REAL NUMBERS, R
One of the most important properties of the real numbers is that points on a straight line that can represent them. As in Fig 3.1, we choose a point, called the origin, to represent 0 and another point, usually to the right, to represent 1.
Then there is a natural way to pair off the points on the line and the real numbers, that is, each point will represent a unique real number and each real number will be represented by a unique point. We refer to this line as the real line. Accordingly, we can use the words point and number interchangeably.
Those numbers to the right of 0, i.e. on the same side as 1, are called the positive numbers and those numbers to the left of 0 are called the negative numbers. The number 0 itself is neither positive nor negative
1. Integers, Z
The integers are those real numbers
…, -3, -2, -1, 0, 1, 2, 3,…
We denote the integers by Z; hence we can write
Z = { …, -2, - 1, 0, 1, 2,…}
The integers are also referred to as the “whole” numbers.
One important property of the integers is that they are “closed” under the operations of addition, multiplication and subtraction; that is, the sum, product and difference of two integers is again in integer. Notice that the quotient of two integers, e.g. 3 and 7, need not be an integer; hence the integers are not closed under the operation of division.