Based on the Historical background of whole numbers, we have an idea about how people have used various symbols or numerals as a method of expressing numbers.

Many of these symbols were drawn from objects that people came across in their everyday life.
The early Egyptians used flowers like lotus or polliwogs or even pointed fingers to symbolize numbers.
The Aztecs used to draw a sack to represent the number 8000 because a sack full of cocoa contained about 8000 beans.

People from the Indian peninsula invented Math symbols such as 1, 2, 3 which we are using today. These are now universally used and were brought in by the Arabs to Europe. Later they were divided into whole numbers and rational numbers, real numbers...etc.

Whole numbers are 0,1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13…

The dots after the numbers signify that the list continues without any specified 'largest' whole number.
Like distances are associated with the markings on the edge of a ruler; the whole number can be associated with points on a line.

While counting is based on the base-ten numbering system that uses ten digits or numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 These numerals are the first whole numbers. When we get to 9 we exhaust all single digit whole numbers and in order to represent the next set of whole numbers we use two digits like 10.Even though (0) zero is written as the first whole number, it is not considered as the first counting number.

Whole number property :

The whole number properties should actually be construed as whole number operations. There are four basic properties or operations for whole numbers; addition (+), subtraction (-), multiplication (x) and division (÷).
The written form of operations or properties is better known as expression.


Property Term Expression Meaning Result
Addition sum 7 + 2 sum of 7 and 2 9
Subtraction difference 8 - 2 difference of 8 and 2 6
Multiplication product 5 × 1 product of 5 and 1 5
Division quotient 6 ÷ 2 quotient of 6 and 2 3

When adding whole numbers, set them in a column so that they can be added correctly.

Step 1: The numbers should be lined up so that the ones column is aligned.
Step 2: The process of adding the numbers starts from the right hand column.
Step 3: Work your way to the left for the final answer.
Step 4: Numbers carried to the next column on the left are then added in that column.

Examples on Adding Whole Numbers

Given below are some examples on adding whole numbers.

1) add 123 + 241

Step 1: Arrange these in an aligned column
Step 2: Add 3 + 1 in the first column
Step 3: Add 4 + 2 in the second column
Step 4: Add 2 + 1 in the third column.

123
+ 241
364

2) Add 46 and 52

Solution: Given 46
+ 52
____

Step 1: First add the one’s place number and carry over the number for the next step. In the above example, we add 6 and 2, because these numbers exist in the one’s place. Add 6 + 2 = 8, since we don't need to carry over any sum we proceed to the next step.

46
+ 52
8
Step 2: Now add the digits in the ten’s place number. Here we add 4 and 5 (4 + 5 = 9).
46
+ 52
98

Answer: 98

Word problem on addition:

Example: In three days Betty ran 4 miles, 7 miles and 2 miles. What was the total distance he ran?

Solution: Operation: addition
Total: 4 + 7 + 2 = 13 miles
So Betty ran 13 miles.

The addition of whole numbers and fractions would give us a mixed number.

Let us take an example.

5 + $\frac{3}{4}$

= [5] + $\frac{3}{[4]}$

= $\frac{20}{4}$ + $\frac{3}{4}$

= $\frac{23}{4}$ = 5 $\frac{3}{4}$

Examples for adding whole numbers and fractions


Add 5 + $\frac{10}{5}$

Solution:

Given 5 + $\frac{10}{5}$

Step 1: We convert the whole number 5 into a fraction. i.e. 5 become $\frac{5}{1}$

Step 2: The bottom numbers (denominators) are not same in the fraction $\frac{5}{1}$ + $\frac{10}{5}$, so we first convert this fraction into an equivalent fraction with the same denominator.

Multiply with 5 for the fraction $\frac{5}{1}$

= $\frac{5\times5}{1\times5}$ = $\frac{25}{5}$

Multiply with 1 for the fraction $\frac{10}{5}$

= $\frac{10\times1}{5\times1}$ = $\frac{10}{5}$

Step 3: Now bottom numbers (denominators) are the same $\frac{25}{5}$ + $\frac{10}{5}$ , so we need to add the above numbers (numerators)

$\frac{25}{5}$ + $\frac{10}{5}$ = $\frac{25+10}{5}$ = $\frac{35}{5}$

Step 4: Simplify the fraction

= $\frac{35}{5}$

= $\frac{7}{1}$

= 7

To subtract one whole number from another we have to line up the numbers by place value. Subtract each number separately beginning from the right.
If I need to subtract 7 ones from 9 ones, I have to regroup and reach into the tens column and take 1 ten.
To begin with, I need to regroup 1 ten with the 8 into the ones column to make it 18 and then subtract 9.

Let us take an example.

Subtract 219 from 348

Step 1: Arrange the numbers in a column.
Step 2: Regroup 1 ten with 8 into the ones column and make it 18.
Step 3: Now subtract 9.
Step 4: There are only 3 ten’s in the tens column of 348, not 4.
Step 5: Now subtract the number in the tens column.
Step 6: Finally, subtract the number in the hundreds column.
The answer to the above example is 129.

318
348
- 219
129

The numbers on the top row show the regrouping.

Examples for Subtracting Whole Numbers


Given below are some examples on subtracting whole numbers

Example 1: Subtract 78 and 35

Solution: Given 78
- 35
___

Step 1: First subtract the number in the one's place. Here 8 and 5 are in the one's place, so 8 - 5 = 3. The minuend number is larger than subtrahend so we continue the process.

78
- 35
3

Step 2: Subtract the ten's place number. Here 7 and 3 are in the ten's place, so 7 - 3 = 4.
The minuend number is larger than subtrahend we continue the process.

78
- 35
43

Answer : 43

Example 2: Subtract 145 and 126

Solution: Given 145
- 126
____

Step 1: First subtract numbers 5 and 6 in the one's place. The minuend number is smaller than the subtrahend, so we borrow a number from the next place. Here 4 turns in to 3 at the ten's place after borrowing and 5 turns 15 at the one's place. So 15 - 6 = 9

135
- 126
9

Step 2: Subtract the ten's place number. 3 and 2 are in the ten's place. So 3 - 2 = 1.
The minuend number is larger than the subtrahend we continue the process.

135
- 126
19

Step 3: Subtract the hundred's place number. 1 and 1 are in the hundred's place. So 1 - 1 = 0.
The minuend number is larger than the subtrahend so we continue the process.

135
- 126
019

Answer : 19

Word Problem on Subtraction:

Example: Sam is swimming in a 1600 meter event. He had already completed 500 meters of the race. How far does he have to swim to complete the race ?

Solution:
Operation : Subtraction
Distance = 1600 - 500
= 1100
Sam still has 1100 meters to swim.

Subtracting a whole number from a fraction is another operation where the rule of fraction comes into force.

The following steps need to be followed for subtracting whole numbers from fraction:

Step 1: To subtract a fraction from a whole number, we need to express the whole number as an equivalent mixed number.
Step 2: The fraction of the mixed number has to have the same denominator as that of the fraction that is getting subtracted.
Step 3: Subtract the numerator of the fraction and write their difference over the common denominator.
Step 4: Subtract the whole number.
Step 5: Combine the whole number and the fraction and express the answer in its lowest term.

Example of subtracting whole number from a fraction:

The following examples explain how to subtract whole numbers from fractions

Example 1: 6 - $\frac{1}{3}$

Solution: Given 6 - $\frac{1}{3}$

Step 1: We convert the whole number 6 into a fraction. i.e. 6 become $\frac{6}{1}$

Step 2: The bottom numbers (denominators) are not the same in the fraction $\frac{6}{1}$ + $\frac{1}{3}$, so we first convert this fraction into an equivalent fraction with the same denominator.

Multiply with 3 for fraction $\frac{6}{1}$

$\frac{6\times3}{1\times3}$ = $\frac{18}{3}$

Multiply with 1 for fraction $\frac{1}{3}$

$\frac{1\times1}{3\times1}$ = $\frac{1}{3}$

Step 3: Now bottom numbers (denominators) are the same $\frac{18}{3}$ - $\frac{1}{3}$ , so we need to subtract the above numbers (numerators)

$\frac{18}{3}$ - $\frac{1}{3}$ = $\frac{18-1}{3}$ = $\frac{17}{3}$

Step 4: Simplify the fraction

= $\frac{17}{3}$

= 5 $\frac{2}{3}$

For whole numbers multiplication refers to repeated addition.
When we multiply whole numbers, we need to line up the numbers correctly.
The numbers in a product are called factors.

So, factor (x) times factor = product
When we come across (5 x 6 = 30). Here 5 and 6 are factors of the product.

Let us take an example:

Multiply 673 x 5

We have to line the number up correctly and so it has to be arranged in this way.

673
× 5
___

Step 1: Multiply 3 in the first row by 5 in the second row.
Step 2: 3 x 5 = 15, as we cannot write 15 in the ones place, we have to follow the addition method. Write 5 and save 1 for the tens place.
Step 3: Go back to the first row and multiply 7 x 5 = 35 and now add the 1 saved earlier.
Step 4: We get 36 and so we need to write 6 and save 3
Step 5: The first row has 6 which is then multiplied with 5 and that gives us 30. We have to add the 3 that we saved earlier and so we have 33.
The product or answer to this multiplication problem is 3365.

Tips: we need to multiply the entire top number by just one bottom digit at a time.
We also need to use a different line for the product of each bottom digit.

Multiplying Whole Number Examples

Multiply 236 × 4

236
× 4
944

Step 1: Multiplying 6 with 4 we get 24. We cannot write 24 in one's place, so we follow the addition method. Write 4 and save 2 for the ten's place.
Step 2: Now multiply 3 with 4 and add 2, we get 3 × 4 + 2 = 14. We cannot write 14 in the ten's place so we have to follow the addition method. Write 4 and save 1 for the hundred's place.
Step 3: Multiply 2 with 4 and add 1; we get 2 × 4 + 1 = 9.
The product or answer to this multiplication problem is 944
Example : Multiply 379 × 6

379
× 6
2274

Step 1: Multiplying 9 with 6 we get 54. We cannot write 54 in the one's place, so we have to follow the addition method. Write 4 and save 5 for the ten's place.
Step 2: Multiply 7 with 6 and add 5 to get 7 × 6 + 5 = 47. We cannot write 47 in the ten's place, so we have to follow the addition method. Write 7 and save 4 for the hundreds place.
Step 3: Now multiply 3 with 6 and add 4 to we get 3 × 6 + 4 = 22
The product or answer to this multiplication problem is 2274

Multiplying whole numbers with fractions could be simplified by using cancellation. Cancellation is the process of determining a common number that would divide the numerators equally and at the same time, any one of the denominators in the fractions gets multiplied.
This process is repeated as often as possible.

Let us try one example:

24 × $\frac{3}{8}$

Step 1: Determine the common factor that will equally divide the numerator and the denominator.
Step 2: 24 is divisible by 8 so we have the numerator getting cancelled by 8 and we get 3.
Step 3: Multiply the resultant quotient 3 with the numerator 3 which gives us 9.

Multiplying Whole Numbers with Fractions Examples

Example 1: $\frac{3}{6}$ × 18

Solution: Given $\frac{3}{6}$ × 18
Step 1: Multiplying 3 with 18, we get 54. The denominator remains the same.
Step 2: Simplifying the fraction $\frac{54}{6}$ = $\frac{9}{1}$ = 9
Answer : 9

Example 2: $\frac{2}{7}$ × 14

Solution: Given $\frac{2}{7}$ × 14

Step 1: Multiplying 2 with 14, we get 28. The denominator remains the same.
Step 2: Simplifying the fraction $\frac{28}{7}$ = $\frac{4}{1}$ = 4
Answer : 4

Division is basically the inverse operation of multiplication and we can observe a circular argument of inverses and it says that (divisor × quotient = dividend)
Its only because of the circular nature of division and multiplication, we can use multiplication to verify the accuracy of our division result or quotient.
The division of two numbers can be represented in any of the three forms.
Standard form, long division form and fraction form.

Standard form: $\frac{Dividend}{Divisor}$ = Quotient

Long division form: quotient
divisor ) dividend

Example : Divide ( 54 ÷ 6 )

Dividing Whole Numbers

Step 1: Divide 54 by 6
Step 2: 6 is contained into 54, i.e. 6 times 9 is 54 ( 6 × 9 = 54 )
Fraction form: $\frac{Dividend}{Divisor}$ = quotient $\frac{54}{6}$ = 9

Dividing whole numbers examples

Example: Divide (375 ÷ 3)

Solution: Given (375 ÷ 3)

Dividing Whole Numbers Example

Step 1: We multiply 3 with 12 and subtract 37 with 36, to get 1
Step 2: Bring down the 1, it turns into 15
Step 3: Multiplying 3 with 5, we get 15 ( 3 × 5 = 15 )
Step 4: The quotient will 125 and remainder will be zero.
Answer : 125

To divide a whole number and fraction we need to first write the whole number as a fraction with a denominator of 1.

Step 1: Convert the whole number into a fraction by inserting 1 in the denominator.
Step 2: Use the reciprocal of the whole number and change the division sign into a multiplication one.
Step 3: Multiply the numerator and the denominator.
Step 4: Reduce the answer if necessary.

Examples for dividing whole numbers and fractions


Example 1: Divide $\frac{6}{8}$ × 4

Solution: Given $\frac{6}{8}$ × 4
Step 1: Convert the whole number (4) into a fraction $\frac{4}{1}$
Step 2: Write the reciprocal of the new fraction $\frac{1}{4}$
Step 3: Multiplying the numerators and denominators across
$\frac{6}{8}$ × $\frac{1}{4}$ = $\frac{6\times1}{8\times4}$ = $\frac{6}{32}$
Step 4: Simplify the fraction $\frac{6}{32}$ = $\frac{3}{16}$

Example 2: Divide $\frac{1}{6}$ × 2

Solution: Given $\frac{1}{6}$ × 2
Step 1: Convert the whole number (2) into a fraction $\frac{2}{1}$
Step 2: Write the reciprocal of the new fraction $\frac{1}{2}$
Step 3: Multiplying the numerators and denominators across
$\frac{1}{6}$ × $\frac{1}{2}$ = $\frac{1\times1}{6\times2}$ = $\frac{1}{12}$

When we have to divide decimals by whole numbers we have to be cautious about putting the decimal point in the correct place in the final answer.
It is always useful to put a decimal point before we start the division operation.
The decimal point would be placed directly above its place in the problem.

Let us try one.

Divide 32.8 by 4
Step 1: Follow the standard division method 4 x 8 = 32.
Step 2: The decimal is placed right above the decimal in the dividend.
Step 3: The answer to the problem is 8.2

Dividing decimals with whole number examples


Example 1: Divide 0.04 ÷ 2

Solution: Given 0.04 ÷ 2

Step 1: Follow the standard division method 2 × 2 = 4
Step 2: The Decimal is placed right above the decimal in the dividend
Step 3: The answer to the problem is 0.02

Example 2: Divide 6.9 ÷ 3

Solution: Given 6.9 ÷ 3

Step 1: Follow the standard division method 3 × 2 = 6 , 3 × 3 = 9
Step 2: The decimal is placed right above the decimal in the dividend
Step 3: The answer to the problem is 2.3

Although we know that the origin of zero is controversial we still believe it was invented by the Indians and the Chinese.
It reached the new world or western world through the Arabs.
Zero is the only integer or whole number that is neither positive nor negative.
As per the rule, any number that has a decimal or is in a fractional form cannot be considered as a whole number and so zero which has neither of these features can be considered as a whole number.

When zero is added to or subtracted from a number it leaves the number at its original value.

Zero is also essential a position holder in the system known as the positional notation.

Rounding whole numbers could often be used to determine the correct answer to multiple choices more quickly.

When we need to approximate the value of a whole number, we could use the following procedure to round off the number to a particular place.

Steps for rounding off whole numbers:

Step 1: Underline the digit in the place to be rounded off.
Step 2: If the digit to the right of the underlined number is less than 5, we leave the underlined digit as it is.
Step 3: In case the digit to the right of the underlined digit is equal to 5 or more, we have to add 1 to the underlined digit.
Step 4: We have to replace all the digits to the right of the underlined digit by zeros.

Rounding Whole Numbers Examples

Example 1: Round off 87,743 to the nearest hundred

Solution: Given 87,743

Step 1: As we are rounding off to the nearest hundred, we have to begin by underlining the digit in the hundreds place which is a 7
Step 2: If we look at the right of the underlined digit, we find the number 4 and so we leave 7 as it is and replace all digits to the right of 7 with zeros.
Step 3: So the number is 87,700

Example 2 : Round off 43,631 to the nearest hundred

Step 1: As we are rounding off to the nearest hundred, we have to begin by underlining the digit in the hundreds place which is a 6.
Step 2: If we look at the right of the underlined digit, we find the number 3 and so we leave 6 as it is and replace all digits to the right of 6 with zeros.
Step 3: So the number is 43,600.

The following problems on whole numbers explain various operations on whole numbers like addition and division of whole numbers.

1) Add 65 and 18

Solution: Given 65
+ 18
__

Step 1: Add the one’s place number and carry it over for the next step. Here we add 5 and 8, because these numbers are existing in the one’s place. Add 5+8 = 13, we carry over 1 that is move it to the ten’s place.
65
+ 18
3
Step 2: Now add the digits in the ten’s place including the number carried over from the one’s place. We add 1+6+1 = 8.
65
+ 18
83

Answer: 83


2) Add 462 and 247

Solution: Given: 462
+ 247
___

Step 1: First add the numbers in the one’s place and carry them over for the next step. Here we add 2 and 7, because these numbers exist in the one’s place. Add 2+7 = 9, we don't carry over in this step and proceed to the next step.

462
+ 247
9

Step 2: Now add the numbers in the ten’s place and carry over for the next step. Here we add 6 and 4, because these numbers exist in the ten’s place. Add 6+4 = 10, we carryover 1 and move it to the hundred’s place.

462
+ 247
09

Step 3: Add the hundred's place number including the number carried over from the ten's place. We get , 1+4+2 = 7

462
+ 247
709

Answer: 709


3)
$\frac{2}{4}$ × 16

Solution: Given, $\frac{2}{4}$ × 16
Step 1: Multiplying 2 with 16, we get 32. The denominator remains the same.
Step 2: Simplifying the fraction $\frac{32}{4}$ = $\frac{8}{1}$ = 8
Answer : 8