Year 8NumberNumber structures and operationsYear 8 Maths

Powers of ten, in both directions

What this lesson covers

In our number system, each place value is a power of 10, and this continues infinitely to the left and right.
Repeated division can be expressed using exponent notation with negative exponents.
Decimals can be represented using negative exponents (i.e. negative powers of ten).

Key idea

The pattern does not stop at the point

You have met place value twice: whole numbers, then decimals.
They are taught as if the decimal point is a border.
It is not. There is one ladder, and it runs both ways.

Key idea

Every column is a power of ten

A thousand is 10 × 10 × 10, which we write as 10³.
A hundred is 10², and ten is 10¹.
The exponent counts how many tens are multiplied together.

Power	Value	Step
10³	1000
10²	100	÷ 10
10¹	10	÷ 10

Key idea

Going down a rung means dividing by ten

1000 ÷ 10 is 100. 100 ÷ 10 is 10.
Every step to the right divides by ten.
So what comes after 10¹?

Power	Value	Step
10³	1000
10²	100	÷ 10
10¹	10	÷ 10
10⁰	1	÷ 10

Key idea

Which is why 10⁰ is 1

10 ÷ 10 is 1. The pattern gives us no choice.
The next rung down from 10¹ is 10⁰, and it is worth 1.
Not a rule to memorise. Just the pattern, carried on.

Power	Value	Step
10³	1000
10²	100	÷ 10
10¹	10	÷ 10
10⁰	1	÷ 10

Key idea

And nothing stops us carrying on

1 ÷ 10 is 0.1. That is the next rung.
The exponent has to keep dropping too, so it goes negative.
0.1 is 10⁻¹. Then 0.01 is 10⁻², and so on for ever.

Power	Value	Step
10⁰	1
10⁻¹	0.1	÷ 10
10⁻²	0.01	÷ 10
10⁻³	0.001	÷ 10

Key idea

What a negative exponent actually means

It does not mean a negative number. 10⁻² is not below zero.
It counts divisions instead of multiplications.
10⁻³ means 1 divided by ten, three times over.

Exponent	Counts	So
Positive	how many tens multiplied	10³ = 10 × 10 × 10
Zero	no tens at all	10⁰ = 1
Negative	how many tens divided by	10⁻³ = 1 ÷ 10 ÷ 10 ÷ 10

Key idea

Three ways to write the same rung

10⁻² is one hundredth.
One hundredth is the fraction 1/100.
And 1/100 is the decimal 0.01. Three names, one number.

Power	Fraction	Decimal
10⁻¹	1/10	0.1
10⁻²	1/100	0.01
10⁻³	1/1000	0.001

Key idea

So the columns have proper names

The ones column is the 10⁰ column.
First past the point is tenths, which is 10⁻¹.
Then hundredths, 10⁻². The point marks where the ones are.

Worked example

Writing 3.61 the long way

Take each digit with the column it sits in.
3 ones, 6 tenths, 1 hundredth.
So 3.61 = 3 × 10⁰ + 6 × 10⁻¹ + 1 × 10⁻².

Digit	Column	Written	Worth
3	ones	3 × 10⁰	3
6	tenths	6 × 10⁻¹	0.6
1	hundredths	1 × 10⁻²	0.01

Key idea

Check it by adding it back up

3 + 0.6 + 0.01.
That is 3.61, which is what we started with.
Expanded form is never a guess. It always adds back.

Digit	Column	Written	Worth
3	ones	3 × 10⁰	3
6	tenths	6 × 10⁻¹	0.6
1	hundredths	1 × 10⁻²	0.01

Worked example

A longer one: 2405.07

Start at the left. 2 is in the thousands, so 2 × 10³.
Work right one column at a time, dropping the exponent by one.
Past the point the exponents go negative. 7 is 7 × 10⁻².

Digit	Written	Worth
2	2 × 10³	2000
4	4 × 10²	400
0	0 × 10¹	0
5	5 × 10⁰	5
0	0 × 10⁻¹	0
7	7 × 10⁻²	0.07

Key idea

The zeros are not optional

Two columns here hold a zero, and contribute nothing.
But they hold the other digits in their places.
Drop them and 2405.07 becomes 245.7, a different number.

Digit	Written	Worth
2	2 × 10³	2000
4	4 × 10²	400
0	0 × 10¹	0
5	5 × 10⁰	5
0	0 × 10⁻¹	0
7	7 × 10⁻²	0.07

Key idea

Now: which is bigger, 0.4 or 0.35?

Plenty of people say 0.35, because 35 is bigger than 4.
Put them on a line and you can see it is not.
0.4 sits to the right of 0.35.

Key idea

Why more digits does not mean bigger

After the point, the digits are not a whole number.
The first one is tenths, and tenths outrank everything after.
4 tenths beats 3 tenths. The hundredths never get a say.

Number	Tenths	Hundredths
0.4	4 tenths	0 hundredths
0.35	3 tenths	5 hundredths
Bigger?	4 beats 3	0.4

Worked example

Ordering four of them

Compare the tenths first. Only if they tie, move right.
0.405 and 0.45 both have 4 tenths, so go to the hundredths.
0 beats nothing, 5 beats 0. Smallest to largest, left to right.

Number	Reading it
0.405	4 tenths, 0 hundredths	smallest
0.45	4 tenths, 5 hundredths
0.5	5 tenths
0.54	5 tenths, 4 hundredths	largest

Your turn

Your turn: four questions

Pause here and try all four.
Question 2 wants powers of ten, not just 4 + 0.5 + 0.02.
Question 3 is the one to slow down on.

	Question
1	Write 0.001 as a power of ten.
2	Write 4.52 in expanded form using powers of ten.
3	Which is bigger, 0.7 or 0.65?
4	Write 10⁻¹ as a fraction and as a decimal.

Show answer

1. 10⁻³. Three divisions by ten below the ones. 2. 4 × 10⁰ + 5 × 10⁻¹ + 2 × 10⁻². 3. 0.7. It has 7 tenths against 6 tenths, and the comparison is settled there. The extra digit in 0.65 never gets a say. 4. 1/10 and 0.1.

Worked example

The answers

0.001 is three divisions below the ones, so 10⁻³.
4.52 is 4 × 10⁰ + 5 × 10⁻¹ + 2 × 10⁻².
0.7 is bigger. And 10⁻¹ is 1/10, which is 0.1.

	Answer
1	10⁻³
2	4 × 10⁰ + 5 × 10⁻¹ + 2 × 10⁻²
3	0.7
4	1/10 and 0.1

Summary

What to take away

One ladder of powers of ten, running both ways for ever.
A negative exponent counts divisions, not subtractions.
To compare, start at the left and stop at the first difference.

	Remember
One ladder	each rung is ten times the one below
10⁰	is 1, because the pattern says so
Negative exponent	counts divisions, not subtractions
10⁻²	is 1/100, which is 0.01
Comparing	start at the left and stop at the first difference

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