Whole numbers include the positive counting numbers \(1, 2, 3\) and so on, along with zero. Consider how many people will attend a party, or how we calculate the age difference between two people, or how we find the number of pencils in a box, or how we compute the total fees paid by the students in a class, or how we divide a package of chocolate biscuits equally among some friends. In all of these cases, we use fundamental mathematical operations on whole numbers such as addition, subtraction, multiplication, and division. In this article, we will learn the operations of whole numbers.

What are Whole Numbers?

Natural numbers are a collection of positive integers, whereas whole numbers are a set of natural numbers plus zero \((0).\) Zero is an undefined identity that denotes a null set or no outcome.

Learn All the Concepts on Whole Numbers

Whole numbers, in basic terms, are a collection of numbers that contain no fractions, decimals, or negative integers. It consists of a set of positive integers and zero. The most significant distinction between natural and whole numbers is the presence of zero in whole numbers.

Whole Number Definition

Whole Numbers are the set of natural numbers along with the number \(0.\) The set of whole numbers in Mathematics is the set \(\{ 0,1,2,3, \ldots \} .\) This set of whole numbers is denoted by the symbol \(W.\)

\(W = \{ 0,1,2,3,4,. \ldots \} \)

Here are some facts about whole numbers, which will help you understand them better:

1. All natural numbers are whole numbers.
2. All counting numbers are whole numbers.
3. All positive integers, and zero, are whole numbers.
4. All whole numbers are real numbers.

Operations on Whole Numbers

1. Addition of Whole Numbers

Of the four arithmetic operations on numbers, addition is the most fundamental and was the first operation developed historically.

Addition on the Number Line The addition of whole numbers can be shown on the number line. Let us see the addition of \(3\) and \(4.\)

Start from \(3.\) Since we add \(4\) to this number, we make \(4\) jumps to the right; from \(3\) to \(4, 4\) to \(5, 5\) to \(6\) and \(6\) to \(7.\) The tip of the last arrow in the fourth jump is at \(7.\) The sum of \(3\) and \(4\) is \(7,\) i.e. \(3+4=7.\)

Properties of Addition of Whole Numbers

(i) Closure Property: If \(a\) and \(b\) are two whole numbers, then \(a+b\) is also a whole number. In other words, the sum of any two whole numbers gives a whole number.

Example: \(5+8=13.\) Here, \(5\) and \(8\) are whole numbers, and their sum \(13\) is also a whole number.

(ii) Commutative Law: If \(a\) and \(b\) are any two whole numbers, then \(a+b=b+a.\) In other words, even if the order of whole numbers is changed, the total of two whole numbers remains the same.

Example: \(4+8=8+4=12\)

(iii) Additive Property of Zero: If \(a\) is a whole number, then \(a+0=a=0+a.\) To put it another way, the number itself is the sum of any whole number and zero. The only whole number that does not affect the value of the number it is added to is zero.

The whole number \(0\) is called the additive identity of the whole number.

Example: \(5+0=5=0+5\)

(iv) Associative Law: If \(a, b, c\) are any three whole numbers, then \((a+b)+c=a+(b+c).\) To put it another way, adding whole numbers is associative.Example: \(5+(2+8)=(5+2)+8=15\)

2. Subtraction of Whole Numbers

It is the reverse process of addition.

Subtraction on the Number Line

The subtraction of two whole numbers can also be shown on the number line. Let us find \(7-5.\)

Start from \(7.\) Since \(5\) is being subtracted, so move towards the left with \(1\) jump of \(1\) unit. Make \(5\) such jumps. We reach point \(2.\) We get \(7-5=2.\)

Properties of Subtraction of Whole Numbers

Below are the properties of subtraction of whole numbers.

1. If \(a\) and \(b\) are two whole numbers such that \(a>b\) or \(a=b,\) then \(a-b\) is a whole number. If \(a<b,\) then subtraction \(a-b\) is not possible in whole numbers. 

2. The subtraction of whole numbers is not commutative; that is, if \(a\) and \(b\) are two whole numbers, then in general \(a-b≠(b-a).\)

3. If \(a\) is any whole number and \(a≠0,\) then \(a-0=a,\) but \(0-a\) is not defined.

4. The subtraction of whole numbers is not associative. If \(a, b, c\) are three whole numbers, then generally, \(a-(b-c)≠(a-b)-c.\)

5. If \(a, b\) and c are whole numbers such that \(a-b=c,\) then \(b+c=a.\)

3. Multiplication of Whole Numbers

Multiplication of whole numbers is a type of addition that is repeated.

Multiplication on the Number Line

We now see the multiplication of whole numbers on the number line. Let us find \(4×3.\)

Start from \(0,\) move \(3\) units at a time to the right, make such \(4\) moves. Where do you reach? You will reach \(12.\) So, we say, \(3×4=12.\)

Properties of Multiplication of Whole Numbers

(i) Closure Property: If \(a\) and \(b\) are two whole numbers, then their product \(a×b\) is also a whole number. In other words, we obtain a whole number when we multiply two whole numbers.

Example: \(4×2=8.\) Here, \(4, 2\) and \(8\) are whole numbers.

(ii) Commutative Law: The multiplication of whole numbers is commutative. In other words, if \(a\) and \(b\) are any two whole numbers, then, \(a×b=b×a\)

Example: \(9×2=2×9=18\)

(iii) Multiplicative Identity: If \(a\) is any whole number, then \(a×1=a=1×a.\) In other words, the product of any whole number with one always gives the number itself.

Here, \(1\) is called the multiplicative identity.

Example: \(8×1=8=1×8\)

(iv) Associative Law: If \(a,b, c\) are any whole numbers, then \(a×b×c=a×b×c.\) In other words, the multiplication of whole numbers is associative; that is, the product of three whole numbers remains the same by altering the order.

Example: \((5×2)×3=5×(2×3)=30\)

(v) Distributive Law of Multiplication over Addition: If \(a,b, c\) are any three whole numbers, then, \(a×(b+c)=a×b+a×c\) and \((b+c)×a=b×a+c×a.\) In other words, whole number multiplication spreads over their addition.Example: \(2×(3+8)=2×3+2×8=22\)

4. Division of Whole Numbers

Dividing whole numbers is the opposite process of multiplying whole numbers. It is the process by which we try to figure out how many times a number (divisor) is contained in another number (dividend).

The answer in the division problem is called a quotient. In the division problem above \((63÷7), 7\) is contained into \(63, 9\) times. \((9×7=63)\)

Properties of Division of Whole Numbers

1. If \(a\) and \(b\) are whole numbers \((b≠0),\) then \(a÷b\) is not necessarily a whole number. In other words, whole numbers are not closed for the division.

2. If a is any whole number, then, \(a÷1=a.\)

Example: \(23÷1=23\)

3. If a is any whole number other than zero, then \(a÷a=1.\) In other words, any whole number other than zero divided by itself gives \(1\) as the quotient.

Example: \(9÷9=1\)

4. Zero divided by any whole number gives the quotient as zero.

Example: \(0÷8=0\)

5. The division of any whole number by \(0\) is not defined.

For example, \(4 \div 0 = \frac{4}{0} = \)not defined

6. Let \(a, b, c\) be whole numbers and \(b≠0, c≠0.\) If \(a÷b=c,\) then \(b×c=a.\)

Example: \(12÷4=3⇒3×4=12\)

Solved Examples – Operation on Whole Numbers

Q.1. Add the given whole number \(462\) and \(823.\)
Ans: We need to add \(462\) and \(823\)
Therefore, \(462+823=1285\)

Q.2. Find the product using distributive property: \(237×103\)
Ans: Given, \(237×103\)
\(237×(100+3)\)
We will use the distributive property: \(a×(b+c)=a×b+a×c\)
Therefore, \(237×(100+3)\)
\(=237×100+237×3\)
\(=23700+711\)
\(=24411\)
Hence, the required product is \(24411.\)

Q.3. Verify the following:
\(537+265=265+537\)
Ans: Given, \(537+265=265+537\)
Consider L.H.S.\(=537+265=802\)
And R.H.S.\(=265+537=802\)
We will use the commutative property: \(a+b=b+a\)
Therefore, L.H.S.=R.H.S.
Hence, it is verified.

Q.4. Find the least number that must be subtracted from \(1000\) so that \(45\) divides the difference exactly.
Ans: We know that the division algorithm
Dividend\(=\)Divisor\(×\)Quotient\(+\)Remainder
\(1000÷45=45×22+10\)
Now \(1000-10=990\)
Therefore, \(10\) should be subtracted from \(1000,\) so that difference \(990\) is divisible by \(45.\)

Q.5. Find the number which, when divided by \(15\) gives \(7\) as the quotient and \(3\) as the remainder.
Ans: We know that the division algorithm
Dividend\(=\)Divisor\(×\)Quotient\(+\)Remainder
\(=15×7+3\)
\(=105+3=108\)
Therefore, the required number is \(108.\)

Summary

In this article, we have discussed whole numbers, the definition of whole numbers, the symbol to represent whole numbers. After that, we discussed the four basic operations: addition, subtraction, multiplication and division on whole numbers, along with their properties and solved examples.

Learn About Patterns in Whole Numbers

FAQs

Q.1. How do you do order of operations with whole numbers?
Ans: We generally use the BODMAS rule.
B–Bracket
O–Order
D–Division
M–Multiplication
A–Addition
S–Subtraction

Q.2. What operations are closed under whole numbers?
Ans: Addition and multiplication operations are closed under whole numbers.

Q.3. What is 0 called in whole numbers?
Ans: \(0\) is the smallest whole number. It is the additive identity.

Q.4. Which is the greatest whole number?
Ans: There is no greatest whole number. Whole numbers extend to negative infinity.

Q.5. Do I divide or multiply first?
Ans: According to the BODMAS rule, the division is done before multiplication.