Conversion of Number Bases

In WAEC maths, you’ll often come across numbers written in different number bases. These are systems like base 2 (binary), base 10 (decimal), base 8 (octal), or base 16 (hexadecimal). Being able to convert between bases is super useful, especially in computing and digital electronics.

 

What Are Number Bases?

A number base tells you how many digits are used in a counting system. For example:

• Base 2 uses only 0 and 1
• Base 10 (what we use every day) uses 0 to 9
• Base 8 uses 0 to 7
• Base 16 uses 0 to 9 and A to F (A = 10, B = 11, ..., F = 15)

 

How to Convert Between Bases

1. Converting from Another Base to Base 10

Use place values to expand the number and convert to base 10.

Example: Convert $1011_2$ to base 10.

Each digit is multiplied by a power of 2 from right to left:

$$1011_2 = 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 8 + 0 + 2 + 1 = 11_{10}$$

 

2. Converting from Base 10 to Another Base

Use successive division by the new base and record the remainders.

Example: Convert $25_{10}$ to base 2.

1. 25 ÷ 2 = 12 remainder 1
2. 12 ÷ 2 = 6 remainder 0
3. 6 ÷ 2 = 3 remainder 0
4. 3 ÷ 2 = 1 remainder 1
5. 1 ÷ 2 = 0 remainder 1

Write the remainders from bottom to top: $11001_2$

$$25_{10} = 11001_2$$

 

Worked Example

 

Practice Question

 

Worked Example (Base 10 to Another Base)

 

Extra Tip: Converting Between Non-Decimal Bases

Sometimes you’ll need to go from one non-decimal base to another. In that case:

1. First convert to base 10
2. Then convert from base 10 to the new base