Standard Form — AQA GCSE Maths Revision Notes

What you'll learn

Standard form — also called standard index form or scientific notation — is a way of writing very large and very small numbers compactly using powers of ten. By the end of this guide you should be able to convert numbers into and out of standard form, decide whether a number has been written in it correctly, and multiply, divide, add and subtract numbers written this way.

You should also be able to compare numbers in standard form, enter them into a calculator correctly, and recognise the two situations in which an answer has to be adjusted before it counts as being in standard form.

The organising idea is that a number in standard form is two separate pieces of information doing two separate jobs. The number in front, always between 1 and 10, carries the digits. The power of ten carries the size. Once you see them as independent, every operation in this topic becomes simple: you handle the digits with ordinary arithmetic and the powers with the index laws, and then check that the front part is still in range.

Key terms and definitions

Standard form — a number written as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer.

A (the mantissa) — the front part. It must have exactly one non-zero digit before the decimal point.

Index (or power) — the value of n. It records how many places the decimal point has moved.

Positive index — used for numbers of 10 or more. 10⁶ means a million.

Negative index — used for numbers between 0 and 1. 10⁻³ means a thousandth.

Ordinary number — a number written out in full, not in standard form.

Core concepts

The two conditions

A number is in standard form only if both conditions hold: the front part is at least 1 and less than 10, and the power of ten is a whole number.

So 4.5 × 10³ qualifies. But 12 × 10⁵ does not, because 12 is not less than 10 — it is a correct value, and it equals 1.2 × 10⁶, but it has not been written in standard form. Nor does 0.7 × 10⁴ qualify, because 0.7 is less than 1; it equals 7 × 10³.

Exam questions test this condition directly, and they also test it indirectly: an answer that comes out as 20 × 10¹⁰ has to be tidied to 2 × 10¹¹ before it earns the mark.

Converting large numbers

Move the decimal point left until exactly one non-zero digit remains in front of it, and count the moves. That count is the index, and it is positive.

For 53 000, the point moves 4 places to reach 5.3, so 53 000 = 5.3 × 10⁴.

A useful sense check: the index tells you how many times the front part has been divided by 10 to get down to size, so a larger number has a larger index. It is not the number of zeros, which only coincides when the front part is a single digit.

Converting small numbers

Move the decimal point right until exactly one non-zero digit sits in front of it, and count the moves. That count is the index, and it is negative.

For 0.00072, the point moves 4 places to reach 7.2, so 0.00072 = 7.2 × 10⁻⁴.

A negative index always means a number between 0 and 1. It never means a negative number: 7.2 × 10⁻⁴ is small but positive. Confusing "small" with "negative" is a common error, and it matters most when comparing numbers.

Converting back to an ordinary number

Reverse the process. A positive index moves the point right, making the number bigger; a negative index moves it left, making it smaller.

So 6.3 × 10⁴ = 63 000, and 5 × 10⁻³ = 0.005.

Multiplying and dividing

This is where the two-pieces idea pays off, because the pieces are handled separately.

To multiply: multiply the front parts, and add the indices. (3 × 10⁵) × (2 × 10³) = 6 × 10⁸.

To divide: divide the front parts, and subtract the indices. (8 × 10⁶) ÷ (4 × 10²) = 2 × 10⁴.

Then check the front part. If multiplying gave 20 × 10¹⁰, the front part is too large, so move one power across: 2 × 10¹¹. If dividing gave 0.5 × 10⁴, the front part is too small: 5 × 10³. Notice which way the index moves in each case — as the front part shrinks by a factor of 10, the index grows by 1, so the value is unchanged.

Adding and subtracting the indices follows the ordinary index laws, including with negatives: (6 × 10⁻⁴) × (3 × 10⁶) has index −4 + 6 = 2, and 18 × 10² tidies to 1.8 × 10³.

Adding and subtracting

The index laws do not apply here, and trying to add the indices is the single biggest error in this topic.

The reliable method is to convert both numbers to ordinary form, add or subtract, then convert back. So (6 × 10⁵) + (7 × 10⁴) = 600 000 + 70 000 = 670 000 = 6.7 × 10⁵.

Where the two numbers already have the same index, they can be added directly: (4 × 10⁵) + (3 × 10⁵) = 7 × 10⁵, because both are counting in the same units. That shortcut is safe; the general case is not.

Comparing numbers

Compare the indices first, since they decide the size. The number with the larger index is larger — provided both are properly in standard form. Only if the indices are equal does the front part decide.

With negative indices, remember that −2 is larger than −5, so 3 × 10⁻² is much larger than 9 × 10⁻⁵. The front part being bigger does not help the second number at all.

Calculators

Use the dedicated standard-form key, usually marked ×10ˣ or EXP. Typing 3 × 10 ^ 5 works too, but typing 3 × 10 EXP 5 enters 3 × 10⁶ by mistake, because the key already supplies the "× 10".

Some calculators display answers in standard form using a small space or an E, so 4.5 × 10⁻⁶ may appear as 4.5⁻⁶. Write the answer out properly in the exam; a calculator display copied literally will not earn the mark.

Worked examples

Example 1: Converting a small number

Write 0.000045 in standard form.

Move the decimal point right until one non-zero digit sits in front of it. Moving it 5 places gives 4.5.

Because the original number is less than 1, the index is negative: 4.5 × 10⁻⁵.

Check by reversing: moving the point 5 places back left from 4.5 restores 0.000045. ✓

Example 2: Multiplying, with an adjustment

Work out (5 × 10⁷) × (4 × 10³), giving the answer in standard form.

Multiply the front parts: 5 × 4 = 20. Add the indices: 7 + 3 = 10. That gives 20 × 10¹⁰.

This is the correct value but it is not in standard form, because 20 is not less than 10. Write 20 as 2 × 10¹ and combine the powers: 2 × 10¹¹.

The final check is always the same: is the front part at least 1 and less than 10?

Example 3: Adding

Work out (6 × 10⁵) + (7 × 10⁴).

The indices differ, so the index laws do not apply. Convert both to ordinary numbers: 600 000 and 70 000.

Add them: 670 000.

Convert back by moving the point 5 places: 6.7 × 10⁵.

Adding the indices would have given 10⁹, an answer roughly ten thousand times too large — a mistake worth being able to spot by size alone.

Common mistakes and how to avoid them

Leaving the front part outside the range. Answers such as 20 × 10¹⁰ or 0.5 × 10⁴ are values, not standard form. Adjust and move one power across.

Adding the indices when adding the numbers. The index laws apply to multiplication and division only. Convert to ordinary numbers to add or subtract.

Reading a negative index as a negative number. 7.2 × 10⁻⁴ is a small positive number.

Counting zeros instead of decimal places. The index counts how far the decimal point moved, which is only the same as the number of zeros in simple cases.

Moving the point the wrong way when converting back. A positive index makes the number bigger; a negative index makes it smaller. Check the answer looks the right size.

Entering "× 10" twice on a calculator. The ×10ˣ or EXP key already includes it.

Exam technique for "Standard Form"

Write the front part and the index as two separate steps when multiplying or dividing. Method marks are available for the correct index even when the front-part arithmetic is wrong.

Finish every calculation by checking that the front part is at least 1 and less than 10. This single check catches the commonest lost mark in the topic.

When a question says "give your answer in standard form", it is telling you the answer will need converting — expect an adjustment.

For comparison questions, line the indices up first and only look at the front parts if the indices match.

Estimate the size of the answer before writing it. An answer that is a thousand times too big usually means indices were added when they should not have been.

Quick revision summary

Standard form is A × 10ⁿ with 1 ≤ A < 10 and n a whole number. Both conditions must hold, so 12 × 10⁵ and 0.7 × 10⁴ are not in standard form.

The index counts how many places the decimal point moved: positive for numbers of 10 or more, negative for numbers between 0 and 1. A negative index never means a negative number.

To multiply, multiply the front parts and add the indices. To divide, divide the front parts and subtract them. Then adjust so the front part is back in range — as it shrinks by a factor of 10, the index grows by 1.

To add or subtract, convert to ordinary numbers first, unless the indices already match. Never add the indices.

Compare by index first; the front part only matters when the indices are equal.

On a calculator, use the ×10ˣ or EXP key, which already supplies the "× 10", and write the answer out in full rather than copying the display.