Multiplying a whole number by ten moves every digit one place to the left. The ones place is left empty, and a zero goes in to hold it. The zero is the result of the shift, not the reason for it.
The pattern
| Multiply by | Digits move | Effect on 37 |
|---|---|---|
| 10 | one place left | 370 |
| 100 | two places left | 3,700 |
| 1,000 | three places left | 37,000 |
The number of zeros in the multiplier is the number of places every digit moves. That is the whole rule, and unlike add a zero it keeps working for every kind of number.
Why the shift happens
37 x 10, written out
37 = 30 + 7 30 x 10 = 300 7 x 10 = 70 300 + 70 = 370
Each part became ten times bigger, so the tens became hundreds and the ones became tens. Every digit ended up one place to the left.
What happens with decimals
This is where add a zero breaks. 4.6 x 10 is 46, not 4.60, which would be the same number it started as. The digits still move one place left, and the decimal point stays where it is.
Decimals moving left
4.6 x 10 = 46 4.6 x 100 = 460 0.375 x 10 = 3.75 0.375 x 1000 = 375
Multiplying by multiples of ten
The same shift handles numbers like 40 and 300. Split the multiplier into its leading digit and its power of ten, multiply by the digit, then shift.
26 times 40
26 x 4 = 104 then shift one place for the ten 26 x 40 = 1,040
15 times 300
15 x 3 = 45 then shift two places for the hundred 15 x 300 = 4,500
This is why mental multiplication of round numbers is so much faster than the written algorithm. Everything reduces to a single-digit fact plus a shift.
Common mistakes
Adding a zero to a decimal
Writing 4.60 for 4.6 x 10 leaves the value unchanged. Move the digits instead.
Counting the wrong number of zeros
100 has two zeros and moves digits two places. 1,000 has three. Count the zeros in the multiplier, not the digits in the number.
Check yourself
58 x 100
5,800. Two places left, two zeros fill the gap.
2.7 x 1,000
2,700. The digits move three places left, so the 7 lands in the hundreds place.
0.049 x 100
4.9. Two places left.
Why does add a zero work for whole numbers but not for decimals?
Because for a whole number the empty ones place has to be filled, and a zero is what fills it. A decimal already has digits after the point, so writing a zero at the end fills nothing and changes nothing.
See also
Sources
- Standard place-value reasoning for powers of ten. Examples on this page are our own.