When you multiply or divide by 10, 100, or 1000, every digit in the number shifts position in the place-value table — left when multiplying, right when dividing. The decimal point never moves; the digits do. Mastering this is the foundation for working with standard form, decimals, and metric unit conversions.
Why do digits shift when you multiply by 10?
Our number system is base 10: each column is worth ten times the column to its right. When you multiply a number by 10, every digit becomes ten times bigger, so it must move one column to the left. An empty column is filled with a zero.
Example: 3.7 × 10
| Tens | Units | . | Tenths | Hundredths |
|---|---|---|---|---|
| 3 | . | 7 | ||
| 3 | 7 | . | 0 |
Each digit shifts one place left: 3 moves from Units to Tens, 7 moves from Tenths to Units. The answer is 37.
How do you multiply by 10, 100, or 1000?
Multiply by 10 → shift every digit 1 place left (multiply by 10¹).
Multiply by 100 → shift every digit 2 places left (multiply by 10²).
Multiply by 1000 → shift every digit 3 places left (multiply by 10³).
| Calculation | Digit shift | Answer |
|---|---|---|
| 4.6 × 10 | 1 left | 46 |
| 4.6 × 100 | 2 left | 460 |
| 4.6 × 1000 | 3 left | 4,600 |
| 0.057 × 10 | 1 left | 0.57 |
| 0.057 × 100 | 2 left | 5.7 |
| 0.057 × 1000 | 3 left | 57 |
Notice that multiplying 0.057 × 1000 gives 57 exactly — no leading zeros needed.
How do you divide by 10, 100, or 1000?
Division is the reverse: every digit shifts to the right.
Divide by 10 → shift every digit 1 place right.
Divide by 100 → shift every digit 2 places right.
Divide by 1000 → shift every digit 3 places right.
| Calculation | Digit shift | Answer |
|---|---|---|
| 830 ÷ 10 | 1 right | 83 |
| 830 ÷ 100 | 2 right | 8.3 |
| 830 ÷ 1000 | 3 right | 0.83 |
| 5 ÷ 10 | 1 right | 0.5 |
| 5 ÷ 100 | 2 right | 0.05 |
| 5 ÷ 1000 | 3 right | 0.005 |
When the digit has nowhere to go on the left of the decimal point, write a zero in the units position (e.g. 0.5, not just .5).
How does this apply to larger powers of 10?
The pattern extends to any power of 10. The exponent tells you how many places to shift.
Worked example: 4.08 × 10,000
10,000 = 10⁴, so shift 4 places left. 4.08 → 40,800. Answer: 40,800.
Worked example: 0.0036 ÷ 10,000
Shift 4 places right. 0.0036 → 0.00000036. Answer: 0.00000036 (or 3.6 × 10⁻⁷ in standard form).
At KS3 you are mainly expected to work with 10, 100, and 1000, but the logic is identical for larger powers.
Where does this skill appear in other topics?
Understanding digit shifts underpins several other KS3 topics:
- Converting metric units: 1 km = 1000 m, so 3.7 km = 3.7 × 1000 = 3,700 m; 450 cm = 450 ÷ 100 = 4.5 m
- Standard form: writing 3,700 as 3.7 × 10³ reverses this process
- Percentage calculations: finding 10% of something is the same as dividing by 10
- Estimating: rounding to the nearest 10 or 100 relies on understanding column values
Worked example (metric conversion): A swimming pool is 25 m long. Convert to centimetres. 1 m = 100 cm, so multiply by 100: 25 × 100 = 2,500 cm.
Frequently asked questions
Does the decimal point move or do the digits move?
The decimal point stays fixed between the units column and the tenths column. It is the digits that move left or right relative to it. A common error is to say "the decimal point moves right when you multiply by 10" — this describes the same result but can cause confusion when there is no visible decimal point (e.g. in whole numbers). Thinking of digit shifts is more reliable.
What happens when you multiply a whole number by 10?
Digits shift one place left and the vacated units position is filled with a zero. For example, 47 × 10 = 470. The zero is a placeholder, not a new digit being added — it marks the empty units column after the 7 has moved to tens.
How can I check my answer quickly?
If you multiply, your answer should be bigger than the original number. If you divide, it should be smaller. Also count the zeros in the power of 10: multiplying 6.5 by 100 (two zeros) should give an answer with two fewer decimal places — 650. If your answer has more decimal places than expected, you have likely shifted the wrong way.
Is this the same as moving the decimal point?
Yes, in effect — but the phrasing matters. When you multiply 4.6 by 10, you can say the decimal point moves one place right (giving 46), or equivalently that every digit moves one place left. Both describe the same outcome. Many teachers prefer "digits shift" to reinforce why the result changes, rather than memorising a mechanical rule.
For Socratic KS3 number and place-value practice with Professor Pi, see aitutors.me.