Integers and Order of Operations

An integer is a whole number. It can be positive ($3$, $58$), negative ($-3$, $-58$) or zero — but it is never a fraction or a decimal.

You already meet negative integers all the time. A game score can drop to $-20$. A cold morning might be $-3\degree\text{C}$. If you owe a friend $5, your "money" is $-5$.

Picture a number line with zero in the middle, positive integers going right and negative integers going left.

The further LEFT you go, the smaller the number — so $-7$ is less than $-3$, even though 7 looks bigger than 3. To compare integers, just ask which sits further right.

The number line makes adding and subtracting easy to picture.

• Adding a positive number moves you RIGHT.
• Subtracting a positive number moves you LEFT.

So if the temperature is $-3\degree\text{C}$ and it rises by $5\degree$, you start at $-3$ and move 5 steps right: the answer is $2\degree\text{C}$.

Two signs next to each other combine into one:

• A plus and a minus together make a minus: $6 + (-4) = 6 - 4 = 2$.
• Two minuses together make a plus: $7 - (-2) = 7 + 2 = 9$.

It helps to read $-(-2)$ as "the opposite of $-2$", which is $+2$. That is why subtracting a negative makes a number bigger.

Multiplying and dividing integers is quick once you know the sign rule. Forget the number line here — just look at the two signs.

• Same signs (both positive or both negative) give a positive answer.
• Different signs (one of each) give a negative answer.

So $-4 \times -3 = 12$ (same signs), but $-4 \times 3 = -12$ (different signs). The exact same rule works for division: $-20 \div -5 = 4$, while $-20 \div 5 = -4$.

When more than two numbers are multiplied, count the negative signs. An even number of negatives gives a positive; an odd number gives a negative. For example, $-2 \times -3 \times -1 = -6$ — three negatives, so the answer is negative.

What is $2 + 3 \times 4$? If you work left to right you get 20. But the correct answer is 14. Maths has a fixed running order so everyone gets the same answer.

That order is BIDMAS:

1. Brackets
2. Indices (powers and roots)
3. Division and Multiplication — left to right
4. Addition and Subtraction — left to right

So in $2 + 3 \times 4$, the multiplication happens first: $3 \times 4 = 12$, then $2 + 12 = 14$.

Brackets let you change the order on purpose. $(2 + 3) \times 4 = 5 \times 4 = 20$.

Worked example — $20 - 2 \times (3 + 1)^2$:

• Brackets first: $3 + 1 = 4$.
• Indices next: $4^2 = 16$.
• Multiplication: $2 \times 16 = 32$.
• Subtraction last: $20 - 32 = -12$.

Most calculations combine both ideas: negative numbers and an order to follow. The trick is to slow down and do one step at a time, writing the line out again each time.

Try $5 + (-3) \times 4$. The multiplication comes first, and it has different signs:

• $(-3) \times 4 = -12$.
• Then $5 + (-12) = 5 - 12 = -7$.

Now a harder one — $(-2)^3 + 6 \div (-3)$:

• Indices first: $(-2)^3 = -2 \times -2 \times -2 = -8$ (three negatives, so negative).
• Division next: $6 \div (-3) = -2$ (different signs).
• Addition last: $-8 + (-2) = -10$.

A neat habit is to deal with each pair of signs before you do the arithmetic, so a line like $7 - (-2) + (-5)$ becomes the much friendlier $7 + 2 - 5 = 4$.

When you finish, glance at your answer and ask "does this feel sensible?" If you expected a small negative and got a huge positive, a sign has probably slipped.

Getting confident with integers and the order of operations now makes everything ahead easier:

• Algebra uses negative numbers and BIDMAS every single time you substitute a value or simplify an expression.
• Coordinates and graphs stretch the number line into two directions, so negative values appear on both axes.
• Geometry and measure often produces negative differences — for instance when comparing temperatures or heights above and below sea level.
• In everyday life, you will use these skills for bank balances, game scores, temperatures and any calculator question with more than one step.

If a topic later feels hard, 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 — there is no prize for rushing.