Standard form = one digit, then a power of 10: Standard form (also called scientific notation) writes a number as a × 10ᵏ.
The coefficient a is between 1 and 10. The exponent k is a whole number.
For example, 45 300 = 4.53 × 10⁴.
- the coefficient — one non-zero digit before the decimal point
- the exponent — an integer (positive, negative or 0)
- Standard form
- A number written as a × 10ᵏ with 1 ≤ a < 10. Example: 6 × 10⁻³.
- Ordinary form
- The everyday way of writing a number. Example: 0.006.
- Coefficient (a)
- The number in front, kept between 1 and 10. Example: the 6 in 6 × 10⁻³.
- Exponent (k)
- The power of 10. Example: the −3 in 6 × 10⁻³.
| Ordinary form | Standard form |
|---|---|
| 8 000 | 8 × 10³ |
| 453 000 | 4.53 × 10⁵ |
| 0.006 | 6 × 10⁻³ |
| 0.000 21 | 2.1 × 10⁻⁴ |
Two quick steps: 1. Put the decimal point just after the first non-zero digit → that is a.
2. Count how many places the point moved → that is k.
For example, 6 040 000 → 6.04 × 10⁶ (the point moves 6 places).
The rule, with examples
- Big number (10 or more) → k is positive, the point moves left. For example, 52 000 = 5.2 × 10⁴.
- Small number (less than 1) → k is negative, the point moves right. For example, 0.0007 = 7 × 10⁻⁴.
- Always check a is between 1 and 10. For example, 48 = 4.8 × 10¹, not 48 × 10⁰.
IB-style question — a big number
Write 45 300 000 in standard form.
Step by step
- Put the point after the first digit to make a coefficient between 1 and 10.
- Count the places the point moved left: 45 300 000 → 4.53 is 7 places.
- Write them together.
Final answer
45 300 000 = 4.53 × 10⁷
IB-style question — a small number
Write 0.000 64 in standard form.
Step by step
- Put the point after the first non-zero digit (6).
- The point moved 4 places right, so the exponent is negative.
- Write them together.
Final answer
0.000 64 = 6.4 × 10⁻⁴
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Common mistake
- 45.3 × 10⁶ — coefficient too big
- 0.7 × 10⁵ — coefficient less than 1
Correct
- 4.53 × 10⁷
- 7 × 10⁴
- Move the point to just after the first digit
Which paper — and how it's asked: Paper 1 (no calculator): write a number in standard form by hand, or give a calculated answer in the form a × 10ᵏ.
Paper 2 (calculator): you work out the value on the GDC, then rewrite it in standard form.
Both have appeared on real exams.
Reading the GDC (Paper 2): Your GDC shows big or small numbers using ᴇ. The display 8.58ᴇ3 means 8.58 × 10³.
For example, a value shown as 2.1ᴇ-4 is 2.1 × 10⁻⁴.
Paper 2 example — compute, then express
A sphere has radius 9.4 cm. Find its volume in the form a × 10ᵏ cm³, where 1 ≤ a < 10.
Step by step
- Use the volume formula and the GDC to work out the value.
- The GDC works out the value.
- Move the point 3 places to write it in standard form.
Final answer
V = 3.48 × 10³ cm³ (3 s.f.)
If you see "in the form a × 10ᵏ, 1 ≤ a < 10": Your final line must have exactly one digit before the decimal point.
For example, finish as 3.6 × 10¹³, never 36 × 10¹².