What you'll learn
Standard form (also called standard index form or scientific notation) is a way of writing very large or very small numbers compactly using powers of ten. In this guide you will learn what standard form is, how to convert numbers to and from standard form, how to multiply and divide numbers in standard form, how to add and subtract them, and how to use your calculator correctly. Standard form appears throughout GCSE Maths and science, especially in measurements and rates.
Key terms and definitions
Standard form — a number written as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer.
Index (power) — the number n in 10ⁿ, telling you how many places the digits move.
Ordinary number — a number written normally, not in standard form.
Positive index — used for large numbers (≥ 10).
Negative index — used for small numbers (between 0 and 1).
Core concepts
What standard form is
A number in standard form is written as A × 10ⁿ, where A is at least 1 but less than 10 and n is a whole number (positive or negative). For example, 4 500 = 4.5 × 10³ and 0.0006 = 6 × 10⁻⁴. The key rule is that the first part must have exactly one non-zero digit before the decimal point.
Converting large numbers
For a large number, count how many places the decimal point moves to the left until there is one digit before it; that count is the positive index. For 53 000: move the point 4 places to get 5.3, so 53 000 = 5.3 × 10⁴.
Converting small numbers
For a small number (less than 1), count how many places the decimal point moves to the right; that count is the negative index. For 0.00072: move the point 4 places to get 7.2, so 0.00072 = 7.2 × 10⁻⁴.
Multiplying and dividing
To multiply numbers in standard form, multiply the A values and add the indices. To divide, divide the A values and subtract the indices. Afterwards, adjust the answer so that 1 ≤ A < 10. For example, (3 × 10⁵) × (2 × 10³) = 6 × 10⁸; (8 × 10⁶) ÷ (2 × 10²) = 4 × 10⁴.
Adding and subtracting
To add or subtract, the easiest method is usually to convert both numbers to ordinary form, add or subtract, then convert back to standard form. Alternatively, make the indices the same first. For example, 3 × 10⁴ + 2 × 10³ = 30 000 + 2 000 = 32 000 = 3.2 × 10⁴.
Worked examples
Example 1: To standard form
Write 0.000045 in standard form.
Move the point 5 places right to get 4.5, so 0.000045 = 4.5 × 10⁻⁵.
Example 2: Multiplying
Work out (5 × 10⁷) × (4 × 10³).
Multiply: 5 × 4 = 20; add indices: 10⁷⁺³ = 10¹⁰. So 20 × 10¹⁰ = 2 × 10¹¹ (adjusting A to be < 10).
Example 3: Adding
Work out (6 × 10⁵) + (7 × 10⁴).
Convert: 600 000 + 70 000 = 670 000 = 6.7 × 10⁵.
Common mistakes and how to avoid them
A not between 1 and 10. The first part must have one non-zero digit before the point; adjust and change the index.
Wrong sign on the index. Large numbers use positive indices; small numbers (less than 1) use negative indices.
Adding indices when adding numbers. You only add/subtract indices when multiplying/dividing, not when adding numbers.
Forgetting to adjust after multiplying. If A comes out ≥ 10 (or < 1), rewrite and change the index.
Calculator slips. Use the ×10ⁿ or EXP/EE button, not "× 10 ^", to avoid errors.
Exam technique for Standard Form
Check the form — exactly one non-zero digit before the decimal point.
Count decimal places carefully to get the index and its sign.
Add indices to multiply, subtract to divide, then adjust A.
Convert to ordinary numbers for addition and subtraction if unsure.
Use the calculator's standard form button and write the final answer in proper standard form.
Quick revision summary
Standard form writes a number as A × 10ⁿ with 1 ≤ A < 10 and n an integer — compact for very large or very small numbers. Large numbers use a positive index (53 000 = 5.3 × 10⁴); small numbers below 1 use a negative index (0.00072 = 7.2 × 10⁻⁴), found by counting how many places the decimal point moves. To multiply, multiply the A values and add the indices; to divide, divide the A values and subtract the indices, then adjust so A stays between 1 and 10. To add or subtract, convert to ordinary numbers (or match the indices), combine, then convert back. Always check the first part has exactly one non-zero digit before the decimal point, get the sign of the index right, and use your calculator's dedicated standard-form button. With these rules, standard form makes awkward numbers easy to handle in both maths and science.