Real Number System

Introduction to Integers and their Absolute Value

You have studied fractional numbers in your earlier classes. Some examples of fractional numbers are.

These numbers are also known as rational numbers.

What comes first to your mind when you hear the word rational?

Yes,you are right. It is something related to the ratios.

The ratio 4:5 can be written as, which is a rational number. In ratios, the numerator and denominator both are positive numbers while in rational numbers, they can be negative also.

Thus, rational numbers can be defined as follows.

“Any number which can be expressed in the form , where p and q are integers and, is called a rational number.”For example, is a rational number in which the numerator is 15 and the denominator is 19.

Now, is −34 a rational number?

Yes, it is a rational number. −34 can be written as. It is in the form of and q ≠ 0.

Thus, we can say that every integer is a rational number.

Now, consider the following decimal numbers.

1.6, 3.49, and 2.5

These decimal numbers are also rational numbers as these can be written as

If in a rational number, either the numerator or the denominator is a negative integer, then the rational number is negative.

For example, are negative rational numbers.

If the numerator and the denominator both are either positive integers or negative integers, then the rational number is positive.

For example, are positive rational numbers.

Conventions used for writing a rational number:

We know that in a rational number, the numerator and denominator both can be positive or negative.

Conventionally, rational numbers are written with positive denominators.

For example, –9 can be represented in the form of a rational number as , but generally we do not write the denominator negative and thus, is eliminated. So, according to the convention, –9 can be represented in the form of a rational number as .

Equality relation for rational numbers:

For any four non-zero integers p, q, r and s, we have

Order relation for rational numbers:

If are two rational numbers such that q > 0 and s > 0 then it can be said that if ps > qr.

Absolute Value of a Rational Number: The absolute value of a rational number is its numerical value regardless of its sign. The absolute value of a rational number pq is denoted as pq. Therefore, -32=32, 12-7=127 etc. Note: The absolute value of any rational number is always non-negative.

Now, let us go through the given example.

Example:

Write each of the following rational numbers according to the convention.

i)

ii)

Solution:

According to the convention used in rational numbers, the denominator must be a positive number.

Let us now write the given numbers according to the convention.

i)

In the number , denominator is negative.

We have,

According to convention, the given number should be written as .

ii)

In the number , denominator is negative.

We have,

According to convention, the given number should be written as . Example: Find the absolute value of the following: (i) -12171 (ii) 1219 Solution: (i) Absolute value = -12171=12171 (ii) Absolute value = 1219=1219

Natural numbers

The counting numbers 1, 2, 3, ... are called natural numbers.

The set of natural number is denoted by the letter N.

∴ N = {1, 2, 3, ...}

1 is the smallest natural number. The set of natural numbers, N is an infinite set.

Whole numbers

The numbers 0, 1, 2, 3, ... are called whole numbers.

The set of whole numbers is denoted by the letter W.

∴ W = {0, 1, 2, 3, ...}

0 is the smallest whole number. The set of whole numbers, W is an infinite set.

Integers

We had observed that adding any two whole numbers always gives a whole number. We can examine whether this case is true for the operation ‘subtraction’. Let us consider the following examples:

13 − 12 = 1

13 − 13 = 0

12 − 13 = ?

We can observe that in the last case, the operation ‘subtraction’ cannot be performed in the system of whole numbers i.e., when a bigger whole number is subtracted from a smaller whole number. In order to solve such type of problems, the system of …

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