Scientific notation and large/small numbers

2026-10-18Numbers & Estimation Mr Chan

Convert between ordinary form and a × 10ⁿ with 1 ≤ |a| < 10 — standard S3 index work.

Key formulas

a \times 10^n with 1 \le |a| < 10

3.6 \times 10^{5} = 360000

4.2 × 10^{−3} = 0.0042

Standard form

Scientific notation writes a number as a × 10ⁿ where the absolute value of a satisfies 1 ≤ |a| < 10 and n is an integer. Move the decimal point until that condition holds; n counts how many places you moved.

Large and small numbers

Numbers greater than or equal to 10 use a positive index. Numbers between 0 and 1 use a negative index. Keep a to a sensible number of significant figures as the question requires.

Operating in scientific notation

Multiply or divide the a-parts, then add or subtract the indices of 10. Renormalise if the new coefficient falls outside [1, 10).

Exam habit

Underline the final decimal point position. Reject answers like 36 × 10⁴ — rewrite as 3.6 × 10⁵.

Worked example

Convert and multiply

Problem. Write 458000 in scientific notation and evaluate (3×10^4)×(2.5×10^{−2}).

Solution. 458000 = 4.58×10^5. Product: 7.5×10^{2} = 750 after renormalising 7.5 already in range.

Review quiz

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0.00058 in scientific notation is

Practice questions

Write 0.00058 in scientific notation.
Write 7.25 × 10⁴ in ordinary form.
Calculate (2 × 10³) × (3.5 × 10⁴) and give the answer in scientific notation.
Express 0.00402 × 10⁵ in scientific notation.