Properties of Rational Numbers

# Closure Property

For two rational numbers, x, and y, the addition, subtraction, and multiplication results always yield a rational number. The Closure Property isn’t applicable for the division as division by zero isn’t defined. In other words, we can say that closure property is applicable for division too other than zero.

• 4/7 + 2/3 =26/21
• 4/3 – 2/4 = 6/12
• 3/5. 2/3 = 6/15

## Commutative Property

Considering two rational numbers x y, the addition and multiplication are always commutative. Subtraction doesn’t obey the commutative property. You can get a clear idea of this property by looking at the solved examples.

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• Commutative Law of Addition: x+y = y+x Ex: 1/3+2/3 = 3/3
• Commutative Law of Multiplication: x.y = y.x Ex: 1/2.2/3 =2/3.1/2 =2/6
• Subtraction x-y≠y-x Ex: 4/3-1/3 = 3/3 whereas 1/3-4/3=-3/3
• Division isn’t commutative x/y ≠y/x Ex: 3/9÷1/2=6/9 whereas 1/2 ÷3/9 =9/6

## Associative Property

Rational Numbers obey the Associative Property for Addition and Multiplication. Let us assume x, y, z to be three rational numbers then for Addition, x+(y+z)=(x+y)+z

whereas for Multiplication x(yz)=(xy)z

Ex: 1/3 + (1/4 + 3/3) = (1/3+ 1/4) + 3/3

⇒19/12 =19/12

## Distributive Property

Let us consider three rational numbers x, y, z then x . (y+z) = (x . y) + (x . z). We will prove the property by considering an example.

Ex: 1/3.(1/4+2/5) =(1/3.1/4)+(1/3.2/5)

1/3. (17/20)= 1/12+2/10

17/60 =17/60

Thus, L.H.S = R.H.S

### Identity and Inverse Properties of Rational Numbers

Identity Property: We know 0 is called Additive Identity, and 1 is called Multiplicative Identity of Rational Numbers.

5/3.1 = 5/3(Multiplicative Identity)

Inverse Property: For a Rational Number, the x/y additive inverse is -x/y, and the multiplicative inverse is y/x.

Ex: Additive Inverse of 2/3 is -2/3

Multiplicative Inverse of 4/5 is 5/4

You need to be aware of a few other properties of Rational Numbers, and they are explained below.

Property 1:

If a/b is a rational number and m is a non-zero integer then a/b =(a*m)/(b*m).

In other words, we can say that the rational number remains unaltered if we multiply both the numerator and denominator with the same integer.

Ex: 2/3 = 2*2/3*2 = 4/6, 2*3/3*3 = 6/9, 2*4/3*4 = 8/12….

Property 2:

If a/b is a rational number and m is a common divisor then a/b = (a÷m)/(b÷m)

The rational number remains unchanged when dividing the numerator and denominator of a rational number with a common divisor.

Ex: 36/42 =36÷6/42÷6 = 6/7

Property 3:

Consider a/b, c/d to be two rational numbers.

Then a/b = c/d ⇒ a*d = b*c

Ex: 2/4 =4/8 ⇒ 2.8=4.4

Property 4:

For every Rational Number n, any of the following conditions hold.

(i) n>0, (ii) n=0, (iii) n<0

Ex: 3/4 is greater than 0.

0/5 is equal to 0.

-3/4 is less than 0.

Property 5:

For any two rational numbers a, b, any one condition is true

(i) a>b, (ii) a=b, (iii) a<b

Ex: 2/3 and 2/5 are two rational numbers, and 2/3 is greater than 2/5

If 4/8 and 8/16 are two rational numbers, then 4/8 = 8/16

If -4/7 and 3/4 are two rational numbers, then -4/7 is less than 3/4

Property 6:

In the case of three rational numbers a > b, b > c, then a>c

If 4/5, 16/30, and -8/15 are three rational numbers, then 4/5 >16/30, and 16/30 is greater than -8/15, then 4/5 is also greater than -8/15.

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