Integers

An integer is any whole number — positive, negative or zero. So $-9$, $0$, $7$ and $1{,}248$ are all integers, but $\tfrac{1}{2}$ and $3.7$ are not.

The number line is the picture that ties them together. Zero sits in the middle, positive integers stretch off to the right and negative integers stretch off to the left. The further right you go the larger the number, and the further left the smaller. So $-5 < -2 < 0 < 3 < 8$ at a glance.

Two ideas about signs come straight from the picture. The opposite of $7$ is $-7$ — it sits the same distance from zero but on the other side. The absolute value of a number is how far it is from zero, ignoring the sign, so $|-6| = 6$ and $|6| = 6$. Reading the line confidently is what makes everything else easier.

Adding and subtracting integers becomes simple once you learn to tidy double signs. Whenever two operation signs sit next to each other, they combine into one:

• $+\,+\,=\,+$ and $-\,-\,=\,+$ (same signs → plus).
• $+\,-\,=\,-$ and $-\,+\,=\,-$ (different signs → minus).

So $8 + (-5)$ becomes $8 - 5 = 3$, and $4 - (-6)$ becomes $4 + 6 = 10$. Tidy the signs first, then add or subtract as normal, working left to right.

For a longer chain like $-3 + 7 - (-2) + (-5)$, deal with each pair of signs and then sweep left to right: $-3 + 7 + 2 - 5 = 1$. Picturing the number line helps when you're learning — adding moves you right, subtracting moves you left — but with practice the rules run on their own.

For multiplying and dividing integers, only the signs of the two numbers matter for the sign of the answer. Look at one pair at a time:

• Same signs → positive answer.
• Different signs → negative answer.

So $-6 \times -4 = +24$, but $-6 \times 4 = -24$ and $24 \div (-6) = -4$. Work out the size of the answer as if both numbers were positive, then attach the right sign.

For a longer product, the quickest trick is to count the negatives. If the count is even, the answer is positive. If the count is odd, the answer is negative.

So $-2 \times -3 \times -1$ has three negatives (odd) → answer is negative: $-6$. And $-1 \times -2 \times -3 \times -5$ has four negatives (even) → answer is positive: $+30$. Decide the sign first, then multiply the digits — fewer slips that way.

When a calculation has several operations, there is one agreed order — BIDMAS:

• Brackets
• Indices (powers and roots)
• Division and Multiplication (equal rank, left to right)
• Addition and Subtraction (equal rank, left to right)

So in $5 + 2 \times 3^2$ you do the index first ($3^2 = 9$), then the multiplication ($2 \times 9 = 18$), then the addition ($5 + 18 = 23$). The answer is not $63$.

Two warnings worth knowing by heart. Division and multiplication rank equally, so $36 \div 6 \times 3$ is worked left to right: $36 \div 6 = 6$, then $6 \times 3 = 18$ — not $36 \div 18 = 2$. The same applies to addition and subtraction. And a fraction bar acts like a hidden bracket — in $\dfrac{8 + 4}{3}$ finish the top first, then divide, giving $4$.

When brackets are nested inside other brackets, peel them from the innermost outwards: $50 - 3 \times (12 - (5 + 2))^2$ becomes $50 - 3 \times (12 - 7)^2 = 50 - 3 \times 5^2 = 50 - 75 = -25$. One step per line — that is the whole game.

Many real-life situations need signed numbers. The skill is deciding what a negative number means in context, and which operation the question is really asking for.

Temperatures are a classic. A reading of $-6\degree\text{C}$ rises by $11\degree$ — that is $-6 + 11 = 5\degree\text{C}$. The difference between $-6\degree\text{C}$ and $5\degree\text{C}$ is $5 - (-6) = 11\degree$, a subtraction of one reading from the other.

The same care helps with bank balances, heights above and below sea level and game scores. A balance of -\40$that receives a$$25$payment becomes$-40 + 25 = -$15$— still overdrawn. A diver at$-18$m who rises$7$m is at$-18 + 7 = -11$ m. Read slowly, decide what the sign means, then choose the operation with confidence.

Solid integer and order-of-operations skills make a huge amount of later maths easier:

• Algebra substitutes negative values into expressions and formulae constantly — $2x - 5$ when $x = -3$ is a sign question in disguise.
• Coordinate geometry uses negative coordinates on both axes, and gradients are differences that can be negative.
• Powers and indices rely on the same sign rules, especially when the base is negative.
• In everyday life, signed calculations describe temperatures, bank balances, elevations and any multi-step calculator problem.

If a later topic feels shaky, it is often a sign rule or a BIDMAS step underneath that needs a quick refresh. Come back to this guide whenever you need it — accuracy beats speed every time.