Integers, place value & order of operations

An integer is a whole number — positive, negative or zero. You must be able to order them and use the four rules (+ − × ÷) with the correct hierarchy of operations.

Work through the operations in that order. Division and multiplication rank equally, as do addition and subtraction — do those left to right.

Common slip: reading left to right gives 3 + 4 = 7, ×2² = 28. That ignores BIDMAS. Indices and × come before the +.

Your turn — order of operations

Prime factors, HCF & LCM

Every integer is a unique product of prime factors. Write it using a factor tree, then use index notation:

To find the HCF (highest common factor) and LCM (lowest common multiple) of two numbers, put their prime factors into a Venn diagram:

Remember which is which: HCF is the small answer (the shared middle only); LCM is the big answer (multiply all the factors in the diagram once).

Your turn — LCM

Working with fractions

Simplify by cancelling common factors, use common denominators to add or subtract, and remember: to divide by a fraction you multiply by its reciprocal (flip it).

Divide ≠ swap both: only the second fraction flips. ¾ ÷ ⅔ becomes ¾ × 3/2, not ⅔ × 4/3.

Your turn — fraction of an amount

Fractions ⇄ decimals ⇄ percentages

These three are the same idea written three ways. You must move between them freely:

Watch out: a terminating decimal (like 0.75) always converts to an exact fraction. Turning a recurring decimal into a fraction is a Higher-tier skill covered later.

Ordering mixed forms

Percentage change

Percentage means "parts per 100". The quickest way to increase or decrease is a multiplier:

Your turn — percentage increase

Reverse percentages

When you're given the final amount and asked for the original, work backwards: divide by the multiplier.

The classic trap: reverse percentage is not "just add the 20% back". Adding 20% to £360 gives £432, which is wrong. You must divide by 0.80 to undo the discount.

Your turn — reverse percentage

Compound interest & depreciation

With compound interest the change is applied repeatedly, each year on the new total. Use the multiplier raised to a power:

Depreciation works the same way but with a decrease multiplier — e.g. a car losing 15% per year uses ×0.85 each year.

Your turn — compound interest

Sharing in a ratio

To divide a quantity in a ratio, add the parts to find the total number of parts, then find one part:

Always check your shares add back to the original total — a quick way to catch arithmetic slips.

Your turn — sharing in a ratio

Direct proportion

Two quantities are in direct proportion when one is a fixed multiple of the other — double one, you double the other. Use the unitary method: find the value of one, then scale up.

Tip: "s varies directly as t" means s = kt for a constant k. Find k from the first pair, then use it.

Your turn — direct proportion

Standard form

Standard form writes very large or very small numbers compactly:

Watch the rule 1 ≤ a < 10: writing 15 × 10⁷ is not standard form because 15 is bigger than 10. Correct it to 1.5 × 10⁸.

Spot the standard form

Ordinary ⇄ standard form

Tap an ordinary number on the left, then its matching standard form on the right.

Rounding: d.p. & significant figures

Round to a given number of decimal places (d.p.) or significant figures (s.f.). The first significant figure is the first non-zero digit.

Misconception: 0.0499 to 2 s.f. is 0.050, not 0.05 "as two digits" or 0.04. Start counting significant figures at the first non-zero digit (the 4), and rounding 0.0499 up carries to 0.050.

Significant figures

Estimating by rounding to 1 s.f.

To estimate a calculation, round each number to 1 significant figure, then work it out. It's a quick sanity-check on a calculator answer.

Your turn — estimate

Upper & lower bounds

A rounded value hides a range. A length given as 12 cm to the nearest cm could really be anywhere from 11.5 up to (but not including) 12.5:

Higher tip: to find the largest possible area of a rectangle, multiply the upper bounds; for the smallest, multiply the lower bounds.

Your turn — lower bound

Surds (Higher tier)

A surd is a root that can't be simplified to a whole number, like √2. Simplify by pulling out square factors, and rationalise a denominator by clearing the root from the bottom.

Higher only: surds, rationalising denominators and fractional/negative indices are on the Higher tier of 4MA1, not Foundation.

Your turn — simplify a surd

Laws of indices (fractional & negative)

The index laws let you simplify and evaluate powers, including negative and fractional ones:

Key facts to memorise: a⁰ = 1 for any non-zero a (not 0!), and a negative power means reciprocal: 2⁻³ = 1/8, it does not make the answer negative.

Your turn — evaluate a power

Rational or irrational?

A rational number can be written as a fraction; an irrational number (like most surds) cannot. Tap a number, then tap the box it belongs in.

The whole of Section 1

You've covered every Number sub-topic in Edexcel 4MA1 — Foundation content plus the Higher-tier surds and indices. Press Finish to see your score.