Number Operations Notes

Q: How do you order and compare numbers?

The number further right on the number line is larger, so -8 < -3. Integers, decimals and fractions can all go on a number line. To compare fractions and decimals, convert them to the same form first, usually decimals.

Q: How do the signs decide the answer?

When multiplying or dividing, the signs decide the sign of the answer. When adding, compare the sizes of the numbers. The four operations work on integers, decimals and fractions, including negatives.

Q: How does place value help with decimals?

Place value tells you what each digit is worth, which keeps big, small and decimal calculations right. Place value matters for very large numbers, very small numbers such as 0.00042, and for lining up decimal points when you add or subtract.

Q: What order do you work in?

Do brackets first, then indices, then division and multiplication, then addition and subtraction. BIDMAS gives the order. Work left to right for division and multiplication, and for addition and subtraction.

Q: How do you count outcomes systematically?

List outcomes in a clear order so none are missed, or use the product rule to count them. Systematic listing means organising outcomes in order, such as alphabetically or numerically, so none are missed or repeated.

Edexcel GCSE Maths: Revision notes

Key facts

On a number line, the number further right is larger, so $-2 > -5$ .
Multiplying or dividing: same signs give positive, different signs give negative.
When multiplying decimals, count the total decimal places in the question.
BIDMAS gives the order: Brackets, Indices, Division and Multiplication, Addition and Subtraction.
The product rule: multiply the number of options at each choice.

Ordering and comparing

The number further right on the number line is larger, so $-8 < -3$ .

Integers, decimals and fractions can all go on a number line. To compare fractions and decimals, convert them to the same form first, usually decimals.

On the line, $-8$ is further left than $-3$ , so $-8 < -3$ .

Equal or not

= equal to
≠ not equal to

Less or greater

< less than
greater than

Or equal to

≤ less than or equal to
≥ greater than or equal to

Which statement is true?

The four operations

When multiplying or dividing, the signs decide the sign of the answer. When adding, compare the sizes of the numbers.

The four operations work on integers, decimals and fractions, including negatives. Use written methods such as long multiplication, long division and column addition without a calculator.

Multiply or divide Add Same signs Positive Add and keep the sign Different signs Negative Subtract and take the sign of the larger

	Multiply or divide	Add
Same signs	Positive	Add and keep the sign
Different signs	Negative	Subtract and take the sign of the larger

Multiply or divide

Same signs:: Positive
Different signs:: Negative

Add

Same signs:: Add and keep the sign
Different signs:: Subtract and take the sign of the larger

Worked example

Work out $-7 \times -3$ and $-7 \times 3$ .

1
$-7 \times -3$ : the signs are the same, so the answer is positive. $7 \times 3 = 21$ .
2
$-7 \times 3$ : the signs are different, so the answer is negative, $-21$ .

What is $-12 \div 4$ ?

Place value and decimals

Place value tells you what each digit is worth, which keeps big, small and decimal calculations right.

Place value matters for very large numbers, very small numbers such as 0.00042, and for lining up decimal points when you add or subtract.

To multiply decimals, multiply as whole numbers, then count the total decimal places in both numbers and place the decimal point.

1
Ignore the decimal points
3 × 4 = 12
2
Count the decimal places
1 + 2 = 3 in total
3
Place the point
3 places from the right: 0.012

Multiplying decimals: 0.3 × 0.04

Multiplying decimalsmultiply as whole numbers, then count all the decimal places

Worked example

Work out $0.4 \times 0.03$ .

1
Ignore the decimal points: $4 \times 3 = 12$ .
2
0.4 has 1 decimal place and 0.03 has 2, so 3 in total.
3
Place the point 3 places from the right.

Work out $0.2 \times 0.5$ .

BIDMAS and inverse operations

Do brackets first, then indices, then division and multiplication, then addition and subtraction.

BIDMAS gives the order. Work left to right for division and multiplication, and for addition and subtraction.

Inverse operations undo each other and help you check answers: addition and subtraction, multiplication and division, squaring and square rooting.

1
Brackets
first
2
Indices
powers and roots
3
Division and multiplication
left to right
4
Addition and subtraction
left to right

The order of operations

Worked example

Work out $3 + 4 \times 2^2$ .

1
Indices first: $2^2 = 4$ , so $3 + 4 \times 4$ .
2
Multiplication: $4 \times 4 = 16$ , so $3 + 16$ .
3
Addition: $3 + 16$ .

Work out $(5 + 3) \times 2$ .

Counting outcomes

List outcomes in a clear order so none are missed, or use the product rule to count them.

Systematic listing means organising outcomes in order, such as alphabetically or numerically, so none are missed or repeated.

The product rule for counting: if there are $m$ ways to do one thing and $n$ ways to do another independent thing, there are $m \times n$ ways to do both.

1
Choose a starter
3 ways
2
Choose a main
2 ways
3
Multiply
3 × 2 = 6 meals in total

The product rule: 3 starters and 2 mains

$m$ ways and $n$ ways $m \times n$ ways in total

Worked example

A café has 4 sandwich fillings and 3 types of bread. How many different sandwiches are possible?

1
Each filling can be paired with each type of bread.
2
Multiply the options: $4 \times 3$ .

A meal has 3 starters, 5 mains and 2 desserts. How many different three-course meals are there?

Try an exam question

Work out the value of $3 + 4 \times 2^2$ . A meal deal has 4 sandwiches, 3 drinks and 2 snacks. Work out the number of different meal deals.

[4 marks]

That's the notes covered.

Carry on to the next subtopic.