Two branches of mathematics

The Greeks had a simple number system, but it was different from the Egyptian system. Egyptians used pictures to represent numbers. 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. Two branches of mathematics resulted with separation of whole numbers and geometry. Eudoxus used geometry to define lengths, angles, areas, and volumes with irrational numbers as lengths or line segments. Converting all mathematics except the theory of whole numbers into geometry had consequences. From Euclid until 1600, geometry was the basis of all rigorous mathematics. Mathematics begins intuitive or empirical and remained so until the 4th century BC. Babylonians used numbers without any understanding of decimal approximations. BC Pythagoras of the 5th century had linked whole numbers and ratios of whole numbers as the way to study nature and insisted the whole numbers were the measure of all things until Hippasus discovered a irrational number or incommensurable ratio. Pythagoreans and the classical Greeks rejected irrational numbers. Classical Greeks believed in the theory of whole numbers with a logical foundation. Deductive mathematic attempts led Euclid to define all his concepts to provide a logical treatment of the whole numbers. Geometric representation of numbers and of operations with numbers was not practical. The deductive orderly proof of the geometry of Euclid, Apollonius, and Archimedes was merged with the indeterminate class of problems in the works of Diophantus. Classical Greek civilization combined with Egyptian and Babylonian in the period of Alexandrian Greek civilization. Various types of undefined numbers continued to be used to provide empirically correct solutions. Heron translated much of the Greek geometrical algebra into arithmetical and algebraic processes and provided a formula for the area of a triangle. Hipparchus and Ptolemy created quantitative astronomy with the creation of trigonometry. Arithmetic and algebra emerged without logical structure since it could be used provide empirical results. Diophantus introduced symbolism in algebraic operations that was entirely arithmetical with no appeal to geometry. He introduced indeterminate equations as a branch of algebra but considered equations producing irrational negative or imaginary roots as unsolvable. Heron accepted irrational quantities; Archimedes put bounds around irrational numbers. Hindus and Arab advanced arithmetic and algebra empirically. They reasoned by analogy and introduced rules for manipulating irrational numbers without providing any logical foundation favored in classical Greek mathematics. The Hindus introduced separate symbols for the numbers 1 to 9, converted from positional notation in base 60 to base 10, recognized 0 as a number, and used negative numbers. Europeans of the Renaissance acquired its knowledge of mathematics via the Arabs and Greek manuscripts. They noticed a dilemma presented by the two disparate states of arithmetic and geometry. They realized the utility of arithmetic and algebra of ancient times lacked a logical foundation. Europeans of the 16th and 17th century considered the negative numbers in Arab texts as absurd. Girard and Harriot used the minus sign to describe negative numbers. A negative number is an independent concept, not the operation of subtraction. Using separate symbols would have been better.

Logarithms, complex numbers, mathematical induction, and the binomial theorem were added to the body of algebra without proofs. Algebra became a study of general types of forms and equations when Vieta introduced literal coefficients separating arithmetic and algebra, types from numbers. By the mid-18th century, algebra included many branches.