**Properties of real numbers :**

There are some properties of real 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 real is always a real number. This is called ‘Closure property of addition’ of real numbers. Thus, R is closed under addition

If a and b are any two real numbers, then (a +b) is also a real number.

**Example : **

2 + 4 = 6 is a real number.

**(ii) Commutative property : **

Addition of two real numbers is commutative.

If a and b are any two real numbers,

then, a + b = b + a

**Example : **

2/9 + 4/9 = 6/9 = 2/3

4/9 + 2/9 = 6/9 = 2/3

Hence, 2/9 + 4/9 = 4/9 + 2/9

**(iii) Associative property :**

Addition of real numbers is associative.

If a, b and c are any three real numbers,

then a + (b + c) = (a + b) + c

**Example : **

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

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

Hence, 2 + (4 + 1) = (2 + 4) + 1

**(iv) Additive identity :**

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

If a/b is any real number,

then a/b + 0 = 0 + a/b = a/b

Zero is the additive identity for real numbers.

**Example : **

2/7 + 0 = 0 + 2/7 = 27

**(v) Additive inverse :**

(- a) is the negative or additive inverse of (a)

If a is a real number,then there exists a real number (-a) such that a + (-a) = (-a) + a = 0

**Example : **

Additive inverse of 5 is (-5)

Additive inverse of (-5) is 5

Additive inverse of 0 is 0 itself.

Let us look at the next stuff on "Properties of real numbers"

**(i) Closure property : **

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

Hence R is closed under subtraction.

If a and b are any two real numbers, then (a - b) is also a real number.

**Example : **

5 - 2 = 3 is a real number.

**(ii) Commutative property : **

Subtraction of two real numbers is not commutative.

If a and b are any two real numbers,

then (a - b) ≠ (b - a)

**Example : **

5 - 2 = 3

2 - 5 = -3

Hence, 5 - 2 ≠ 2 - 5

Therefore, Commutative property is not true for subtraction.

**(iii) Associative property :**

Subtraction of real numbers is not associative.

If a/b, c/d and e/f are any three real 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

Hence, 2/9 - (4/9 - 1/9) ≠ (2/9 - 4/9) - 1/9

Therefore, Associative property is not true for subtraction.

Let us look at the next stuff on "Properties of real numbers"

**(i) Closure property :**

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

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

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

**Example : **

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

**(ii) Commutative property :**

Multiplication of real numbers is commutative.

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

then (a/b)x (c/d) = (c/d)x(a/b).

5/9 x 2/9 = 10/81

2/9 x 5/9 = 10/81

Hence, 5/9 x 2/9 = 2/9 x 5/9

Therefore, Commutative property is true for multiplication.

**(iii) Associative property :**

Multiplication of real numbers is associative.

If a/b, c/d and e/f are any three real 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

Hence, 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 real number and 1 is the rational number itself. ‘One’ is the multiplicative identity for real numbers.

If a/b is any rational number,

then a/b x 1 = 1 x a/b = a/b

**Example : **

5/7 x 1 = 1x 5/7 = 5/7

**(v) Multiplication by 0 :**

Every real number multiplied with 0 gives 0.

If a/b is any real number,

then a/b x 0 = 0 x a/b = 0

**Example : **

5/7 x 0 = 0x 5/7 = 0

**(vi) Multiplicative Inverse or Reciprocal :**

For every real number a/b, a≠0, there exists a real 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 real number,

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

**Example : **

The reciprocal of 2/3 is 3/2

The reciprocal of 1/3 is 3

The reciprocal of 3 is 1/3

The reciprocal of 1 is 1

The reciprocal of 0 is undefined

Let us look at the next stuff on "Properties of real numbers"

**(i) Closure property :**

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

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

then a/b ÷ c/d is always a real number.

**Example : **

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

**(ii) Commutative property :**

Division of real numbers is not commutative.

If a/b and c/d are two real 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

Hence, 2/3 ÷ 1/3 ≠ 1/3 ÷ 2/3

Therefore, Commutative property is not true for division.

**(iii) Associative property :**

Division of real numbers is not associative.

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

then a/b ÷ (c/d ÷ 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

Hence, 2/9 ÷ (4/9 ÷ 1/9) ≠ (2/9 ÷ 4/9) ÷ 1/9

Therefore, Associative property is not true for division.

**(i) Distributive property of multiplication over addition :**

Multiplication of real numbers is distributive over addition.

If a/b, c/d and e/f are any three real 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

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

Hence, 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 real numbers is distributive over subtraction.

If a/b, c/d and e/f are any three real 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 - 1/5) = 1/3 x 2/5 - 1/3 x 1/5 = (2 - 1) / 15 = 1/15

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

Therefore, Multiplication is distributive over subtraction.

After having gone through the stuff given above, we hope that the students would have understood "Properties of real numbers".

Apart from the stuff given above, if you want to know more about "Properties of real numbers", please click here

Apart from "Properties of real numbers", if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

**WORD PROBLEMS**

**HCF and LCM word problems**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**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**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**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 **

**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 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**