whole numbers

whole numbers

We learn numbers from a very young age. Here is an attempt to know about Whole numbers and their properties. We shall discuss closure property, commutativity, associativity and distributivity of whole numbers. We also discuss about the factors and multiples, perfect number, prime and composite numbers, twin primes and  co-prime numbers.

Natural numbers

Natural numbers, otherwise known as counting numbers, start from 1. Successive numbers are obtained by adding 1 . So the natural numbers are 1, 2 , 3, 4, 5, 6 etc.

Whole numbers

Unlike natural numbers, whole numbers start from 0. Other numbers are obtained by adding 1 successively. So the whole numbers are 0, 1, 2, 3, 4, 5, 6, 7 etc.

There is no upper limit to these numbers.

Let us say, to the contrary, that  a is the largest whole number.

We know that a+1 is also a whole number by the property of whole numbers.

As a+1>a, this is a contradiction. This contradiction has arisen  because of our wrong assumption that ‘a’ is the largest whole number. Thus we conclude that there is no largest whole number number.

Properties of Whole Numbers

Closure property: When we add or multiply two whole numbers, we always get a whole number. So we say that the whole  numbers are closed under the operations of addition and multiplication.

e.g. 2+3=5-> a Whole number

5+10=15 -> a Whole number

3×15=45 -> a Whole number

6×11=66 -> a Whole number

But when we subtract or divide, we will not get a Whole number always. So we can say that Whole numbers are not closed under the operations of subtraction and division.

e.g. 15-11= 4 -> a Whole number

6-13 = -7 -> not a Whole number

15 / 3 = 5 -> a Whole number

5/ 7 = 0.714285 …  -> not a Whole number

 Commutativity of addition and multiplication

Addition and multiplication are commutative for whole numbers. i.e. We can add or multiply two numbers in any order.

e.g. 2+3=3+2=5

5×7=7×5= 35

Associativity of addition and multiplication

We can associate any two whole numbers while adding three or more whole numbers.

e.g.  (2+3)+4=2+(3+4)=(2+4)+3=9

We can associate any two whole numbers while multiplying  three or more whole numbers.

e.g. (5×2)x4=5x(2×4)=(5×4)x2=40

Distributivity of multiplication over addition and subtraction

6x(5+7)=6×5+6×7=72

This is distributivity of multiplication over addition

4x(6-3)=4×6-4×3= 12

This is distributivity of multiplication over subtraction

Identity for addition

When we add 0 to any number, we get the same number.

6+0=6

8+0=8

Therefore,  0 is called the identity element for addition

Identity for multiplication

When we multiply any number by 1, we get the same number.

7×1=7

15×1=15

Therefore, 1 is called the identity element for multiplication.

Do you know-

that division by 0 is not defined in whole numbers?

that when we multiply any number by 0, we get 0?

Factors and multiples

A factor of a number is an exact divisor of that number.

1, 2, 3, 4, 6, 8, 12 and 24 are the factors of 24. i.e. each one of these numbers divides 24 exactly.

In other words, 24 is a multiple of each one of its factors.

The number of factors of a given number are finite. There are exactly 8 factors of 24.

Prime number

The numbers others than 1 whose only factors are 1 and the number itself are called prime numbers.

2, 3, 5, 7, 11, 13 etc are prime numbers.

Numbers having more than two factors are called composite numbers.

4, 6, 8, 9, 10, 12 etc are composite numbers

The number 1 is neither prime not composite. 1 is a unique number.

There are infinitely many primes.

Twin primes

Two prime numbers whose difference is 2 are called twin primes.

e.g. 3 and 5 are twin primes

11 and 13 are twin primes

17 and 19 are twin primes

Co primes

Two co-prime numbers have no common factor other than 1.

11 and 12 are co-prime.

Any two consecutive whole number are co-prime.

The co-prime numbers need not be primes.

Perfect numbers

A number for which sum of all its factors is equal to twice the number is
called a perfect number.

6 is a perfect number. The factors of 6  are 1, 2, 3 and 6. The sum of the factors is 12 which is twice the number 6.

Similarly 28 is another perfect number. The factors of 28 are 1, 2, 4, 7, 14 and 28. The sum of the factors is 56 which is twice the number 28.

Are you ready for few questions?

Choose the right option from the bracket and fill in the blanks.

1. The smallest whole number is ______. (1/0)
2. The only whole number which does not belong to natural numbers is _______. (1/0)
3. ________is the smallest prime number. (1/2)
4. ________ is the smallest composite number. (2/4)
5. ________ is the only even prime. (2/4)
6. 2 and 3 are __________. (twin primes/ co-prime)
7. 19 and 21 are ________. (twin primes/ co-prime)
8. ________ is the identity element of addition.(0/1)
9. ________ is the identity element of multiplication.(0/1)
10. Subtraction and division are ________. (commutative/ non-commutative)

Answers

1. 0
2. 0
3. 2
4. 4
5. 2
6. co-prime
7. co-prime
8. 0
9. 1
10. non-commutative

Hope you enjoyed the lesson.