track weather patterns. After he
ran the equations several times, Lorenz decided to save some time and paper by
starting in the middle of the sequence of calculations rather than at the beginning. He
also printed the results out to three decimal places instead of six. He expected to get a
similar graph as before.
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When Lorenz came back to check his printout an hour later, he saw something
unusual. The graph was vastly different from earlier graphs! Lorenz thought about this
new graph for a long time. He realized that he couldn't accurately predict the weather,
but he wondered why the graph was so different. He realized that even though the
difference between using three or six decimal places to run his equations seemed tiny,
it had a huge effect on his results. Scientists eventually called this unusual
phenomenon the "butterfly effect."
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When a butterfly flaps its wings, the change in the atmosphere is small, but over
time, that little difference can affect the entire planet. For example, after a month, the
little change in the atmosphere from the butterfly's wings might cause a tornado off
the Indonesian coast! The butterfly effect means that small differences in starting
conditions can mean big changes in results. The butterfly effect, which affects chaotic
systems, led Edward Lorenz to discover other elements of chaos theory. (Bugs of
different kinds, like computer bugs, do seem to flit in and out of mathematics.)