Writing in Standard Form –

Writing numbers in standard form is a key Number topic at GCSE. You need to convert both very large and very small numbers into the form A x 10^n, where 1 ≤ A < 10 and n is an integer. This appears on Foundation and Higher papers.

What Is Standard Form?

Standard form (also called standard index form or scientific notation) is a way of writing numbers that would otherwise require many digits. It expresses any number as a value between 1 and 10 multiplied by a power of 10.

For large numbers, the power of 10 is positive. For example, 5,800,000 = 5.8 x 10^6. For small numbers (between 0 and 1), the power is negative. For example, 0.00042 = 4.2 x 10^-4.

The key rule is that A must satisfy 1 ≤ A < 10. If your value of A is 10 or more, or less than 1, the number is not in correct standard form.

Key Formulas

Step-by-Step Method

1. Place the decimal point after the first non-zero digit to create a number between 1 and 10 — this is A.
2. Count how many places the decimal point moved from its original position — this gives the value of n.
3. If the original number is 10 or larger, n is positive.
4. If the original number is less than 1, n is negative.
5. Write the answer as A × 10^n.

Worked Example 1 — Foundation Level

Question: Write 72,000 in standard form.

Working:

Step 1 — Place the decimal after the first non-zero digit: 7.2.

Step 2 — The decimal moved 4 places to the left (72000.0 → 7.2000).

Step 3 — The original number is large, so n is positive: n = 4.

Answer: 7.2 × 10⁴

Worked Example 2 — Higher Level

Question: Write 0.0000508 in standard form.

Working:

Step 1 — The first non-zero digit is 5. Place the decimal after it: 5.08.

Step 2 — The decimal moved 5 places to the right (0.0000508 → 5.08).

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

Answer: 5.08 × 10⁻⁵

Worked Example 3 — Exam Style

Question: A virus is 0.000000125 m long. Write this in standard form.

Working:

Step 1 — First non-zero digit is 1. A = 1.25.

Step 2 — Count decimal shifts: the point moves 7 places to the right.

Step 3 — Number is less than 1, so n = −7.

Answer: 1.25 × 10⁻⁷ m

Common Mistakes

• Writing A outside the range 1 to 10. For example, 45 × 10³ is not in standard form because 45 ≥ 10. The correct form is 4.5 × 10⁴.
• Getting the sign of n wrong. Large numbers have positive powers; small numbers (less than 1) have negative powers. Think: "big number = big positive power."
• Miscounting the number of places the decimal point moves. Count carefully, especially with numbers that have many zeros.

Exam Tips

• Always double-check that A is between 1 (inclusive) and 10 (exclusive).
• Convert your answer back to an ordinary number to verify it matches the original.
• On ordering questions, compare the powers of 10 first — a higher power means a larger number for positive values.

Practice Questions

Q1 (Foundation): Write 3,400,000 in standard form.

Q2 (Foundation): Write 0.0072 in standard form.

Q3 (Higher): The distance from the Earth to the Sun is approximately 149,600,000 km. Write this in standard form.

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Related Topics

• Calculating with Standard Form
• Standard Form
• Indices and Index Laws

Summary

• Standard form writes a number as A × 10^n where 1 ≤ A < 10.
• For large numbers (≥ 10), n is positive.
• For small numbers (< 1), n is negative.
• Count how many places the decimal point moves to find n.
• Always check that A is in the correct range.