Standard form

Standard form is a Paper 1 and Paper 2 topic on OCR J560. It appears in science contexts (very large/small quantities). Master the conversion rules and the four operations.

What is standard form?

A number written as A × 10ⁿ where:

• A is a number such that 1 ≤ A < 10.
• n is an integer (positive for large numbers, negative for small numbers).

Converting to standard form

Large numbers (positive power): move the decimal point left until A is between 1 and 10.

Example: 6,400,000 → 6.4 × 10⁶ (moved 6 places left).

Small numbers (negative power): move the decimal point right.

Example: 0.00047 → 4.7 × 10⁻⁴ (moved 4 places right).

Operations in standard form

Multiplication

(A × 10ⁿ) × (B × 10ᵐ) = (A × B) × 10^(n+m)

Example: (3.2 × 10⁴) × (2.0 × 10³) = 6.4 × 10⁷.

If A × B ≥ 10 or < 1: adjust. Example: (6 × 10⁵) × (5 × 10³) = 30 × 10⁸ = 3.0 × 10⁹.

Division

(A × 10ⁿ) ÷ (B × 10ᵐ) = (A ÷ B) × 10^(n−m)

Example: (8.4 × 10⁷) ÷ (2 × 10³) = 4.2 × 10⁴.

Addition and subtraction

Convert to the same power of 10 first (or convert both to ordinary numbers).

Example: (3 × 10⁵) + (4 × 10⁴) = 30 × 10⁴ + 4 × 10⁴ = 34 × 10⁴ = 3.4 × 10⁵.

Ordering in standard form

Compare powers first; if powers are equal, compare A values.

Example: order 3.6 × 10⁸, 4.2 × 10⁷, 5.1 × 10⁸. Answer: 4.2 × 10⁷ < 3.6 × 10⁸ < 5.1 × 10⁸.

Common OCR exam mistakes

1. A not being between 1 and 10: writing 34 × 10³ — should be 3.4 × 10⁴.
2. Sign error with negative powers: 0.003 = 3 × 10⁻³ (not 3 × 10³).
3. Adding indices when multiplying — but forgetting to check if the result's A is still valid.