"Is not equal to", for example 0.3 0.03.

N1, N2NumberFoundationGrades 1–3

Place Value & Ordering

Q: Is -0.5 bigger or smaller than -0.25?

Smaller. -0.5 is further from zero on the negative side, so -0.5 < -0.25.

Place value is the idea that a digit's value depends on its position: the 7 in 7,305 is worth 7,000, but in 0.07 it is worth seven hundredths. Everything in arithmetic — lining up columns, multiplying by 10, ordering decimals, rounding — rests on it.

30 min AQA GCSE Maths

The big picture

Our number system is built on tens. Each column is worth ten times the one to its right, so multiplying by 10 moves every digit one place to the left, and dividing by 10 moves every digit one place to the right. The decimal point never moves; the digits do.

Ordering is a place-value skill too. To compare 0.45 and 0.405, compare column by column from the left — not by counting digits. And on the negative side of zero, the order reverses: $- 8$ is smaller than $- 3$ .

What you'll learn

State the value of any digit in a whole number or decimal
Multiply and divide by 10, 100 and 1000, including decimals
Order integers, decimals and negative numbers
Use the symbols $=$ , $\neq =$ , $<$ , $>$ , $\leq$ , $\geq$

The value of a digit

Columns to the left of the decimal point are ones, tens, hundreds, thousands…; to the right are tenths, hundredths, thousandths…

In $52, 836.47$ : the 5 is worth 50,000, the 8 is worth 800, the 4 is worth 4 tenths ( $0.4$ ) and the 7 is worth 7 hundredths ( $0.07$ ).

1The columns after the point are tenths (0), hundredths (6), thousandths (8).

Answer6 hundredths, or

0.06

Your turnSame method, new numbers — try it before moving on.

Write down the value of the 9 in $41.293$ as a decimal.

Your answer

Multiplying and dividing by powers of 10

Multiplying by 10, 100 or 1000 moves every digit 1, 2 or 3 places to the left. Dividing moves them to the right. Fill empty places before the decimal point with zeros.

Multiplying by 0.1 is the same as dividing by 10, and dividing by 0.01 is the same as multiplying by 100.

1(a) Digits move 2 places left:

372

2(b) Digits move 3 places right:

0.045

3(c) Dividing by 0.1 is multiplying by 10:

6

Answer(a)

372

(b)

0.045

(c)

6

Your turnSame method, new numbers — try it before moving on.

Work out $8.4 \div 100$ .

Your answer

Your turnSame method, new numbers — try it before moving on.

Given $37 \times 52 = 1924$ , work out $3.7 \times 520$ .

Your answer

Tip — Estimate to check: $2.3 \times 4.1$ is roughly $2 \times 4 = 8$ , so $9.43$ is sensible and $94.3$ is not.

Ordering numbers

For decimals, line up the decimal points and compare digits column by column from the left. Adding zeros to the end can help: $0.5 = 0.500$ , which is larger than $0.405$ .

For negative numbers, picture a number line: the further left, the smaller. $- 12 < - 5 < 0 < 3$ .

Symbols: $<$ less than, $>$ greater than, $\leq$ less than or equal to, $\geq$ greater than or equal to, $\neq =$ not equal to.

- 8

is to the left of

- 3

, so

- 8 < - 3

— even though 8 is bigger than 3.

1Write with three decimal places:

0.700, 0.070, 0.707, 0.770, 0.077

2Compare:

70 < 77 < 700 < 707 < 770

(thousandths).

Answer

0.07, 0.077, 0.7, 0.707, 0.77

Your turnSame method, new numbers — try it before moving on.

Which is the smallest: $- 2.5$ , $- 2.05$ , $- 0.25$ or $- 25$ ?

Your answer

Think like an examiner

In "use the fact that…" questions, you must use the given calculation; an answer found by long multiplication alone may not score.
Show a common number of decimal places when ordering — it makes your reasoning visible.
Answers like "7 hundredths" and " $0.07$ " are both accepted for the value of a digit; "7" on its own is not.

Place value

\times 1 0^{n} : digits move n places left

Dividing moves them right.

\times 0.1 = \div 10

Multiplying by a decimal can make a number smaller.

- 8 < - 3

Negatives: further from zero means smaller.

Watch out for these

0.45 > 0.5

because 45 is bigger than 5.

Compare tenths first: 4 tenths < 5 tenths, so

0.45 < 0.5

3.7 \times 10 = 3.70

The digits move, not the zero:

37

- 8 > - 3

because 8 > 3.

On a number line

- 8

is further left:

- 8 < - 3

6 \div 0.1 = 0.6

Dividing by 0.1 multiplies by 10:

60

Stretch yourself

Using each of the digits 3, 0, 7 and 5 exactly once, and one decimal point, make the number closest to 50.

Hint — Try numbers just below and just above 50 in the form __.__ .

Questions students ask

Does the decimal point move when I multiply by 10?

It is clearer to think of the digits moving left while the point stays still. Either way you get the same answer.

- 0.5

bigger or smaller than

- 0.25

Smaller.

- 0.5

is further from zero on the negative side, so

- 0.5 < - 0.25

What does

\neq =

mean?

"Is not equal to", for example

0.3 \neq = 0.03

Key takeaways

A digit's value depends on its column.
Multiply by $1 0^{n}$ : digits move $n$ places left; divide: right.
Compare decimals column by column from the left, padding with zeros.
For negatives, the further from zero, the smaller.
Use given facts and estimation to check decimal calculations.

Exam-style questions

Practise it the way it is examined

2 original questions · 6 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 13 marks

(a)Write down the value of the 4 in $6.842$ .[1]
(b)Work out $0.36 \times 1000$ .[1]
(c)Write these numbers in order, starting with the smallest: $- 4, 2, - 9, 0, - 1$ [1]

Question 2Grade 33 marks

You are told that $148 \times 26 = 3848$ .

(a)Work out $14.8 \times 2.6$ .[1]
(b)Work out $3848 \div 2.6$ .[2]

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Place Value & Ordering

The value of a digit

Multiplying and dividing by powers of 10

Ordering numbers

Practise it the way it is examined

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