Whole numbers are numbers that start from zero and continue without end. They do not have fractions or decimal parts. Examples of whole numbers are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and so on.

Whole numbers are used in counting and ordering things in everyday life. For example, when we count the number of books, chairs, or people in a class, we use whole numbers.

Whole numbers are written without any fraction or decimal. For instance, 12 is a whole number, but 12½ or 12.5 are not whole numbers.

Set of Whole Numbers

The set of whole numbers is written as

W = {0, 1, 2, 3, 4, 5, 6, 7, …}

The dots (…) mean that the numbers continue endlessly. So, whole numbers have no end. They are infinite.

Natural numbers start from 1, while whole numbers start from 0.

1. Natural numbers: {1, 2, 3, 4, 5, …}


2. Whole numbers: {0, 1, 2, 3, 4, 5, …}

So, the only difference is that whole numbers include zero (0), but natural numbers do not.

Place Value and Face Value

When we write a number, each digit has a place value and a face value.

1. Face value: The face value of a digit is the number itself. For example, in the number 526, the face value of 5 is 5.


2. Place value: The place value of a digit depends on its position in the number. For example, in 526:

1. The place value of 6 is 6 ones or 6.


2. The place value of 2 is 2 tens or 20.


3. The place value of 5 is 5 hundreds or 500.

So, 526 = 500 + 20 + 6.

This way of breaking down numbers is called expanding numbers or expanded form.

We can write whole numbers in figures or in words.

Examples:

1. 9 — Nine


2. 15 — Fifteen


3. 124 — One hundred and twenty-four


4. 1,207 — One thousand, two hundred and seven


5. 10,056 — Ten thousand and fifty-six

When reading large numbers, it helps to group the digits in threes from the right (using commas).

For example: 1,234,567 is read as One million, two hundred and thirty-four thousand, five hundred and sixty-seven.

Comparison of Whole Numbers

We can compare two or more whole numbers to know which is greater or smaller.

1. 8 is less than 12 → 8 < 12


2. 15 is greater than 9 → 15 > 9


3. 20 is equal to 20 → 20 = 20

When comparing, the number with more digits is the greater number. For example, 125 is greater than 97 because 125 has three digits while 97 has two digits.

Ordering of Whole Numbers

Ordering means arranging numbers in a particular order.

We can arrange numbers in:

1. Ascending order – from smallest to biggest. Example: 12, 18, 23, 45, 56, 89.


2. Descending order – from biggest to smallest. Example: 89, 56, 45, 23, 18, 12.

Basic Operations on Whole Numbers

There are four basic operations we can do with whole numbers: Addition, Subtraction, Multiplication, and Division.

1. Addition of Whole Numbers
   When we add two or more whole numbers, we find their total or sum.
   Example: 45 + 23 = 68.
   Addition of whole numbers always gives another whole number.


2. Subtraction of Whole Numbers
   Subtraction means taking away one number from another.
   Example: 70 – 25 = 45.
   If the number being subtracted is bigger, the answer may not be a whole number.


3. Multiplication of Whole Numbers
   Multiplication means repeated addition.
   Example: 4 × 3 means 4 added three times → 4 + 4 + 4 = 12.
   Multiplication of whole numbers also gives a whole number.


4. Division of Whole Numbers
   Division means sharing equally or grouping.
   Example: 20 ÷ 5 = 4.
   Division may sometimes not give a whole number. For example: 7 ÷ 2 = 3½ (which is not a whole number).

Properties of Whole Numbers

1. Closure Property:
   If we add or multiply two whole numbers, the result is also a whole number. Example: 3 + 4 = 7, 3 × 4 = 12.


2. Commutative Property:
   Changing the order of numbers does not change the answer in addition or multiplication. Example: 2 + 3 = 3 + 2, and 4 × 5 = 5 × 4.


3. Associative Property:
   When adding or multiplying three or more whole numbers, the grouping does not change the answer. Example: (2 + 3) + 4 = 2 + (3 + 4).


4. Distributive Property:
   a × (b + c) = (a × b) + (a × c).
   Example: 2 × (3 + 4) = (2 × 3) + (2 × 4) = 14.


5. Identity Property:
   For addition: 0 is the identity element. (a + 0 = a).
   For multiplication: 1 is the identity element. (a × 1 = a).

Uses of Whole Numbers in Everyday Life

We use whole numbers to:

1. Count objects, people, money, and time.


2. Read phone numbers, house numbers, or page numbers.


3. Measure distance or quantity.


4. Record scores, ages, or number of students.


5. Tell dates and years.

Examples

1. Write in words: 6,430 → Six thousand, four hundred and thirty.


2. Arrange in ascending order: 35, 12, 47, 5, 20 → 5, 12, 20, 35, 47.


3. Add: 245 + 37 = 282.


4. Subtract: 520 – 175 = 345.


5. Multiply: 12 × 4 = 48.


6. Divide: 25 ÷ 5 = 5.

Evaluation Questions

1. What are whole numbers?


2. Write the first ten whole numbers.


3. What is the difference between whole numbers and natural numbers?


4. Expand 7,245 according to their place values.


5. Write 50,204 in words.


6. Arrange these numbers in descending order: 18, 42, 9, 35, 27.


7. Solve: 325 + 87 – 62.


8. Multiply 46 by 5.


9. Divide 84 by 7.


10. Mention three uses of whole numbers in daily life.

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Meaning of Standard Form

When numbers are very large or very small, it is not easy to write or read them in their full form. To make them easier to write, we use a short form called the standard form or scientific notation.

The standard form of a number is written as:
A × 10ⁿ,

where:

1. A is a number greater than or equal to 1 but less than 10.


2. n is a whole number (positive or negative power of 10).

This means we write the number as a product of a number between 1 and 10, and a power of 10.

1. It helps us to write very large or very small numbers in a shorter way.


2. It makes calculation easier, especially in science and technology.


3. It helps to compare large quantities quickly.

Example of Very Large Numbers

For example,

1. The population of a country may be 250,000,000.


2. The distance from the Earth to the Sun is about 150,000,000 km.

These numbers are large and not easy to write many times. So, we write them in standard form like this:

250,000,000 = 2.5 × 10⁸

150,000,000 = 1.5 × 10⁸

Put a decimal point at the end of the whole number (if it is not already shown).

Move the decimal point left until there is only one non-zero digit to the left of the point.

Count how many places you moved the decimal — that number is the power n.

If you moved the point left, the power is positive (10ⁿ).

The standard form is A × 10ⁿ, where A is the number you got after moving the point.

Examples (to standard form)

Example 1 — Write 4,500,000 in standard form

Start: 4,500,000. Put the decimal at the end: 4,500,000.

Move the decimal left so there is one digit before it: we want 4.5 then the rest.

1. Move 1 place: 450,000.0


2. Move 2 places: 45,000.00


3. Move 3 places: 4,500.000


4. Move 4 places: 450.0000


5. Move 5 places: 45.00000


6. Move 6 places: 4.500000 ← now one non-zero digit before the point.

We moved the point 6 places, so n = 6.

Standard form: 4.5 × 10⁶.

Check (multiply back):
4.5 × 10⁶ = 4.5 × 1,000,000.
4 × 1,000,000 = 4,000,000 and 0.5 × 1,000,000 = 500,000.
Add: 4,000,000 + 500,000 = 4,500,000 (correct).

Example 2 — Write 75,000 in standard form

Start: 75,000.

Move decimal left to get one digit before the point: 7.5 (moves = 4 places).

1. 1: 7,500.0


2. 2: 750.00


3. 3: 75.000


4. 4: 7.5000

Moved 4 places → n = 4.

Standard form: 7.5 × 10⁴.

Check: 7.5 × 10⁴ = 7.5 × 10,000 = (7 × 10,000) + (0.5 × 10,000) = 70,000 + 5,000 = 75,000.

Example 3 — Write 9,200,000,000 in standard form

Start: 9,200,000,000.

Move decimal left to get 9.2 (count moves).

The number has 10 digits; moving gives 9.200000000 — moved 9 places.

So n = 9.

Standard form: 9.2 × 10⁹.

Check: 9.2 × 10⁹ = 9.2 × 1,000,000,000 = 9,000,000,000 + 200,000,000 = 9,200,000,000.

Example 4 — Write 86,000 in standard form

Start: 86,000.

Move decimal left until one digit left: 8.6 (moves = 4).

1. 1: 8,600.0


2. 2: 860.00


3. 3: 86.000


4. 4: 8.6000

n = 4.

Standard form: 8.6 × 10⁴.

Check: 8.6 × 10⁴ = 8.6 × 10,000 = 80,000 + 6,000 = 86,000.

Changing from standard form back to ordinary number (step-by-step)

Rule: If power is positive, move the decimal point to the right that many places. If power is negative, move the point to the left that many places.

Example 5 — Change 3.5 × 10⁴ to ordinary form

Start: 3.5 × 10⁴. Power = 4 → move decimal 4 places right.

Move point: 3.5 → 35.0 (1 move) → 350.0 (2 moves) → 3500.0 (3) → 35000.0 (4).

Ordinary number: 35,000.

Check: 3.5 × 10⁴ = 3.5 × 10,000 = 30,000 + 5,000 = 35,000.

Example 6 — Change 7.2 × 10³

Power = 3 → move decimal 3 places right: 7.2 → 72 → 720 → 7,200.

Check: 7.2 × 1,000 = 7,200.

Example 7 — Change 2.45 × 10⁵

Power = 5 → move decimal 5 places right:

1. 2.45 → 24.5 (1)


2. 24.5 → 245.0 (2)


3. 245.0 → 2,450.0 (3)


4. 2,450.0 → 24,500.0 (4)


5. 24,500.0 → 245,000.0 (5)

Ordinary number: 245,000.

Check by splitting: 2 × 10⁵ = 200,000; 0.45 × 10⁵ = 45,000; total = 245,000.

Decimal numbers to standard form (negative powers) — short examples

Example 8 — Write 0.005 in standard form

Start: 0.005 (decimal already shown).

Move decimal right until one non-zero digit is left of the point: we want 5.0. How many moves?

1. 0.005 → 0.05 (1 move)


2. 0.05 → 0.5 (2 moves)


3. 0.5 → 5.0 (3 moves)

We moved 3 places to the right, so exponent = -3.

Standard form: 5 × 10^-3.

Check: 5 × 10^-3 = 5 ÷ 1000 = 0.005.

Example 9 — Write 45.3 in standard form

Start: 45.3. Move the decimal left 1 place to get one digit before point: 4.53 (moved 1 place).

Exponent = +1.

Standard form: 4.53 × 10¹.

Check: 4.53 × 10 = 45.3.

Expanded form → Standard form (worked example)

Example 10 — Write 84,512 in standard form using expanded form

Expanded form:

84,512 = (8 × 10⁴) + (4 × 10³) + (5 × 10²) + (1 × 10¹) + (2 × 10⁰).

Factor out 10⁴ (the highest power):

= 10⁴ × (8 + 0.4 + 0.05 + 0.001 + 0.0002)

Add the bracket carefully:

1. 8 + 0.4 = 8.4


2. 8.4 + 0.05 = 8.45


3. 8.45 + 0.001 = 8.451


4. 8.451 + 0.0002 = 8.4512

So number = 10⁴ × 8.4512 = 8.4512 × 10⁴.

Check: 8.4512 × 10⁴ → move decimal right 4 → 84,512.

Class activity

1. 120,000
   Put decimal: 120,000. → want 1.2 → moves = 5 → 1.2 × 10⁵.
   Check: 1.2 × 100,000 = 120,000.


2. 5,000,000
   5,000,000. → want 5.0 → moves = 6 → 5 × 10⁶.
   Check: 5 × 1,000,000 = 5,000,000.


3. 960,000,000
   960,000,000. → want 9.6 → moves = 8 → 9.6 × 10⁸.
   Check: 9 × 10⁸ = 900,000,000 and 0.6 × 10⁸ = 60,000,000; sum = 960,000,000.


4. 35,000
   35,000. → want 3.5 → moves = 4 → 3.5 × 10⁴.
   Check: 3.5 × 10,000 = 35,000.

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Meaning of Decimal Numbers

Decimal numbers are numbers that have a decimal point (.) separating the whole part from the fractional part.

Examples:
3.5, 0.08, 12.45, 0.0004

The numbers to the right of the decimal point are parts of one whole — that is, they represent tenths, hundredths, thousandths, etc.

Decimal	Word Form	Fraction Form
0.1	One-tenth	1/10
0.01	One-hundredth	1/100
0.001	One-thousandth	1/1000

Meaning of Standard Form

A number is written in standard form (or scientific notation) when it is written as:

A × 10ⁿ

where

1. A is a number between 1 and 10, and


2. n is an integer (positive or negative) that shows how many times the decimal point has been moved.

Difference Between Whole Number and Decimal in Standard Form

Type	Decimal Point Moved	Power of 10
Whole Number	Move left	Positive power
Decimal Number (less than 1)	Move right	Negative power

Steps for Writing Decimal Numbers in Standard Form

1. Write down the decimal number.


2. Move the decimal point to the right until only one non-zero digit remains to the left of the point.


3. Count how many places you moved the point.


4. Write the number as A × 10⁻ⁿ, where n is the number of places moved.


5. The power is negative because the decimal is small (less than 1).

Step-by-Step Examples

Example 1 — Write 0.4 in standard form

Start: 0.4
Move the decimal point to make it 4.0 (1 move to the right).
Since the point moved 1 place, n = 1.
Write as 4 × 10⁻¹.
Check: 4 × 10⁻¹ = 4 ÷ 10 = 0.4
Answer: 0.4 = 4 × 10⁻¹

Example 2 — Write 0.05 in standard form

Start: 0.05
Move the decimal until you have one non-zero digit left of the point → 5.0
Count moves: 0.05 → 0.5 (1 move) → 5.0 (2 moves)
n = 2
Standard form: 5 × 10⁻²
Check: 5 ÷ 100 = 0.05
Answer: 0.05 = 5 × 10⁻²

Example 3 — Write 0.0003 in standard form

Start: 0.0003
Move decimal right until you get 3.0
0.0003 → 0.003 (1 move)
0.0003 → 0.03 (2 moves)
0.0003 → 0.3 (3 moves)
0.0003 → 3.0 (4 moves)
n = 4
Standard form: 3 × 10⁻⁴
Check: 3 ÷ 10,000 = 0.0003
Answer: 0.0003 = 3 × 10⁻⁴

Example 4 — Write 0.00008 in standard form

Start: 0.00008
Move decimal right → 8.0 (5 moves)
n = 5
Standard form: 8 × 10⁻⁵
Check: 8 ÷ 100,000 = 0.00008
Answer: 0.00008 = 8 × 10⁻⁵

Example 5 — Write 0.00000027 in standard form

Start: 0.00000027
Move decimal right until 2.7
Moves = 7
n = 7
Standard form: 2.7 × 10⁻⁷
Check: 2.7 ÷ 10,000,000 = 0.00000027
Answer: 0.00000027 = 2.7 × 10⁻⁷

Example 6 — Write 34.5 in standard form

This number is greater than 10, so it will have a positive power.
Start: 34.5
Move decimal left until one digit is before the point: 3.45 (1 move).
n = 1
Standard form: 3.45 × 10¹
Check: 3.45 × 10 = 34.5
Answer: 34.5 = 3.45 × 10¹

Example 7 — Write 0.00892 in standard form

Start: 0.00892
Move decimal to the right to get 8.92
Moves = 3
n = 3
Standard form: 8.92 × 10⁻³
Check: 8.92 ÷ 1,000 = 0.00892
Answer: 0.00892 = 8.92 × 10⁻³

Converting Back to Decimal Number

To change from standard form to decimal form:

1. If the power is positive, move the decimal point to the right.


2. If the power is negative, move the decimal point to the left.

Example 8 — Change 4 × 10⁻³ to ordinary number

Power = -3 → move decimal 3 places left.
4 → 0.004
Answer: 0.004

Example 9 — Change 6.1 × 10⁻²

Power = -2 → move decimal 2 places left.
6.1 → 0.061
Answer: 0.061

Example 10 — Change 7.4 × 10³

Power = 3 → move decimal 3 places right.
7.4 → 74 → 740 → 7,400.
Answer: 7,400

Important Notes

1. For very large numbers, powers of 10 are positive.


2. For very small numbers (less than 1), powers of 10 are negative.


3. The first number (A) must always be between 1 and 10, not less or greater.

Example: 27 × 10⁻³ ❌ → Correct form is 2.7 × 10⁻²

Class Practice

Change the following to standard form:

1. 0.0005


2. 0.0026


3. 0.0000007


4. 45.7


5. 0.0043

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Conclusion

Whole numbers are the basic counting numbers, including zero, without fractions or decimals.

They help us to count, measure, and perform everyday mathematical operations easily.

Understanding whole numbers builds a strong foundation for learning higher topics like fractions, decimals, and algebra.