Multiplying by Powers of Ten

Change £5 into pence and you get 500\text{p}; measure 7\,\text{cm} in millimetres and it is 70\,\text{mm}. Swapping between money units or measuring units almost always means multiplying by 10, 100 or 1000 — and there is a lovely shortcut so you never do it the long way.

Once you know multiplication and place value, something neat falls out. Multiplying a number by 10, 100 or 1000 doesn't change which digits you have — it just slides every digit one, two, or three places to the left in the place-value columns.

Why left? Each step to the left makes a digit worth ten times as much: ones become tens, tens become hundreds, hundreds become thousands. So multiplying by ten bumps the whole number up one column at a time.

23 \times 10 = 230

Multiplying by 100 slides every digit two places, and by 1000 it slides three places:

23 \times 100 = 2300 \qquad 23 \times 1000 = 23000

It looks like you are just "adding zeros," but really you are sliding the digits up the columns. The zeros are simply the empty places the digits left behind — they hold those columns open so each digit keeps its new, bigger value.

See it in the place-value table

Put a number in the place-value columns, then multiply. Watch the digits themselves jump to the left, and fresh zeros grow in to fill the empty places they leave. Press Refresh for a new number and a new power of ten.

Press play to watch the same idea animate. A number appears in its place-value columns; as we multiply by 10, 100 or 1000, every digit slides left and zeros grow in to fill the empty places. Replay it — a different number and power each time.

Worked examples

Read each one as a slide, not as "stick a zero on":

23 \times 10: the 2 (two tens) becomes two hundreds, the 3 (three ones) becomes three tens, and a zero fills the empty ones place — 230.
7 \times 100: the lone 7 slides two places, from ones to hundreds, and two zeros hold the tens and ones open — 700.
40 \times 1000: the 4 slides three places to ten-thousands and the 0 slides with it, leaving three new zeros — 40000.

The same slide works for decimals — and here is where "add a zero" really falls apart. Take 3.4 \times 10. The 3 (three ones) slides up to the tens place and the 4 (four tenths) slides up to the ones place. There is no empty place left over, so no zero appears:

3.4 \times 10 = 34 \quad(\text{not } 3.40)

"Just add a zero" is a trap — it only looks right for whole numbers.

It is the digits that shift left, not the zeros that get tacked on. For 3.4 \times 10 the answer is 34, not 3.40 (which is just 3.4 again).
Count the places to match the power: \times 10 is one place, \times 100 is two, \times 1000 is three. Slide one column too few and you are out by a factor of ten.

Going the other way: dividing

Dividing by a power of ten does the exact opposite — it makes each digit worth less, so every digit slides to the right. Divide by 10 and hundreds become tens, tens become ones, ones become tenths:

230 \div 10 = 23 \qquad 700 \div 100 = 7

So multiplying slides left and dividing slides right — they undo each other, column for column.

rocket A rocket might be 2 metres tall — that is 200 centimetres, because 2 \times 100 = 200. To swap metres for centimetres you just slide the digits two places left; to go back you slide them right. Metric units are built in powers of ten on purpose, so changing units is never harder than sliding digits along the place-value columns. (Old units like feet and inches make you multiply by twelve — much messier!)

coin A ten-cent coin is worth ten times a one-cent coin, and a dollar is ten times a dime. Money climbs in tens, just like place value. Ten 10-cent coins make 10 \times 10 = 100 cents — one dollar. So if a sticker costs 15 cents and you buy 100 of them, the digits slide two places: 15 \times 100 = 1500 cents, which is fifteen dollars.

Khan Academy works through multiplying by 10, 100 and 1000 here: