Short History Notes About Algebra and Geometry
The origins of algebra date back over a thousand years. Muhammad Ibn Musa al-Khwarizmi, a mathematician and astronomer, lived in Baghdad during the golden age of science. He wrote a book about mathematics describing the Hindu-Arabic numeral system, the system we use today. The word algebra comes from the Arabic word ab-jabr, which was in the title of the book. In the 1200s, Europeans became interested in the book. They learned about the Hindu decimal system and Arabic notation of numerals, which is our present-day number system. The book gave them the word algebra.
The concept of zero that we use in modern mathematics was invented in India by Hindu mathematicians during the Gupta Empire (A.D. 232 to A.D. 550). However, the Hindus were not the first people to come up with the concept of zero. The ancient Mayas of Central America were probably the first people to use zero in a mathematical system. Unlike the decimal system that we use today, the Mayas used a base 20 number system. The Mayan system only had three symbols: a shell-shaped symbol that represented zero, a dot for one, and a horizontal bar for five. Place value was shown by reading from bottom to top, with higher symbols standing for larger numbers. Each level increased the place value by 20.
Every student of mathematics should be familiar with the ancient Greek mathematician Pythagoras, who lived in the sixth century B.C. He became famous for his theorems and thought that all things were based in numbers. We use numbers to describe geometric figures. Sometimes the numbers occur in patterns. These patterns can help us learn even more about the figures they describe. One of the most useful number patterns includes positive integers and their squares.
The formula we use for determining the length of the hypotenuse of a triangle-c2 = a2 + b2-is named after Pythagoras. The Pythagorean theorem allows us to find the length of one side of a right triangle when the other two sides are known. Although the formula for finding the length of a triangle’s side was named for a Greek, the Chinese also discovered and used the same theorem. The classic Chinese mathematics book, Arithmetic in Nine Sections, was written during the Han Dynasty (202 B.C. to A.D. 220). One section of the book includes a problem and a solution that mirror the Pythagorean theorem.
No single individual or civilization was responsible for the development of algebra. Almost every ancient culture was fascinated by mathematics. Some used mathematics and geometry to create monumental structures such as the Egyptian pyramids in Africa and the Mayan pyramids on the Yucatan Peninsula of Central America. In ancient China, scholars of the Han Dynasty extended the knowledge of theoretical mathematics. Both ancient India and the Mayan civilizations gave us the concept of zero. And the ancient Indians also gave us the basis for the decimal system. Ancient Arabia created the numbers we write today-Arabic numerals. The Greeks also came up with many of the mathematical concepts that we use today, including Euclidian geometry. The accomplishments of many cultures around the globe are based on modern mathematical theory and practice.
Euclid wrote the first complete geometry book. He lived in ancient Greece about 300 B.C. Euclid began his book with five “facts” or assumptions about geometry in a plane. He assumed these statements were true without proving them. These statements, called Euclid’s postulates, form the building blocks for plane Euclidean geometry. They explain why certain constructions can be made. The knowledge of these postulates will be the bases for how successful a student will be in geometry. He designed his postulates so that it is a statement assumed to be true without a proof. He then developed an axiom, like a postulate, is a statement taken to be true without proof. They are statements about properties of equality. Axioms give us a formal way to refer to facts that seem obviously true. In addition to his postulates, Euclid included five axioms or common notions (ideas) in his geometry text. The fact that Euclid included statements about algebra in a book about geometry shows that he saw a clear connection between the two subjects.
Francois Viete (1540-1603) was an influential French mathematician, although he claimed mathematics was more of a hobby than a vocation. He practiced law and held various political appointments before turning his mind toward mathematics. Viete made many important contributions to the field of algebraic notation. In 1591 his book In Artem Analyticam Isagoge was published in which Viete introduced the first systematic algebraic notation. Among his contributions to the field of algebra are the introduction of letters for variables, a new method of notation for exponents, a substitution method for solving polynomials (named for him), and the introduction of the term “coefficient.”
In 1536, the Spanish brought a printing press to New Spain, now Mexico. This press was used to print the first mathematics text written and published in the Americas. Summario Compendiso de las quentas de plata y re by Juan Diez Freyle was published in 1556. The main part of the book concerned the conversion of gold ore into the value equivalents of different kinds of coins used in New Spain. A section of the book was devoted to algebra and included problems that required people to solve the quadratic equation. An example of one of these problems is, “A man traveling on a road asks another how many leagues it is to a certain place. The other replies, ‘There are so many leagues that, squaring the number and dividing the product by 5, the quotient will be 80.’” Freyle’s book is believed to be the first non-religious book published in the Americas.
One of the greatest
sources of statistical data about the United States comes from the U.S. Census
Bureau. 
However, taking a census is not new. The ancient Romans were among the first
government or culture to take a census.
Roman census-takers recorded information about people and their property. The
information was used to figure the amount of taxes that a family or person would
have to pay the Roman government. In fact, the word
census comes from a Latin word meaning “to tax.” The United States Constitution
requires that a census information gathered by census-takers, questionnaires,
and other means. Federal and state governments, businesses, academic
institutions, and other organizations use census data and statistics for making
projections of population trends and for
decision-making.
The symbols for mathematical and algebraic operations come from various sources. Many were used in practice long before they appeared in any printed or published form. One of the earliest documents showing a plus sign (+) is a German manuscript written in 1456. The Latin word et, which means “and” indicated addition as in “5 et 7.” The person who wrote the manuscript used a shortened form that looked somewhat like +. Most scholars agree, however, that the first published use of the plus sign was in a Dutch book, Een sonderlinghe boeck in dye edel conste Arithmetica, written by Giel Vander Hoecke and printed in 1514. The book also used the minus sign (-) to indicate subtraction.
Carl Friedrich Gauss was the first mathematician to prove definitively the fundamental theorem of algebra, which states: An equation to the first power has one root. An equation to the second power has two roots. An equation to the third power has three roots. An equation to the nth power has n roots. Born in 1777, Gauss achieved great success in both mathematics and science and was considered a mathematical prodigy. He applied mathematics to many disciplines including astronomy, electricity, and magnetism. He published over a hundred articles, essays, and books. Many scholars consider his manuscript Disquisitiones Arithmeticae, published in 1801, as one of the most important books on number theory in the history of mathematics.
Leonhard Euler was a Swiss mathematician who advanced the concepts of pure mathematics in geometry, algebra, and number theory. Euler, sometimes referred to as “the most productive mathematician in history,” was born in 1707 in Basel, Switzerland. After graduating from the University of Basel, Euler taught at the St. Petersburg Academy of Science in St. Petersburg, Russia. Euler introduced many familiar algebraic and geometric notations in use today, including a symbol for sum, the letters a, b, and c for the sides of a triangle, the symbol pi for the ratio of a circle’s circumference to its diameter, and the notation f(x) for a function.
Benjamin Banneker is most often remembered for the work he did surveying the boundaries of Washington, D.C. However, Banneker’s accomplishments extended far beyond surveying. Benjamin Banneker was born in 1731, the son of former slaves. Although he had little formal schooling, he became an accomplished inventor, writer, mathematician, astronomer, and surveyor. As a young man, Banneker constructed a clock with carved wooden gears. The clock kept perfect time for over 50 years. Banneker was also an astronomer and could accurately predict the occurrence of solar and lunar eclipses and when the sun and moon would rise and set.
“Magic” squares have been around a long time. According to Chinese legend, the first magic square, or lo-shu, was given to Emperor Yu around 2200 B.C. As he stood by the side of the Huang River, Emperor Yu was visited by a giant tortoise. The lo-shu was on the tortoise’s shell. The shell was divided into nine equal sections forming a 3 X 3 square. Numbers were arranged in each square so that their sum equaled 15 no matter which way the Emperor added-across, down or diagonally. Since then, many people have created lo-shu puzzles in squares. One of America’s founding fathers, Benjamin Franklin, was fascinated by magic squares and created them in his spare time.
Follow up:
Find the definition of the word “algebra”_______________________________________________________________________________________________________________________________________________________________________________________________________________
Complete this “zero” activity: Zero
Explore more about the Pythagorean Theory:
Magic Squares:
http://grapevine.net.au/~grunwald/une/KLAs/maths/magic3x3.html
http://www.cut-the-knot.org/game_st.shtml
http://www.grogono.com/magic/index.php
More about Francois Viete:
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Viete.html
More about the Census:
http://www.census.gov/main/www/popclock.html
http://www.education-world.com/a_lesson/TM/WS_census_pets.shtml
http://www.education-world.com/a_lesson/TM/WS_census_states.shtml