What ordinary form and standard form mean
Big Idea: Ordinary form means the number is written out normally. For example, 4 300 000 and 0.000081.__LINEBREAK__Standard form writes the same values in a shorter way. For example, 4.3 × 10⁶ and 8.1 × 10⁻⁵ .
IB wording: You may hear scientific notation, but IB usually says standard form. For example, in exam commands you will typically see "Write in standard form".
| Number | Form | Example meaning |
|---|---|---|
| 4 300 000 | ordinary form | Written out with zeros |
| 4.3 × 10⁶ | standard form | Same value, shorter form |
| 0.000081 | ordinary form | Tiny value written out |
| 8.1 × 10⁻⁵ | standard form | Same tiny value, compact form |
Fast validity check: If the coefficient is not between 1 and 10, it is not finished standard form.
Converting large numbers to standard form (positive exponent)
Big Idea: Large ordinary numbers become standard form with a positive exponent. For example, 5 840 000 becomes 5.84 × 10⁶.
Method: move the decimal point left until the coefficient is between 1 and 10. For example, 32 000 000 becomes 3.2 × 10⁷.
| Ordinary number | Standard form | Why |
|---|---|---|
| 5 840 000 | 5.84 × 10⁶ | Decimal moved 6 places left, so exponent is +6 |
| 31 500 | 3.15 × 10⁴ | Decimal moved 4 places left, so exponent is +4 |
| 900 000 000 | 9.0 × 10⁸ | Large number means positive exponent |
Safe positive-exponent routine
- Move the decimal so only the first non-zero digit stays before it.
- Count how many places you moved it left.
- Write that count as a positive exponent.
- Check the coefficient is between 1 and 10.
Direction check: Large number in ordinary form means decimal moved left and exponent is positive.
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Converting small numbers to standard form (negative exponent)
Big Idea: Small decimals become standard form with a negative exponent. For example, 0.00409 becomes 4.09 × 10⁻³.
Method: move the decimal point right until the coefficient is between 1 and 10. For example, 0.000000085 becomes 8.5 × 10⁻⁸.
Safe negative-exponent routine
- Move the decimal so only the first non-zero digit stays before it.
- Count how many places you moved it right.
- Write that count as a negative exponent.
- Check the coefficient is between 1 and 10.
| Ordinary number | Standard form | Why |
|---|---|---|
| 0.00409 | 4.09 × 10⁻³ | Decimal moved 3 places right, so exponent is -3 |
| 0.00052 | 5.2 × 10⁻⁴ | Decimal moved 4 places right, so exponent is -4 |
| 0.000000085 | 8.5 × 10⁻⁸ | Very small number means negative exponent |
Exam Tips:
- Tiny decimal in ordinary form means the exponent will be negative.
- Do not drop placeholder zeros when converting.
- Always do a final coefficient check: 1 <= |a| < 10.