RATIONAL NUMBERS

Any number in the form p/q, where q ≠ 0 is called rational number.

Even after coining integers, one could not relax! 10 ÷ 5 is no doubt fine, giving the answer 2 but is 8¸5 comfortable?

Numbers between numbers are needed. 8 ÷ 5 seen as 1.6, is a number between 1 and 2. But, where does (–3)¸4 lie? Between 0 and –1. Similarly, where do you find -12/5 on the number line? Between –2 and –3.

Thus, a ratio made by dividing an integer by another integer is called a rational number. (Remember, we should not divide by zero!)

Formally speaking, a rational number is a number of the fractional (ratio) form a/b, where a and b are integers and b ≠ 0. The collection of all rational numbers is denoted by Q.

Non-negative rational numbers may be thought of as fractions. They can also be expressed as decimals and percentages.

Rational Numbers on a Number Line

Locating the rational numbers on a number line is an important skill. For example, to represent the number -3/4 on the number line, -3/4 being negative would be marked to the left of 0 and it is between 0 and –1.

We know that the integers, 1 and –1 are equidistant from 0 and so are the numbers 2 and –2, 3 and –3 from 0. This concept remains the same for rational numbers too. Now, as we mark 3/4 to the right of zero, at 3 parts out of 4 between 0 and 1, the same way, we will mark -3/4 to the left of zero, at 3 parts out of 4 between 0 and –1 as shown below.

Similarly, it is easy to find -3/2 between -1 and -2 since

-³⁄₂ = -1½

Decimal Representation of a Rational Number

A rational number can be nicely represented in decimal form rather than in the usual fractional form. Given a rational number in the form a/b where b ≠ 0. just divide the numerator a by the denominator b and we can see that it can be expressed as a terminating or non-terminating, recurring decimal.

Examples :

1/4 = 25/100 = 0.25

1³⁄₂₀ = 23/20 = 115/100 = 1.15

3 = 3.0

-5⅘ = -29/5 = -58/10 = -5.8

1/3 = 0.3333......

Properties of Rational Numbers

Closure Property :

Addition or subtraction or multiplication or division of two rational numbers is always closed under addition.

That is, addition or subtraction or multiplication or division of two rational numbers is also a rational number.

Closure Property is true for addition, subtraction, multiplication and division of two rational numbers.

Commutative Property :

Addition or multiplication of two rational numbers are always commutative.

Commutative Property is true for addition or multiplication of two rational numbers.

Subtraction and division of two rational number are NOT commutative.

Commutative Property is NOT true for subtraction or division of two rational numbers.

Associative Property :

Addition and multiplication of two rational numbers are always associative.

Associative Property is true for addition or multiplication of two rational numbers.

Subtraction and division of two rational number are NOT associative.

Associative Property is NOT true for subtraction or division of two rational numbers.

Distributive Property :

(i) Distributive Property of Multiplication over Addition :

Multiplication of rational numbers is distributive over addition.

(ii) Distributive Property of Multiplication over Subtraction :

Multiplication of rational numbers is distributive over subtraction.

Identity Property :

(i) Additive Identity :

The sum of any rational number and zero is the rational number itself.

Zero is the additive identity for rational numbers.

(ii) Multiplicative Identity :

The product of any rational number and 1 is the rational number itself. ‘1’ is the multiplicative identity for rational numbers.

Inverse Property :

(i) Additive Inverse :

Consider the rational number a/b, where b ≠ 0.

Then, -a/b is the additive inverse of a/b.

In other words, a/b and -a/b are additive inverse to each other.

(ii) Multiplicative Inverse or Reciprocal :

Consider the rational number p/q, where q ≠ 0.

Then, q/p is the multiplicative inverse of p/q.

In other words, p/q and q/p are multiplicative inverse to each other.

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