# PROPERTIES OF RATIONAL NUMBERS

There are some properties of rational numbers like closure property, commutative property and associative property.

Let us explore these properties on the four binary operations (Addition, subtraction, multiplication and division) in mathematics.

(i) Closure Property :

The sum of any two rational numbers is always a rational number. This is called ‘Closure property of addition’ of rational numbers. Thus, Q is closed under addition

If a/b and c/d are any two rational numbers, then

(a/b) + (c/d) is also a rational number

Example :

2/9 + 4/9  =  6/9  =  2/3 is a rational number

(ii) Commutative Property :

Addition of two rational numbers is commutative.

If a/b and c/d are any two rational numbers, then

(a/b) + (c/d)  =  (c/d) + (a/b)

Example :

2/9 + 4/9  =  6/9  =  2/3

4/9 + 2/9  =  6/9  =  2/3

So,

2/9 + 4/9  =  4/9 + 2/9

(iii) Associative Property :

Addition of rational numbers is associative.

If a/b, c/d and e/f  are any three rational numbers, then

a/b + (c/d + e/f)  =  (a/b + c/d) + e/f

Example :

2/9 + (4/9 + 1/9)  =  2/9 + 5/9  =  7/9

(2/9 + 4/9) + 1/9  =  6/9 + 1/9  =  7/9

So,

2/9 + (4/9 + 1/9)  =  (2/9 + 4/9) + 1/9

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

If a/b is any rational number, then

a/b + 0  =  0 + a/b  =  a/b

Zero is the additive identity for rational numbers.

Example :

2/7 + 0  =  0 + 2/7  =  2/7

(-a/b) is the negative or additive inverse of (a/b).

If a/b is a rational number,then there exists a rational number (-a/b) such that

a/b + (-a/b)  =  (-a/b) + a/b  =  0

Example :

Additive inverse of 3/5 is (-3/5).

Additive inverse of (-3/5) is 3/5.

Additive inverse of 0 is 0 itself.

## Subtraction

(i) Closure Property :

The difference between any two rational numbers is always a rational number.

Hence Q is closed under subtraction.

If a/b and c/d are any two rational numbers, then

(a/b) - (c/d) is also a rational number.

Example :

5/9 - 2/9  =  3/9  =  1/3 is a rational number.

(ii) Commutative Property :

Subtraction of two rational numbers is not commutative.

If a/b and c/d are any two rational numbers, then

(a/b) - (c/d)    (c/d) - (a/b)

Example :

5/9 - 2/9  =  3/9  =  1/3

2/9 - 5/9  =  -3/9  =  -1/3

And,

5/9 - 2/9    2/9 - 5/9

Therefore, Commutative property is not true for subtraction.

(iii) Associative Property :

Subtraction of rational numbers is not associative.

If a/b, c/d and e/f  are any three rational numbers, then

a/b - (c/d - e/f)    (a/b - c/d) - e/f

Example :

2/9 - (4/9 - 1/9)  =  2/9 - 3/9  =  -1/9

(2/9 - 4/9) - 1/9  =  -2/9 - 1/9  =  -3/9

And,

2/9 - (4/9 - 1/9)    (2/9 - 4/9) - 1/9

Therefore, Associative property is not true for subtraction.

## Multiplication

(i) Closure Property :

The product of two rational numbers is always a rational number. Hence Q is closed under multiplication.

If a/b and c/d are any two rational numbers, then

(a/b) x (c/d)  =  ac/bd is also a rational number.

Example :

5/9 x 2/9  =  10/81 is a rational number.

(ii) Commutative Property :

Multiplication of rational numbers is commutative.

If a/b and c/d are any two rational numbers, then

(a/b) x (c/d)  =  (c/d) x (a/b)

Example :

5/9 x 2/9  =  10/81

2/9 x 5/9  =  10/81

So,

5/9 x 2/9  =  2/9 x 5/9

Therefore, Commutative property is true for multiplication.

(iii) Associative Property :

Multiplication of rational numbers is associative.

If a/b, c/d and e/f  are any three rational numbers, then

a/b x (c/d x e/f)  =  (a/b x c/d) x e/f

Example :

2/9 x (4/9 x 1/9)  =  2/9 x 4/81  =  8/729

(2/9 x 4/9) x 1/9  =  8/81 x 1/9  =  8/729

So,

2/9 x (4/9 x 1/9)  =  (2/9 x 4/9) x 1/9

Therefore, Associative property is true for multiplication.

(iv) Multiplicative Identity :

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

If a/b is any rational number, then

a/b x 1 = 1 x a/b  =  a/b

Example :

5/7 x 1  =  1 x 5/7  =  5/7

(v) Multiplication by 0 :

Every rational number multiplied with 0 gives 0.

If a/b is any rational number, then

a/b x 0  =  0 x a/b  =  0

Example :

5/7 x 0  =  0 x 5/7  =  0

(vi) Multiplicative Inverse or Reciprocal :

For every rational number a/b, b ≠ 0, there exists a rational number c/d such that a/b x c/d = 1.

Then,

c/d is the multiplicative inverse of a/b.

If b/a is a rational number, then

a/b is the multiplicative inverse or reciprocal of it.

Example :

The multiplicative inverse of 2/3 is 3/2.

The multiplicative inverse of 1/3 is 3.

The multiplicative inverse of 3 is 1/3.

The multiplicative inverse of 1 is 1.

The multiplicative inverse of 0 is undefined.

## Division

(i) Closure Property :

The collection of non-zero rational numbers is closed under division.

If a/b and c/d are two rational numbers, such that c/d ≠ 0, then

a/b ÷ c/d is always a rational number.

Example :

2/3 ÷ 1/3  =  2/3 x 3/1  =  2 is a rational number.

(ii) Commutative Property :

Division of rational numbers is not commutative.

If a/b and c/d are two rational numbers, then

a/b ÷ c/d  ≠  c/d ÷ a/b

Example :

2/3 ÷ 1/3  =  2/3 x 3/1  =  2

1/3 ÷ 2/3  =  1/3 x 3/2  =  1/2

And,

2/3 ÷ 1/3    1/3 ÷ 2/3

Therefore, Commutative property is not true for division.

(iii) Associative Property :

Division of rational numbers is not associative.

If a/b, c/d and e/f  are any three rational numbers, then

a/b ÷ (c/÷ e/f)    (a/b ÷ c/d) ÷ e/f

Example :

2/9 ÷ (4/9 ÷ 1/9)  =  2/9 ÷ 4  =  1/18

(2/9 ÷ 4/9) ÷ 1/9  =  1/2 - 1/9  =  7/18

And,

2/9 ÷ (4/9 ÷ 1/9)    (2/9 ÷ 4/9) ÷ 1/9

Therefore, Associative property is not true for division.

## Distributive Property

(i) Distributive Property of Multiplication over Addition :

Multiplication of rational numbers is distributive over addition.

If a/b, c/d and e/f  are any three rational numbers, then

a/b x (c/d + e/f)  =  a/b x c/d  +  a/b x e/f

Example :

1/3 x (2/5 + 1/5)  =  1/3 x 3/5  =  1/5 -----(1)

1/3 x 2/5 + 1/3 x 1/5 :

=  2/15 + 1/15

=  (2 + 1)/15

=   3/15

=  1/5 -----(2)

From (1) and (2),

1/3 x (2/5 + 1/5)  =  1/3 x 2/5 + 1/3 x 1/5

Therefore, Multiplication is distributive over addition.

(ii) Distributive Property of Multiplication over Subtraction :

Multiplication of rational numbers is distributive over subtraction.

If a/b, c/d and e/f  are any three rational numbers, then

a/b x (c/d - e/f)  =  a/b x c/d  -  a/b x e/f

Example :

1/3 x (2/5 - 1/5)  =  1/3 x 1/5  =  1/15

1/3 x 2/5 - 1/3 x 1/5 :

=  2/15 - 1/15

=  (2 - 1)/15

=   1/15 -----(2)

From (1) and (2),

1/3 x (2/5 - 1/5)  =  1/3 x 2/5 - 1/3 x 1/5

Therefore, Multiplication is distributive over subtraction.

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

You can also visit the following web pages on different stuff in math.

WORD PROBLEMS

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6