The Babylonian Empire and the Arabian civilization were fundamental civilization that developed important math finding that modern civilizations still use and consider them fundamental to complete basic math tasks.
Current political map of the Arabian Peninsula
Arabic mathematics began with the Abbasid dynasty, which promoted cultural and scientific growth. The establishment of cultural and learning centers like the House of Wisdom in Baghdad. Their mathematics consisted at first of only common language and they only used positive numbers due to their interests in trigonometry and geometry. Their first works consisted of extending the Hundi to include fractions, they developed combinatorics and knew the first three rows of what is now known as the "Pascal Triangle", and learned to manipulate polynomials. Arabic mathematician Al-Khwarizmi introduced the term "algebra" or "al-jabr", and this branch became a very important part of Arabic mathematics.
Map of the Babylonian Empire during 560 B.C
Most knowledge of Babylonian mathematics comes from mathematical stones in cuneiform that have been found. One important characteristic of their mathematics is that their value system had 60 as its base and all numbers were "decimals". For example: 1.7 was 67; 21.49 was 1309 (21 * 60 (whole number) + 49 (decimal)). Additionally , they were able to find positive roots of any quadratic equation, roots of certain cubic equations and they used the Pythagorean theorem to solve problems. They had tables for multiplication, division and squares.
Muḥammad ibn Mūsā al-Khwārizm
Al-Khwarizmi, later Latinized as Algoritmi (born c. 780—died c.850) was a Muslim astronomer and mathematician that lived in Baghdad and worked “House of Wisdom”. In the "House of Wisdom" scientific and philosophic treatises mostly Greek that were acquired and then translated. It also published original research.
He published his first book ("The Compendious Book on Calculation by Competition and Balancing") based on his work and studies of algebra. It consists of a rules and demonstrations that finds systematic solutions of linear and quadratic equations based on geometric arguments.
“Al-Khwārizmī Concerning the Hindu Art of Reckoning”, his second work introduced the Hindu-Arabic numbers and arithmetic to western culture. This is the system used today it consists of ten symbols: 1,2,3,4,5,6,7,8,9,0 that represents numbers in the decimal system. "The Image of the Earth" is al-Khwārizmī's third major book. It shows coordinates of places in the world with better values for length of the Mediterranean Sea and places in Asia and Africa.
Al-Kindi was a philosopher of Arabic tradition that contributed to many fields including philosophy, astronomy, and mathematics. He applied mathematics to his other fields. For example he used geometry in optics studying the previous implication that light and vision can be formalized into lines. His four volumes "On the Use of the Indian Numeral" had a great impact on the spread of that numerical system in the Middle-East and the West. He also wrote on the theory of parallels and stated that infinity is only possible in mathematics.
Broken Numbers: Writing Fractions
The first knowledge we have of fractions is from the Egyptians. By 1800 B.C.E. they were using what is called unit fractions. In this method 1 is the numerator, the hieroglyph of a mouth represents the vinculum and below it is the denominator. Other fractions (those with a numerator greater than one) were expressed as a sum of fractions but the repetition of a denominator was not allowed. Yet, with this system it was very hard to do calculations proving to be a huge disadvantage.
In Ancient Rome, fractions were written in words and based on the unit of weight "as". The fractions were centered on twelfth but like the Egyptian system, the words made it hard to calculate.
The first civilization to create a better way of representing fractions were the Babylonians. They based numbers on 60 and extended their numbers to include fractions in the sixtieths but the lack of a decimal point led to confusion and misinterpretation
The present day format comes from the Indian civilization. By around 500 A.D. they had developed the brahmi, a writing system with 9 symbols and a zero . Their fractions were written with a numerator and denominator (much like today) but without the horizontal line separating them. It was the Arabs who added the line called vinculum to separate the numerator and the denominator.
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