Math Core

Our number system is built on tens. Every time you move one place to the left, a digit is worth 10 times as much. Once you see that pattern, multiplying and dividing by 10, 100 or 1,000 becomes something you can do in your head.

Each place is 10 times the place to its right

Look at the number 5,555. Every digit is a 5, but each 5 has a different value.

thousands	hundreds	tens	ones
5	5	5	5
5,000	500	50	5

Start at the right and move left: 5, then 50, then 500, then 5,000. Each value is 10 times the one before it. Now go the other way, from left to right: 5,000, then 500, then 50, then 5. Each value is $\dfrac{1}{10}$ of the one before it.

The pattern keeps going past the ones place. The place to the right of the ones is the tenths place, so a 5 there is worth $\dfrac{5}{10}$ , or 0.5. That is $\dfrac{1}{10}$ of 5.

The place value pattern

A digit in one place is worth 10 times as much as the same digit in the place to its right.

It is worth $\dfrac{1}{10}$ of what the same digit is worth in the place to its left.

Worked example: Comparing two digits

In 7,730, how many times as much is the first 7 worth as the second 7?

The first 7 is in the thousands place, so it is worth 7,000. The second 7 is in the hundreds place, so it is worth 700. Since $7{,}000 = 10 \times 700$ , the first 7 is worth 10 times as much.

Powers of ten and exponents

Numbers like 10, 100 and 1,000 are called powers of ten, because you make them by multiplying 10 by itself. You can write them in a short way with an exponent.

Definition

Exponent

An exponent tells how many times to use a number as a factor. In $10^3$ , the 10 is the base and the 3 is the exponent:

10^3 = 10 \times 10 \times 10 = 1{,}000

You read $10^3$ as "ten to the third power."

power of ten	as a product	standard form
$10^1$	$10$	10
$10^2$	$10 \times 10$	100
$10^3$	$10 \times 10 \times 10$	1,000
$10^4$	$10 \times 10 \times 10 \times 10$	10,000

Notice the pattern: the exponent tells you how many zeros follow the 1. So $10^6$ is a 1 followed by six zeros, or 1,000,000.

Multiplying by a power of ten

When you multiply by 10, every digit moves one place to the left, so each digit is worth 10 times as much. For a whole number, that puts a 0 on the end: $36 \times 10 = 360$ .

Multiplying by 100 moves the digits two places left, and multiplying by 1,000 moves them three places. For whole numbers, count the zeros in the power of ten and write that many zeros on the end.

Worked example: Multiplying whole numbers

$48 \times 100 = 4{,}800$ (two zeros)
$25 \times 10^3 = 25 \times 1{,}000 = 25{,}000$ (three zeros)
$6 \times 10^4 = 60{,}000$ (four zeros)

With a decimal, you can't just add zeros. Instead, the digits shift left, which makes the decimal point look like it moves to the right. Multiplying by 10 moves it one place; multiplying by 100 moves it two places.

Worked example: Multiplying a decimal

Find $3.7 \times 100$ .

Multiplying by 100 moves every digit two places left. The 3 ones become 3 hundreds and the 7 tenths become 7 tens. Moving the decimal point two places right gives $3.7 \to 37. \to 370.$ You need a 0 to fill the empty ones place.

$3.7 \times 100 = 370$ .

Dividing by a power of ten

Dividing does the opposite. Dividing by 10 moves every digit one place to the right, so each digit is worth $\dfrac{1}{10}$ as much. The decimal point looks like it moves left.

Worked example: Dividing by powers of ten

$420 \div 10 = 42$
$65 \div 100$ : move the decimal point in $65.$ two places left to get 0.65.
$8 \div 10^2 = 8 \div 100$ : write 8 as $8.$ and move the point two places left. You need a 0 as a placeholder: $8. \to 0.8 \to 0.08$ . The answer is 0.08.

Common mistake

Don't add zeros to the end of a decimal when you multiply. $2.5 \times 10$ is not 2.50 (that is the same number as 2.5). Move the digits instead: $2.5 \times 10 = 25$ .

Tip

Check the size of your answer. Multiplying by a power of ten should make a number bigger, and dividing should make it smaller. If $56 \div 100$ gave you 5,600, you moved the wrong way.

Practice

Practice 1

Write $10^5$ in standard form.

Type a number.

Practice 2

Write 10,000 as a power of ten. What is the exponent?

Type a number.

Practice 3

Find $73 \times 100$ .

Type a number.

Practice 4

In 3,330, how many times as much is the 3 in the hundreds place worth as the 3 in the tens place?

Type a number.

Practice 5

Find $4.6 \times 10$ .

Type a number.

Practice 6

Find $29 \div 100$ .

Type a number.

Practice 7

Which is equal to $5.2 \times 10^2$ ?

Practice 8

Which power of ten makes this true? Write it in standard form.

0.75 \times \underline{\quad\quad} = 750

Type a number.