Meaning of Whole Numbers

Whole numbers are the set of numbers that start from zero and continue without end in the positive direction.

They are written as:

{0,1,2,3,4,5,6,7,…}

They do not include:

1. negative numbers (like –1, –2)


2. fractions (like 1/2)


3. decimals (like 3.75)

We sometimes write the set of whole numbers as W.

History and Development of Numbers

Ancient counting: Early humans counted using stones, sticks or marks on bones.

Tally marks: groups of five strokes.

1. Tally marks (e.g. |||||)


2. Roman numerals: I, V, X, L, C, D, M


3. Hindu–Arabic numerals: digits 0–9 and place value (system used today)

Place Value and Face Value

Face value of a digit is the digit itself.
Place value is the value of the digit according to the position it occupies in the number.

Positions from right to left: ones (units) → tens → hundreds → thousands → ten thousands → hundred thousands → millions → billions → trillions.

Example: in 7 534 218

1. 8 has face value 8 and place value 8 (ones)


2. 1 has face value 1 and place value 10 (tens)


3. 2 has face value 2 and place value 200 (hundreds)


4. 4 has face value 4 and place value 4 000 (thousands)

To read large numbers we group digits in threes from the right.

Reading and Writing Whole Numbers

(a) Writing in words

Break the number into groups of three digits and attach the class name:

1. 2 345 = Two thousand, three hundred and forty-five.


2. 10 203 001 = Ten million, two hundred and three thousand and one.

(b) Writing in figures

Reverse the process: listen for “million”, “thousand”, etc., write the correct number of zeros or placeholders.

Be careful to put zeros where a group is empty:

1. “One million and five” = 1 000 005.

Comparing and Ordering

To compare two whole numbers:

1. Check the number of digits – the one with more digits is larger.


2. If they have the same number of digits, compare from the left-most digit.

Ascending order: smallest to largest.

Descending order: largest to smallest.

Example: Arrange 32 415; 29 480; 33 002 in ascending order:

1. 29 480


2. 32 415


3. 33 002

Basic Operations on Whole Numbers

Addition

1. Line up digits according to place value.


2. Add column by column starting from the right.


3. Carry over if the sum is greater than 9.

Subtraction

1. Line up digits.


2. If a column is smaller than the one you are subtracting, borrow from the next left column.

Multiplication

1. Use times tables.


2. Multiply each digit of the lower number by each digit of the upper, carrying where necessary.

Division

1. Use long division.


2. Dividend ÷ Divisor = Quotient (and sometimes a remainder).

Order of Operations

When an expression has more than one operation use BODMAS:

1. Brackets


2. Order (powers)


3. Division


4. Multiplication


5. Addition


6. Subtraction

Factors and Multiples

A factor of a number divides it exactly with no remainder.

1. Example: Factors of 12 = 1, 2, 3, 4, 6, 12.

A multiple of a number is obtained by multiplying it by any whole number.

1. Example: Multiples of 5 = 5, 10, 15, 20, …

Highest Common Factor (HCF)

The greatest whole number that divides two or more numbers exactly.

1. Method: list factors or use prime factorisation.

Lowest (Least) Common Multiple (LCM)

The smallest number that is a multiple of two or more given numbers.

1. Method: list multiples or use prime factorisation.

Powers (Indices)

The power of a whole number is repeated multiplication.

1. Square of 6 is 6² = 36.


2. Cube of 4 is 4³ = 64.

Rules:

1. a^m × aⁿ = a^m+n


2. a^m ÷ aⁿ = a^m−n

Quantitative Reasoning with Whole Numbers

Word problems use whole numbers in:

1. counting money,


2. population,


3. distances,


4. school scores, etc.

Steps:

1. Understand the problem and write the important numbers.


2. Choose the correct operation(s).


3. Calculate carefully.


4. Check your answer makes sense.

Key Points to Remember

1. Whole numbers start from 0 and continue without end.


2. Understanding place value is essential for every operation.


3. Group digits in threes for easy reading.


4. BODMAS guides the order in which operations are carried out.


5. Factors, multiples, HCF and LCM are tools for simplifying and comparing numbers.

Workings

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(i) Counting — step-by-step

Units (add 1 each time)

Start = 1

1. 1
2. 1 + 1 = 2


3. 2 + 1 = 3


4. 3 + 1 = 4


5. 4 + 1 = 5


6. 5 + 1 = 6

(Each step: previous number + 1 → next number.)

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Tens (add 10 each time)

Start = 10

1. 10
2. 10 + 10 = 20


3. 20 + 10 = 30


4. 30 + 10 = 40


5. 40 + 10 = 50


6. 50 + 10 = 60

(Each step: previous number + 10 → next number.)

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Hundreds (add 100 each time)

Start = 100

1. 100
2. 100 + 100 = 200


3. 200 + 100 = 300


4. 300 + 100 = 400


5. 400 + 100 = 500


6. 500 + 100 = 600

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Thousands (add 1 000 each time)

Start = 1 000

1. 1 000
2. 1 000 + 1 000 = 2 000


3. 2 000 + 1 000 = 3 000


4. 3 000 + 1 000 = 4 000


5. 4 000 + 1 000 = 5 000


6. 5 000 + 1 000 = 6 000

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Ten-thousands (add 10 000 each time)

Start = 10 000

1. 10 000
2. 10 000 + 10 000 = 20 000


3. 20 000 + 10 000 = 30 000


4. 30 000 + 10 000 = 40 000


5. 40 000 + 10 000 = 50 000


6. 50 000 + 10 000 = 60 000

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Hundred-thousands (add 100 000 each time)

Start = 100 000

1. 100 000
2. 100 000 + 100 000 = 200 000


3. 200 000 + 100 000 = 300 000


4. 300 000 + 100 000 = 400 000


5. 400 000 + 100 000 = 500 000


6. 500 000 + 100 000 = 600 000

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Millions (add 1 000 000 each time)

Start = 1 000 000

1. 1 000 000
2. 1 000 000 + 1 000 000 = 2 000 000


3. 2 000 000 + 1 000 000 = 3 000 000


4. 3 000 000 + 1 000 000 = 4 000 000


5. 4 000 000 + 1 000 000 = 5 000 000


6. 5 000 000 + 1 000 000 = 6 000 000

(You can also view these as: 1 million = 1 000 thousands; add one million = add 1 000 thousands.)

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Billions (add 1 000 000 000 each time)

Start = 1 000 000 000

1. 1 000 000 000
2. 1 000 000 000 + 1 000 000 000 = 2 000 000 000


3. 2 000 000 000 + 1 000 000 000 = 3 000 000 000


4. 3 000 000 000 + 1 000 000 000 = 4 000 000 000


5. 4 000 000 000 + 1 000 000 000 = 5 000 000 000


6. 5 000 000 000 + 1 000 000 000 = 6 000 000 000

(Or: 1 billion = 1 000 million; so add 1 billion = add 1 000 million.)

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Trillions (add 1 000 000 000 000 each time)

Start = 1 000 000 000 000

1. 1 000 000 000 000
2. 1 000 000 000 000 + 1 000 000 000 000 = 2 000 000 000 000


3. 2 000 000 000 000 + 1 000 000 000 000 = 3 000 000 000 000


4. 3 000 000 000 000 + 1 000 000 000 000 = 4 000 000 000 000


5. 4 000 000 000 000 + 1 000 000 000 000 = 5 000 000 000 000


6. 5 000 000 000 000 + 1 000 000 000 000 = 6 000 000 000 000

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(ii) Quantitative Reasoning — ( examples)

Example 1 — Addition (units/tens)

Problem: A class has 28 boys and 32 girls. How many pupils in total?

Work:

1. Break into tens and units: 28 = 20 + 8 ; 32 = 30 + 2.
2. Add tens: 20 + 30 = 50.
3. Add units: 8 + 2 = 10.
4. Combine: 50 + 10 = 60.

Answer: 60 pupils

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Example 2 — Subtraction with borrowing (hundreds)

Problem: A shop had 850 pencils. It sold 475. How many left?

Work (column method):

          850
        - 475
         -----
        

1. Units: 0 − 5 → cannot, borrow 1 ten from tens column (tens becomes 4), units become 10.


2. Units: 10 − 5 = 5.


3. Tens: now 4 − 7 → cannot, borrow 1 hundred from hundreds column (hundreds becomes 7), tens become 14.


4. Tens: 14 − 7 = 7.


5. Hundreds: 7 − 4 = 3.

Result = 375.

Answer: 375 pencils

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Example 3 — Multiplication (thousands + digit multiplication)

Problem: A factory makes 1 250 nails per day. In 7 days it makes how many?

Work (expand then multiply):

1. 1250 × 7 = (1000 + 200 + 50 + 0) × 7


2. = (1000×7) + (200×7) + (50×7) + (0×7)


3. = 7000 + 1400 + 350 + 0


4. Now add: 7000 + 1400 = 8400 ; 8400 + 350 = 8750.

Answer: 8 750 nails

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Example 4 — Division with large numbers (millions)

Problem: A country has 9 000 000 people and ₦27 000 000 to share equally. How much per person?

Work (simplify by dividing by 1 000 000):

1. 27 000 000 ÷ 9 000 000 = (27 ÷ 9) after canceling 1 000 000 from both.


2. 27 ÷ 9 = 3.

Answer: ₦3 per person

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Example 5 — Subtraction with billions (aligning units)

Problem: World population 8 000 000 000; a country has 220 000 000. What is the difference?

Work (convert to same unit — millions):

1. 8 000 000 000 = 8 000 million.


2. 220 000 000 = 220 million.


3. Subtract in millions: 8 000 − 220 = 7 780 million.


4. Convert back: 7 780 million = 7 780 000 000.

Answer: 7 780 000 000

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Example 6 — Subtraction with trillions (decimal/trillion form)

Problem: Budget = 4 trillion naira. Spent 2.35 trillion. How much remains?

Work:

1. Method A (decimal): 4.00 − 2.35 = 1.65 → 1.65 trillion.


2. Method B (full figures):

         4 000 000 000 000
        -2 350 000 000 000
        -------------------
        

3. (4 000 000 000 000 − 2 000 000 000 000) = 2 000 000 000 000


4. (2 000 000 000 000 − 350 000 000 000) = 1 650 000 000 000


5. Result = 1 650 000 000 000 = 1.65 trillion naira.

Answer: 1.65 trillion naira