1550078481-Ordinary_Differential_Equations__Roberts_

190 Ordinary Differential Equations

that the various sets of numbers- natural, rational, real, and complex- were

themselves in the process of being developed. The ancient Babylonians, Chi-

nese, Egyptians, Greeks, and Romans each had their own distinct numerals

and systems of representing numbers. The Babylonians used wedge-shaped

characters and base 60. The Egyptians used hieroglyphics. The Greek alpha-

betic numeral system was derived from the initial letters of the number name.

And the Romans also used a letter-based numeral system with which most of

us are familiar. These systems of numeration did not represent numbers com-

pactly nor were they easy systems in which to develop calculating algorithms.

The Hindu-Arabic numeral system- the system which we, and the majority

of the world, currently use to represent numbers and for computing- was in-

vented by the Hindus and preserved and transmitted to the Western World

by the Arabs. The earliest existing examples of our present numeral system

appeared in the inscriptions of King Asoka who ruled most of India in the

third century B.C. Early examples do not contain a symbol for zero or the

idea of positional notation. Exactly when these crucial ideas were introduced

in India is not certain. But it is clear that they were introduced prior to

800 A.D. It was not until the thirteenth century that our current computing

patterns used in conjunction with the Hindu-Arabic numeral system reached

its present form.

The development of the various sets of numbers proceeded as follows: First,

man invented the natural numbers for counting. Then, he invented the pos-

itive rational numbers for the purpose of measurement. Next, the positive

real numbers were devised to accommodate irrational numbers such as ,/2.

Later, negative numbers were accepted and finally the complex numbers were

devised to incorporate imaginary numbers.

Until the beginning of written history, the natural numbers seem to have

served the purpose of mankind adequately. The idea of unit fractions- that

is, a fraction with numerator one-arose early in both Babylon and Egypt.

About 2000 B.C. the Babylonians conceived the idea of fractions with numer-

ators greater than one. However, no acceptable treatment of fractions which

predates the Egyptian Ahmes Papyrus of approximately 1550 B.C. has yet

been discovered. The followers of Pythagoras (c. 540 B.C.) knew and demon-

strated the incommensurability of the diagonal and the side of a square. That

is, the Pythagoreans knew that v'2 was irrational. About 375 B.C. Theaete-

tus developed a general theory of quadratic irrationals. The first mention

of negative numbers other than as subtrahends appears in the Arithmetica

of Diophantus in about 275 A.D. where the equation 4x + 20 = 4 is called

absurd since the solution is x = -4. In India, negative numbers were viewed

as distinct entities by at least 628. In his Ars Magna (1545), Cardan first
accepted negative numbers as roots of polynomials. Also, Cardan was the
first person to use the square root of a negative number in computations. He
demonstrated that x = 5 + F-15 and x = 5 - F-15 were both solutions of
the equation x^2 + 40 = lOx. In 1637 , Descartes coined the terms real and