Multiplying by Powers of Ten

Multiplying by 10 makes every digit worth ten times more, so each moves one place left.

An example

3.42 × 10 = 34.2. The 3 was worth 3 and is now worth 30.

Dividing goes the other way

3.42 ÷ 10 = 0.342. Every digit is now worth a tenth of what it was.

The point does not move

It is easier to picture the digits shifting. The point marks where the ones are, and the ones do not move.

Digits shift; the point stays put

Multiplying 3.42 by 10 gives 34.2. It is more accurate to say every digit moved one place left than to say the point moved right, because place value is about where digits sit. Either description gets the right answer; the first keeps working when the numbers get strange.

Where "add a zero" breaks

3.42 × 10 is not 3.420. The shortcut you learned for whole numbers fails outright for decimals, because appending a zero to a decimal changes nothing at all. This is the clearest example this year of a rule expiring.

Dividing shifts the other way

3.42 ÷ 10 = 0.342. Every digit moves one place right. Dividing by 100 moves them two places. Sometimes a zero must be written in front to hold the ones place: 3.42 ÷ 100 = 0.0342.

This is why metric conversion is easy

Converting metres to centimetres multiplies by 100; grams to kilograms divides by 1,000. Every metric conversion is a digit shift, which is precisely why the metric system was designed around powers of ten.