Number Toolkit Notes

Q: How do you add, subtract, multiply and divide negative numbers?

Two negative signs together make a positive. Adding a negative is subtracting: 5 + (-3) = 2. Subtracting a negative is adding: 5 - (-3) = 8. Multiplying or dividing the same signs gives a positive: (-4) \times (-2) = 8. Different signs give a negative: (-4) \times 2 = -8.

Edexcel IGCSE Maths: Revision notes

Key facts

An integer is a whole number: positive, negative or zero.
Two negative signs together make a positive; same signs when multiplying or dividing give a positive.
HCF is the largest shared factor; LCM is the smallest shared multiple.
1 is not prime; 2 is the only even prime.
Square numbers are $n^2$ , cube numbers $n^3$ ; roots reverse them.
BIDMAS: Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right).

Integers and place value

Integers are whole numbers, and a digit's value depends on its position.

An integer is any whole number: positive, negative or zero. Place value tells you what a digit is worth. In $4207$ , the 4 is worth 4000, the 2 is 200, the 0 is 0 tens and the 7 is 7. Values increase left to right on a number line, so negatives get smaller as their size increases: $-8 < -3 < 0 < 5$ .

Values increase to the right: −8 < −3 < 0 < 5

$<$

less than

$>$

greater than

$\leq$

less than or equal to

$\geq$

greater than or equal to

Which is the smallest: $-4$ , $-40$ , $0$ or $3$ ?

Negative numbers

Two negative signs together make a positive.

Adding a negative is subtracting: $5 + (-3) = 2$ . Subtracting a negative is adding: $5 - (-3) = 8$ . Multiplying or dividing the same signs gives a positive: $(-4) \times (-2) = 8$ . Different signs give a negative: $(-4) \times 2 = -8$ .

Subtracting a negativeis adding: $5 - (-3) = 8$
Same signs, multiply or dividepositive: $(-4) \times (-2) = 8$
Different signsnegative: $(-4) \times 2 = -8$

Worked example

Work out $-7 - (-10)$ .

1
Rewrite the double sign: $-7 + 10$ .
2
Calculate: $10 - 7 = 3$ .

Work out $(-6) \times (-3)$ .

Factors, multiples and primes

A factor divides exactly into a number; a prime has exactly two factors.

A factor divides into a number exactly. A multiple is the number multiplied by an integer. Factors of 12: 1, 2, 3, 4, 6, 12; multiples of 12: 12, 24, 36, 48, ... A prime number has exactly two factors, 1 and itself (2, 3, 5, 7, 11, 13). 1 is not prime and 2 is the only even prime. Prime factorisation writes a number as a product of primes, for example $60 = 2^2 \times 3 \times 5$ .

Factor tree for 60: the prime ends give 2 × 2 × 3 × 5

HCF

Largest number that divides into both

LCM

Smallest number that is a multiple of both

Which of these is a prime number?

Squares, cubes and roots

Squaring multiplies a number by itself; roots reverse powers.

A square number is $n^2$ : 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. A cube number is $n^3$ : 1, 8, 27, 64, 125. The square root reverses squaring ( $\sqrt{49} = 7$ ) and the cube root reverses cubing ( $\sqrt[3]{27} = 3$ ). The symbol $\sqrt{25}$ means the positive root, 5, but $x^2 = 25$ has two solutions, $x = 5$ and $x = -5$ .

y = x²: the square numbers are the heights at x = 1, 2, 3 ...

What is $\sqrt[3]{125}$ ?

Order of operations

Use BIDMAS: brackets, indices, then division and multiplication, then addition and subtraction.

Use BIDMAS. Division and multiplication have equal priority, so work left to right. The same applies to addition and subtraction.

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

Order of operations

Worked example

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

1
Indices first: $2^2 = 4$ .
2
Multiplication: $4 \times 4 = 16$ .
3
Addition: $3 + 16 = 19$ .

Work out $6 + 12 \div 3 \times 2$ .

Try an exam question

Work out $3 + 4 \times (7 - 5)^2$ and write 84 as a product of prime factors.

[4 marks]

That's the notes covered.

Carry on to the next subtopic.