Standard Form –

Standard form — also called standard index form or scientific notation — is a compact way of writing very large or very small numbers. It appears on both Foundation and Higher GCSE Maths papers, typically in questions about space, biology, or physics contexts. You need to convert numbers into and out of standard form, and perform calculations (addition, subtraction, multiplication, division) with them. This guide takes you through each skill methodically, with worked examples at both tiers and targeted exam advice. For a broader revision overview, visit our complete GCSE Maths topics list.

What Is Standard Form?

Standard form expresses a number as a value between 1 and 10 (inclusive of 1 but not 10) multiplied by a power of 10.

• For large numbers, n is positive. For example, 4,500,000 = 4.5 × 10⁶.
• For small numbers, n is negative. For example, 0.00032 = 3.2 × 10⁻⁴.

The value of n tells you how many places the decimal point has moved from the original number to reach a.

Why Use Standard Form?

Writing 602,200,000,000,000,000,000,000 every time you mention Avogadro's number is impractical. Standard form gives us 6.022 × 10²³ — far easier to read, compare, and calculate with.

Step-by-Step Method

Converting an Ordinary Number to Standard Form

1. Place the decimal point after the first non-zero digit to get a number between 1 and 10.
2. Count how many places you moved the decimal point — this is the value of n.
3. If the original number is 1 or greater, n is positive. If the original number is between 0 and 1, n is negative.
4. Write the result as a × 10^n.

Converting Standard Form to an Ordinary Number

1. Look at the power of 10.
2. If n is positive, move the decimal point n places to the right, adding zeros as needed.
3. If n is negative, move the decimal point |n| places to the left, adding zeros as needed.

Multiplying in Standard Form

1. Multiply the a values together.
2. Add the powers of 10.
3. If the resulting a value is not between 1 and 10, adjust it and change the power accordingly.

Dividing in Standard Form

1. Divide the a values.
2. Subtract the powers of 10 (first power minus second power).
3. Adjust if necessary so that a is between 1 and 10.

Adding and Subtracting in Standard Form

1. Convert both numbers to ordinary numbers (or to the same power of 10).
2. Add or subtract.
3. Convert the result back to standard form.

Worked Example 1 — Foundation Level

Question: Write 0.000047 in standard form.

Working:

Step 1 — The first non-zero digit is 4. Place the decimal point after it: 4.7.

Step 2 — Count how many places the decimal point moved from its original position (after the leading 0 in 0.000047) to after the 4. It moved 5 places to the right.

Step 3 — The original number is less than 1, so n is negative: n = −5.

Answer: 4.7 × 10⁻⁵

Worked Example 2 — Higher Level

Question: Calculate (3.2 × 10⁴) × (4.5 × 10⁷). Give your answer in standard form.

Working:

Step 1 — Multiply the a values: 3.2 × 4.5 = 14.4.

Step 2 — Add the powers: 4 + 7 = 11. So the result so far is 14.4 × 10¹¹.

Step 3 — 14.4 is not between 1 and 10. Adjust: 14.4 = 1.44 × 10¹. So 14.4 × 10¹¹ = 1.44 × 10¹².

Answer: 1.44 × 10¹²

Common Mistakes

• Writing a value outside the range 1 ≤ a < 10. For example, writing 45 × 10³ instead of 4.5 × 10⁴. Examiners will not award the mark if a is not between 1 and 10.
• Getting the sign of n wrong. Large numbers have positive powers; small numbers (less than 1) have negative powers. Double-check by asking: "Is this number big or tiny?"
• Forgetting to adjust after multiplying or dividing. If multiplying the a values gives 24.8, you must write it as 2.48 × 10¹ and add 1 to the power.
• Adding the powers when you should be adding the ordinary numbers. You cannot add standard form numbers by simply adding powers — you must convert to the same power of 10 or to ordinary numbers first.

Exam Tips

• On non-calculator papers, break the multiplication into manageable parts. For example, 3.2 × 4.5 = 3 × 4.5 + 0.2 × 4.5 = 13.5 + 0.9 = 14.4.
• Use standard form with other topics. You might see it in compound interest, area and volume, or probability questions. Recognise when an answer needs converting.
• When comparing numbers in standard form, look at the power first. A larger power of 10 always means a larger number (for positive numbers). Only compare the a values when the powers are equal.
• Remember the key formulas for multiplying and dividing — these are listed in our formulas guide.

Practice Questions

Q1 (Foundation): Write 3,600,000 in standard form.

Q2 (Foundation): Write 8.1 × 10⁻³ as an ordinary number.

Q3 (Higher): Work out (6.4 × 10⁸) ÷ (1.6 × 10³). Give your answer in standard form.

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Related Topics

• Powers and Roots
• Indices and Index Laws
• Decimals
• Bounds and Error Intervals

Summary

• Standard form writes numbers as a × 10^n where 1 ≤ a < 10 and n is an integer.
• Positive n means a large number; negative n means a small number less than 1.
• To multiply, multiply the a values and add the powers.
• To divide, divide the a values and subtract the powers.
• To add or subtract, convert to ordinary numbers (or the same power of 10) first.
• Always adjust your final answer so a is between 1 and 10.
• Check your answer makes sense — large number means positive power, small number means negative.