# PROPERTIES OF WHOLE NUMBERS

## About "Properties of whole numbers"

Properties of whole numbers :

In math, whole numbers are the numbers which are positive integers including zero.

They are,

0, 1, 2, 3, 4, 5, 6, .....................................

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

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

Example :

2 + 4  =  6 is a whole number.

(ii) Commutative property :

Addition of two whole numbers is commutative.

If a and b are any two whole numbers,

then,  a + b  =  b + a

Example :

2+ 4  =  6

4 + 2  =  6

Hence, 2 + 4  =  4 + 2

(iii) Associative property :

Addition of whole numbers is associative.

If a, b and c  are any three whole 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

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

If a is any whole number,

then a + 0 = 0 + a  =  a

Zero is the additive identity for whole numbers.

Example :

7 + 0 = 0 + 7 = 7

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

## Subtraction

(i) Closure property :

The difference between any two whole numbers need not be a whole number.

Hence W is not closed under subtraction.

Example :

2 - 5  =  -3 is a not whole number.

(ii) Commutative property :

Subtraction of two whole numbers is not commutative.

If a and b are any two whole 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 whole numbers is not associative.

If a, b, c and d are any three whole numbers,

then a - (b - d)    (a - b) - d

Example :

2 - (4 - 1)  =  2 - 3  =  -1

(2 - 4) - 1  =  -2 - 1  =  -3

Hence, 2 - (4 - 1)    (2 - 4) - 1

Therefore, Associative property is not true for subtraction.

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

## Multiplication

(i) Closure property :

The product of two whole numbers is always a whole number. Hence W is closed under multiplication.

If a and b are any two whole numbers,

then a x b = ab is also a whole number.

Example :

5 x 2  =  10 is a whole number.

(ii) Commutative property :

Multiplication of whole numbers is commutative.

If a and b are any two whole numbers,

then a x b = b x a.

5 x 9  =  45

9 x 5  =  45

Hence, 5 x 9  =  9 x 5

Therefore, Commutative property is true for multiplication.

(iii) Associative property :

Multiplication of whole numbers is associative.

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

then a x (b x c)  =  (a x b) x c

Example :

2 x (4 x 5)  =  2 x 20  =  40

(2 x 4) x 5  =  8 x 5  =  40

Hence, 2 x (4 x 5)  =  (2 x 4) x 5

Therefore, Associative property is true for multiplication.

(iv) Multiplicative identity :

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

If a is any whole number,

then a x 1 = 1 x a  =  a

Example :

5 x 1 = 1 x 5  =  5

(v) Multiplication by 0 :

Every whole number multiplied with 0 gives 0.

If a is any whole number,

then a x 0 = 0 x a  =  0

Example :

5 x 0 = 0 x 5  =  0

## Division

(i) Closure property :

When we divide of a whole number by another whole number, the result does not need to be a whole number.

Hence, W is not closed under multiplication.

Example :

When we divide the whole number 3 by another whole number 2, we get 1.5 which is not a whole number.

(ii) Commutative property :

Division of whole numbers is not commutative.

If a and b are two whole numbers,

then a ÷ b   ≠  b ÷ a

Example :

÷ 1  =  2

÷ 2  =  1.5

Hence, ÷ 1    1 ÷ 2

Therefore, Commutative property is not true for division.

(iii) Associative property :

Division of whole numbers is not associative.

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

then a ÷ (b ÷ c)    (a ÷ b) ÷ c

Example :

÷ (4 ÷ 2)  =  3 ÷ 2  =  1.5

(3 ÷ 4) ÷ 2  =  0.75 ÷ 2  =  0.375

Hence, ÷ (4 ÷ 2)    (÷ 4) ÷ 2

Therefore, Associative property is not true for division.

## Distributive Property

(i) Distributive property of multiplication over addition :

Multiplication of whole numbers is distributive over addition.

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

then a x (b + c)  =  ab + ac

Example :

2 x (3 + 4)  =  2x3 + 2x4  =  6 + 8  =  14

2 x (3 + 4)  =  2x (7)  =  14

Hence, 2 x (3 + 4)  =  2x3 + 2x4

Therefore, Multiplication is distributive over addition.

(ii) Distributive property of multiplication over subtraction :

Multiplication of whole numbers is distributive over subtraction.

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

then a x (b - c)  =  ab - ac

Example :

2 x (4 - 1)  =  2x4 - 2x1  =  8 - 2  =  6

2 x (4 - 1)  =  2x (3)  =  6

Hence, 2 x (4 - 1)  =  2x4 - 2x1

Therefore, Multiplication is distributive over subtraction.

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

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

WORD PROBLEMS

HCF and LCM  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