Mathematics · Lesson 1

Number bases made simple

JAMBWAECNECO

What the syllabus asks

JAMB: Number and numeration: number bases, operations in different bases from 2 to 10, conversion of numbers including decimals

WAEC: Number bases: conversion of numbers from one base to another (bases 2 to 10, including fractional parts in base 2), basic operations (addition, subtraction, multiplication) in number bases, application to computer (binary) arithmetic, finding an unknown base

NECO: Number bases: conversion of numbers from one base to another (bases 2 to 10, including fractional parts in base 2), basic operations (addition, subtraction, multiplication) in number bases, application to computer (binary) arithmetic, finding an unknown base

What you will learn

Explain place value in any base from 2 to 10.
Convert numbers between base ten and other bases, including binary fractions.
Convert between two non-ten bases.
Add, subtract and multiply in a given base.
Solve equations to find an unknown base.
Relate binary numbers to computers.

Key points

In base b, digits run from 0 to b − 1 and place values are 1, b, b², b³, …
To base ten: multiply each digit by its place value and add.
From base ten: divide repeatedly by b and read remainders from the bottom up.
Between two non-ten bases: go through base ten.
Binary fractions: places are ½, ¼, ⅛, …; to convert a decimal fraction, multiply by 2 repeatedly and read the whole parts downward.
Add by carrying groups of b; subtract by borrowing b.
Multiply directly with carries, or via base ten.
For an unknown base, form an equation; reject bases that are negative or not larger than every digit.
1 byte = 8 bits = 256 values; hex digits A–F stand for 10–15.

Start here: Why different bases?

We count in tens because humans have ten fingers. After 9 we run out of single digits, so we write 10: one group of ten and nothing left over. This is base ten (denary).

But there is nothing special about ten. Imagine a market woman who packs oranges in bags of 5. If she has 23 oranges, she has 4 full bags and 3 loose oranges. If she then packs every 5 bags into a carton, she still has 4 bags. Writing her stock as "4 bags and 3 loose" is the number 43 in base five: 43₅ = 4×5 + 3 = 23 in base ten.

In base b, only the digits 0, 1, …, b − 1 are used. Each place is worth b times the place to its right: …, b³, b², b, 1.

So in base 5, only digits 0 to 4 are allowed. A numeral like 253₅ is impossible, because 5 cannot appear in base 5 (five units make one group, which moves to the next place).

Bases in daily life

Time: 60 seconds make a minute, 60 minutes make an hour (base 60 ideas).
Dozens: eggs sold in crates of 30, or by the dozen (12).
Computers: everything is stored in base 2 using switches that are on (1) or off (0).

Worked example 1.1

Which of these numerals is not valid: 1201₃, 456₇, 382₈, 1102₂?

Base 3 allows 0–2 ✓. Base 7 allows 0–6 ✓. Base 8 allows 0–7 ✓.
Base 2 allows only 0 and 1, but 1102₂ contains a 2.

1102₂ is invalid

⚠️ Common mistake: reading 43₅ as "forty-three". It is "four-three base five", worth 23 in base ten.

Try it yourself. What is the largest digit allowed in base 6?

Show solution

A short history

Base ten is not the only system people have used. The ancient Babylonians counted in base 60, which is why we still have 60 minutes in an hour and 360 degrees in a circle. The Maya of Central America used base 20, counting fingers and toes. Several Nigerian counting systems also show base-20 features: traditional Yoruba numerals, for example, build many numbers from groups of twenty (ogún) and use subtraction, so that numbers near a multiple of twenty are named by counting back from it. Seeing these systems helps you understand that a "base" is simply a choice about how big each group is.

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Practice questions

1. Convert 3021₄ to base ten.

Show worked answer

192 + 0 + 8 + 1 = 201.

2. Convert 87 to base 2.

Show worked answer

1010111₂ (64 + 16 + 4 + 2 + 1).

3. Convert 245₆ to base 4.

Show worked answer

245₆ = 72 + 24 + 5 = 101 → 1211₄ (64 + 32 + 4 + 1).

4. Add 2341₅ + 1434₅.

Show worked answer

346 + 244 = 590 = 4330₅.

5. Subtract 1011₂ from 11010₂.

Show worked answer

26 − 11 = 15 = 1111₂.

Key words

Base (radix): The number of different digits used, and the value of each place relative to the next.
Denary / decimal: Base ten.
Binary: Base two, using only 0 and 1.
Octal: Base eight.
Hexadecimal: Base sixteen, using 0–9 and A–F.
Place value: The value of a position in a numeral.
Digit: A single symbol in a numeral.
Bit: A binary digit, 0 or 1.
Carry: A group moved to the next column in addition.
Borrow: A group taken from the next column in subtraction.
Subscript: The small number showing the base, e.g. 101₂.

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