Standard form — writing very big and very small numbers tidily
Standard form (a.k.a. scientific notation) is the standard way to write extremely large or extremely small numbers without trailing zeros or tiny decimals. It's used everywhere in science, finance and computing.
The form
A number in standard form looks like:
A × 10ⁿ
where:
- A is at least 1 and less than 10 (i.e. 1 ≤ A < 10).
- n is an integer (positive, negative, or zero).
So 4.7 × 10⁵ is valid; 47 × 10⁴ is not (A = 47 is too big).
Converting to standard form
Big numbers (n positive): count how many places the decimal point moves left to leave one non-zero digit before it.
58,000 = 5.8 × 10⁴ (four places).
Small numbers (n negative): count how many places the decimal point moves right.
0.000037 = 3.7 × 10⁻⁵ (five places).
⚠ The sign of n is positive when the original is ≥ 10 and negative when it's < 1.
Converting back to ordinary numbers
Move the decimal point. If n is positive, move right; if negative, move left. Pad with zeros.
9.2 × 10⁶ = 9,200,000. 6.04 × 10⁻³ = 0.00604.
Multiplying and dividing in standard form
Use index laws (see N7).
- Multiply: multiply the A's, add the powers, then re-adjust if the new A is outside [1, 10).
- Divide: divide the A's, subtract the powers, re-adjust if needed.
Worked example: (3 × 10⁴) × (4 × 10⁵).
- 3 × 4 = 12.
- 10⁴ × 10⁵ = 10⁹.
- = 12 × 10⁹.
- A = 12 is too big; rewrite: 1.2 × 10¹⁰.
Worked example: (8 × 10⁻³) ÷ (2 × 10⁵).
- 8 ÷ 2 = 4.
- 10⁻³ ÷ 10⁵ = 10⁻⁸.
- = 4 × 10⁻⁸.
Adding and subtracting in standard form
You cannot simply add the A's unless the powers match. Two strategies:
- Convert to ordinary form, add, then convert back.
- Match the powers by adjusting one number, then add the A's.
Worked example: 3.4 × 10⁵ + 5 × 10⁴.
- Rewrite 5 × 10⁴ as 0.5 × 10⁵.
- 3.4 + 0.5 = 3.9.
- Sum = 3.9 × 10⁵.
Calculator tips
- Use the EE or ×10ˣ button.
3.4 EE 5enters 3.4 × 10⁵. - After a calculation, your calculator may display
5.6 × 10⁻⁴as5.6 E−4or similar — convert to the standard form notation before writing the answer.
⚠Common mistakes— Common mistakes (examiner traps)
- A outside [1, 10). Writing 12 × 10⁹ when the question requires standard form. Adjust to 1.2 × 10¹⁰.
- Wrong sign on n for small numbers.
0.0042 = 4.2 × 10⁻³, not4.2 × 10³. - Adding the powers when adding the numbers. Powers only add for multiplication. For + and −, match powers first.
- Off-by-one count of decimal places. Recount carefully.
- Calculator answers with × button typed manually.
3 × 10⁵typed as 3 × 10 ^ 5 is fine;3 EE 5is faster and avoids errors.
➜Try this— Quick check
Without a calculator: (6 × 10⁻⁴) × (7 × 10⁹).
- 6 × 7 = 42.
- 10⁻⁴ × 10⁹ = 10⁵.
- 42 × 10⁵ → adjust → 4.2 × 10⁶.
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