Number bases made easy: converting, adding and subtracting in any base

Number bases come up in almost every JAMB, WAEC and NECO Mathematics paper, usually as a quick conversion or a simple calculation. They are some of the most reliable marks on the paper, because once you understand the method, the questions are entirely mechanical.

What a number base is

We normally count in base ten (the denary system), using ten digits: 0 to 9. Each position in a number is worth ten times the position to its right:

4,375 = (4 × 1000) + (3 × 100) + (7 × 10) + (5 × 1) = (4 × 10³) + (3 × 10²) + (7 × 10¹) + (5 × 10⁰)

In base two (binary), each position is worth two times the position to its right, and only the digits 0 and 1 are used. In base five, positions are worth powers of five, and the digits are 0 to 4.

Rule: in base b, the digits run from 0 to b − 1. So base 8 uses 0–7, and a digit 8 or 9 can never appear in a base 8 number. Examiners sometimes test this directly.

We write the base as a subscript: 1011₂ means "1011 in base two". Some papers write 1011two instead.

Converting from another base to base ten

Multiply each digit by its place value (a power of the base), then add.

Worked example 1

Convert 1011₂ to base ten.

8 + 0 + 2 + 1 = 11

Worked example 2

Convert 243₅ to base ten.

(2 × 5²) + (4 × 5¹) + (3 × 5⁰) = (2 × 25) + (4 × 5) + (3 × 1) = 50 + 20 + 3 = 73

Worked example 3

Convert 1201₃ to base ten.

(1 × 27) + (2 × 9) + (0 × 3) + (1 × 1) = 27 + 18 + 0 + 1 = 46

Converting from base ten to another base

Divide repeatedly by the new base, writing down the remainders. Then read the remainders from bottom to top.

Worked example 4

Convert 45 to base two.

Reading remainders from bottom to top: 101101₂

Check: 32 + 0 + 8 + 4 + 0 + 1 = 45 ✓

Worked example 5

Convert 200 to base five.

Bottom to top: 1300₅

Check: (1 × 125) + (3 × 25) + 0 + 0 = 125 + 75 = 200 ✓

Converting between two non-ten bases

Go through base ten: convert to base ten first, then to the new base.

Worked example 6

Convert 213₄ to base six.

Step 1: to base ten. (2 × 16) + (1 × 4) + (3 × 1) = 32 + 4 + 3 = 39.

Step 2: to base six.

Bottom to top: 103₆

Adding in a base

Add column by column from the right, as usual. But whenever a column total reaches the base, carry to the next column.

Worked example 7

Add 101₂ and 111₂.

    1 0 1
  + 1 1 1
  -------
  1 1 0 0

• Right column: 1 + 1 = 2 = 10₂ → write 0, carry 1.
• Middle column: 0 + 1 + 1 (carry) = 2 = 10₂ → write 0, carry 1.
• Left column: 1 + 1 + 1 (carry) = 3 = 11₂ → write 11.

Answer: 1100₂. Check: 5 + 7 = 12, and 1100₂ = 8 + 4 = 12 ✓

Worked example 8

Add 342₅ and 213₅.

• Right: 2 + 3 = 5 = 10₅ → write 0, carry 1.
• Middle: 4 + 1 + 1 = 6 = 11₅ → write 1, carry 1.
• Left: 3 + 2 + 1 = 6 = 11₅ → write 11.

Answer: 1110₅. Check: 342₅ = 97 and 213₅ = 58; 97 + 58 = 155. 1110₅ = 125 + 25 + 5 = 155 ✓

Subtracting in a base

When you need to "borrow", you borrow one unit of the base, not ten.

Worked example 9

Subtract 165₈ from 432₈.

• Right column: 2 − 5 can't be done. Borrow 1 from the next column, which is worth 8: 2 + 8 = 10; 10 − 5 = 5.
• Middle column: 3 became 2 after lending. 2 − 6 can't be done. Borrow 8: 2 + 8 = 10; 10 − 6 = 4.
• Left column: 4 became 3. 3 − 1 = 2.

Answer: 245₈. Check: 432₈ = 282 and 165₈ = 117; 282 − 117 = 165, and 245₈ = 128 + 32 + 5 = 165 ✓

Multiplying in a base

The safest method is to convert both numbers to base ten, multiply, then convert back.

Multiply 12₃ by 2₃. 12₃ = 5 and 2₃ = 2. 5 × 2 = 10. 10 in base three: 10 ÷ 3 = 3 r 1; 3 ÷ 3 = 1 r 0; 1 ÷ 3 = 0 r 1 → 101₃.

Finding an unknown base

Some questions give a conversion and ask for the base.

Worked example 10

If 23ₓ = 17₁₀, find x.

Expand: 2x + 3 = 17 → 2x = 14 → x = 7.

Check that the digits are valid: base 7 allows digits 0–6, and 2 and 3 are fine ✓

Common mistakes

• Reading remainders top to bottom instead of bottom to top.
• Carrying or borrowing 10 instead of the base.
• Using an invalid digit, such as a 5 in a base 5 number.
• Forgetting zero place values when expanding, as with the 0 in 1011₂.
• Not checking. Converting your answer back to base ten takes seconds and catches most errors.

Practise until it's automatic

Number bases are pure method, so practise until every step is automatic. Levia's Mathematics levels include number base conversions and calculations from Foundations upwards, each with a full worked explanation.