What Are Number Bases?: WASSCE Mathematics

Every number you write is in a base. When you write $245$ , you mean two hundred and forty-five — but only because you are using base ten, the decimal system. The digit $2$ doesn't mean two; it means two hundreds. The digit $4$ means four tens. The digit $5$ means five units. The position of each digit matters.

That is how all number bases work. The same digits can mean very different values depending on which base you are using. In base ten you have ten digits ( $0$ to $9$ ). In base two you have only two digits ( $0$ and $1$ ). In base eight you have eight digits ( $0$ to $7$ ). Each base counts a different way — but the underlying logic of place value is the same.

This topic covers number bases from start to finish. By the end, you will be able to convert between bases, do arithmetic in any base, and solve the equations WAEC asks every year.

Definition

A number base (or radix) is the number of distinct digits used to represent numbers in a positional system.

Base ten (decimal): digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$
Base two (binary): digits $0, 1$
Base eight (octal): digits $0, 1, 2, 3, 4, 5, 6, 7$
Base sixteen (hexadecimal): digits $0$ to $9$ , then $A, B, C, D, E, F$

The base is written as a subscript: $245_{10}$ means $245$ in base ten. $101_2$ means $101$ in base two.

The decimal system — base 10

You already use base 10 every day. Look at how the number $3{,}471$ breaks down:

3471_{10} = 3 \times 1000 + 4 \times 100 + 7 \times 10 + 1 \times 1

\begin{aligned}&3471_{10} \\ &= 3 \times 1000 + 4 \times 100 + 7 \times 10 + 1 \times 1\end{aligned}

Each position represents a power of 10:

Position	Thousands	Hundreds	Tens	Units
Power	$10^3$	$10^2$	$10^1$	$10^0$
Value	$1000$	$100$	$10$	$1$
Digit	$3$	$4$	$7$	$1$

The rightmost digit is the units place ( $10^0 = 1$ ). Each position to the left is ten times the previous. This is the place value system, and it works the same way in every base — only the multiplier changes.

Other bases — same idea, different multiplier

In any base $b$ , the rightmost position is $b^0 = 1$ . The next position is $b^1$ . Then $b^2$ , $b^3$ , and so on.

Base two (binary)

In base 2, you only have two digits: $0$ and $1$ . The place values are powers of 2.

Position	$2^3$	$2^2$	$2^1$	$2^0$
Value	$8$	$4$	$2$	$1$

So the binary number $1011_2$ means:

1011_2 = 1 \times 8 + 0 \times 4 + 1 \times 2 + 1 \times 1 = 11_{10}

\begin{aligned}&1011_2 \\ &= 1 \times 8 + 0 \times 4 + 1 \times 2 + 1 \times 1 \\ &= 11_{10}\end{aligned}

The binary digits $1011$ represent the decimal number $11$ .

Base eight (octal)

In base 8, you have digits $0$ to $7$ (eight in total). The place values are powers of 8.

Position	$8^3$	$8^2$	$8^1$	$8^0$
Value	$512$	$64$	$8$	$1$

So the octal number $245_8$ means:

245_8 = 2 \times 64 + 4 \times 8 + 5 \times 1 = 128 + 32 + 5 = 165_{10}

\begin{aligned}245_8 &= 2 \times 64 + 4 \times 8 + 5 \times 1 \\ &= 128 + 32 + 5 \\ &= 165_{10}\end{aligned}

Base five

In base 5, you have digits $0, 1, 2, 3, 4$ (five in total). Powers of 5.

Position	$5^3$	$5^2$	$5^1$	$5^0$
Value	$125$	$25$	$5$	$1$

So $342_5 = 3 \times 25 + 4 \times 5 + 2 \times 1 = 75 + 20 + 2 = 97_{10}$ .

The key rule for any base

In base $b$ :

You use $b$ different digits, from $0$ to $b-1$ .
The place values are $b^0, b^1, b^2, b^3, \ldots$ from right to left.
The digit $b$ itself is never used in base $b$ — it gets written as $10_b$ (one of the next place, zero of this place).

For example, in base 5, the digit $5$ does not exist. You write five as $10_5$ .

Why number bases matter

Different bases are useful in different contexts:

Base 10 is what humans use because we have ten fingers.
Base 2 is what computers use, because electronic circuits are either on ( $1$ ) or off ( $0$ ).
Base 8 and base 16 are used in computing because they convert easily to and from base 2.
Other bases (3, 4, 5, 7) are tested in WAEC to make sure you understand place value.

WAEC objectives most commonly use bases 2, 3, 5, 7, and 8. The principles are the same for any base.

Worked example

Write the place-value breakdown for each number.

(a) $342_5$

342_5 = 3 \times 5^2 + 4 \times 5^1 + 2 \times 5^0 = 3 \times 25 + 4 \times 5 + 2 \times 1

\begin{aligned}342_5 &= 3 \times 5^2 + 4 \times 5^1 + 2 \times 5^0 \\ &= 3 \times 25 + 4 \times 5 + 2 \times 1\end{aligned}

(b) $1101_2$

1101_2 = 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 1 \times 8 + 1 \times 4 + 0 \times 2 + 1 \times 1

\begin{aligned}&1101_2 \\ &= 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 \\ &= 1 \times 8 + 1 \times 4 + 0 \times 2 + 1 \times 1\end{aligned}

(c) $463_8$

463_8 = 4 \times 8^2 + 6 \times 8^1 + 3 \times 8^0 = 4 \times 64 + 6 \times 8 + 3 \times 1

\begin{aligned}463_8 &= 4 \times 8^2 + 6 \times 8^1 + 3 \times 8^0 \\ &= 4 \times 64 + 6 \times 8 + 3 \times 1\end{aligned}

(d) $2034_5$

Notice the digit $0$ in the tens place. It still has a place value — it just contributes nothing.

2034_5 = 2 \times 5^3 + 0 \times 5^2 + 3 \times 5^1 + 4 \times 5^0 = 2 \times 125 + 0 + 3 \times 5 + 4 \times 1

\begin{aligned}&2034_5 \\ &= 2 \times 5^3 + 0 \times 5^2 + 3 \times 5^1 + 4 \times 5^0 \\ &= 2 \times 125 + 0 + 3 \times 5 + 4 \times 1\end{aligned}

Spotting invalid numbers

A digit must be smaller than the base. So:

$235_4$ is invalid — the digit $5$ does not exist in base 4 (digits are $0, 1, 2, 3$ only).
$182_8$ is invalid — the digit $8$ does not exist in base 8 (digits are $0$ to $7$ ).
$109_8$ is valid — all digits ( $1, 0, 9$ ) — wait, $9$ is too large for base 8. So actually invalid. The valid digits in base 8 are $0$ to $7$ .
$1011_2$ is valid — all digits are $0$ or $1$ .

This is a common WAEC objective: spot the invalid number.

Worked example

Which of these numbers are invalid? State the reason.

(a) $234_5$ — Valid. All digits are below $5$ .

(b) $342_3$ — Invalid. The digit $4$ does not exist in base 3 (digits are $0, 1, 2$ ).

(c) $1102_2$ — Invalid. The digit $2$ does not exist in base 2 (digits are $0, 1$ ).

(d) $678_8$ — Invalid. The digit $8$ does not exist in base 8 (digits are $0$ to $7$ ).

(e) $4506_7$ — Invalid. The digit $6$ is valid in base 7, but check all digits — $5$ is fine, $0$ is fine, $4$ is fine. Wait — all four digits ( $4, 5, 0, 6$ ) are less than $7$ . Actually valid.

In Sierra Leone

Number bases come up in everyday life more than you think.

When you count using your fingers up to ten and start again, you are working in base 10. If you only had your two hands as the unit and counted pairs of hands — ten, twenty, thirty — you would still be in base 10, just thinking in tens.

A Sierra Leonean trader at Lumley market who keeps track of jollof bowls sold by making a mark for every five — one mark = five bowls — is effectively using base 5. Each cluster of five becomes one new unit at the next level.

A cash counter who deals only in $5{,}000$ leone notes and counts how many bundles of ten notes she has is grouping into tens of fives — a hybrid system. The key insight is the same: humans naturally group things, and the group size is what we call the base.

Common mistake

Watch out for these three errors:

1. Forgetting that the digit equal to the base doesn't exist.

❌ Writing $182_8$ — the digit $8$ is not valid in base 8.
✓ The largest digit in base 8 is $7$ .

2. Confusing place values with digit values.

The digit $4$ in $463_8$ is not $4$ in our decimal sense — it means $4 \times 64 = 256$ at that place.

3. Forgetting the subscript.

$1011$ alone is ambiguous. $1011_{10}$ is one thousand and eleven. $1011_2$ is eleven. Always write the base unless it is clearly base 10.

WAEC tests this

WAEC tests number bases in two main patterns:

Place value identification: "What is the value of the digit $5$ in $4567_8$ ?" Answer: $5 \times 8 = 40$ in base 10 terms (or $50_8$ in base 8 — depending on how the question is phrased).
Validity questions: "Which of the following is not a valid number?" — pick the one with a digit equal to or greater than the base.

Get comfortable with reading the subscript and identifying the digit range for each base. This is the foundation for all the conversion work in Lessons 2 and 3.

Quick self-check

1. Write the place-value expansion of $521_6$ .

2. Which of the following is invalid? (a) $231_4$ (b) $345_5$ (c) $1102_3$ (d) $617_8$

3. What digits are allowed in base 7?

4. In $345_8$ , what does the digit $4$ represent in base 10?

Answers

1. $521_6 = 5 \times 6^2 + 2 \times 6^1 + 1 \times 6^0 = 5 \times 36 + 2 \times 6 + 1 \times 1$ .

2. (b) $345_5$ is invalid — the digit $5$ does not exist in base 5.

3. $0, 1, 2, 3, 4, 5, 6$ (seven digits in total).

4. $4 \times 8 = 32$ in base 10.

What's next

In the next lesson we cover converting from base 10 to other bases — the most common direction WAEC tests. The method is repeated division by the new base, reading the remainders from bottom to top. We will do many worked examples in base 2, base 5, and base 8.

What Are Number Bases?