Number Bases: Conversion and Arithmetic
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Lesson overview
Converting between base ten and other bases (including fractional parts), arithmetic in a given base and finding an unknown base.
Go to lesson quiz →Exam guidance · JAMB UTME Speed Lens
Core idea (Acids, Bases, pH & Neutralisation): Track proton transfer (H⁺ vs OH⁻), distinguish strength (degree of ionisation) from concentration (mol/dm³), and match indicator to titration curve.
Pacing
~40 seconds per MCQ (180 questions in 120 minutes, no negative marking).
How JAMB marks this
Fast dimensional/sign elimination and standard formula shortcuts — skip multi-line algebra when substitution tests options faster.
Watch out for
UTME distractors routinely include the intermediate value (e.g., forgetting to convert cm³ to dm³ or °C to Kelvin).
Lesson content
Overview
We count in tens because we have ten fingers, but nothing in mathematics forces that choice. Computers count in twos (binary), old clocks and angles carry the trace of base sixty, and a market woman who packs oranges in heaps of five is, without knowing it, using base five. JAMB sets one or two number-base questions every year. They are some of the cheapest marks on the paper because the methods never change: expand to convert into base ten, divide repeatedly to convert out of base ten, and carry or borrow in the base you are working in. In this lesson you will convert whole numbers and fractions between bases, add, subtract and multiply in a given base, and solve equations in which the base itself is the unknown.
Key concept: base, digits and place value
The base (or radix) of a number system is the number of different digits it uses. Base b uses the digits 0, 1, 2, …, b − 1. Base two uses only 0 and 1; base five uses 0 to 4; base eight uses 0 to 7. Each position in a numeral is worth a power of the base. Reading from the right, the places are worth b⁰ = 1, b¹, b², b³, and so on. To the right of the point the places are worth b⁻¹, b⁻², b⁻³, …. The base is written as a subscript: 324₅ means 'three-two-four in base five'. A number with no subscript is in base ten. A digit can never equal or exceed its base. The numeral 352₄ is impossible, because 5 is not a base-four digit. This one fact settles many unknown-base questions.
Methods: converting and calculating
Into base ten: multiply each digit by its place value and add. For example, 213₄ = 2 × 4² + 1 × 4 + 3 × 1 = 32 + 4 + 3 = 39. Out of base ten: divide by the new base again and again, writing down each remainder, until the quotient is 0. The remainders read from the bottom up give the answer. Fractional parts: digits after the point are worth negative powers. 10.11₂ = 2 + 0 + ½ + ¼ = 2.75. To turn a base-ten fraction into binary, multiply the fraction by 2 repeatedly and record the whole-number parts in order: 0.375 × 2 = 0.75 (0), 0.75 × 2 = 1.5 (1), 0.5 × 2 = 1.0 (1), so 0.375 = 0.011₂. Addition: add column by column, but carry every time a column total reaches the base, not ten. In base six, 345₆ + 234₆: 5 + 4 = 9 = 1 × 6 + 3, so write 3 and carry 1; 4 + 3 + 1 = 8 = 1 × 6 + 2, write 2 and carry 1; 3 + 2 + 1 = 6 = 10₆, write 0 and carry 1. The answer is 1023₆. Subtraction: when you borrow, you borrow one lot of the base. In base two, borrowing gives 10₂ = 2 in the column. Multiplication and division: the safest method under exam pressure is to convert both numbers to base ten, calculate, and convert the answer back. Unknown base: expand both sides in terms of the base, form an equation and solve it. Reject any solution that is negative or that is not larger than every digit used.
Formulae to know
Whole number: (dₙ…d₂d₁d₀)ᵦ = dₙbⁿ + … + d₂b² + d₁b + d₀ Fraction: (0.d₁d₂d₃)ᵦ = d₁b⁻¹ + d₂b⁻² + d₃b⁻³ Digits allowed in base b: 0, 1, …, b − 1 Largest n-digit number in base b: bⁿ − 1 (for example 111₂ = 2³ − 1 = 7)
Worked example 1: binary to base ten
Convert 101101₂ to a number in base ten.
Active Worked Example · Predict Step 1
4 stepsWrite down the formula, rule, or first move you would make before revealing Step 1.
Worked example 2: base ten to base five
Express 89 in base five.
Active Worked Example · Predict Step 1
5 stepsWrite down the formula, rule, or first move you would make before revealing Step 1.
Worked example 3 (UTME standard): finding the base
If 132ₓ = 42, find the base x.
Active Worked Example · Predict Step 1
5 stepsWrite down the formula, rule, or first move you would make before revealing Step 1.
Common traps
• Reading the remainders from the top down. The first remainder is the units digit, so read from the bottom up. • Carrying at ten instead of at the base. In base five, 3 + 4 = 7 = 12₅, so you write 2 and carry 1. • Accepting an impossible base. If the numeral contains the digit 6, the base must be at least 7, whatever your equation says. • Calling the units place 'b¹'. The units place is b⁰ = 1; the next place is b¹. • Multiplying in a strange base digit by digit. Convert to base ten, multiply, and convert back; it is slower to read but much safer.
Summary
• Base b uses the digits 0 to b − 1, and the places are worth powers of b: 1, b, b², … (and b⁻¹, b⁻², … after the point). • To base ten: multiply each digit by its place value and add. • From base ten: divide repeatedly by the base and read the remainders upwards; for fractions, multiply repeatedly and read the whole-number parts downwards. • In arithmetic, carry or borrow one lot of the base, or work in base ten and convert back. • For an unknown base, form an equation in the base and keep only a whole-number solution larger than every digit.
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