You're likely comfortable with adding and subtracting. But things get more interesting when negative numbers join the party. Think of a number line. Moving right is addition; moving left is subtraction. Adding a negative number is the same as moving left (subtracting). Subtracting a negative number is like cancelling a debt – you actually move right (add).

Imagine your bank account has £50. You spend £70 on a new game. Your new balance is £50 - £70 = -£20. You're overdrawn. Later, you return an item and get a £15 credit. Your balance becomes -£20 + £15 = -£5. A number line helps visualise these financial ups and downs as movements back and forth.

The rules for multiplying and dividing signed numbers are straightforward. The key is to look at the signs. If the signs are the same, the result is always positive. If the signs are different, the result is always negative. This applies to both multiplication and division.

Why does a negative times a negative make a positive? Think of it as removing a debt. If someone removes 3 debts of £4 from you ($-4$), your net worth has effectively increased by £12. So, $-3 \times -4 = 12$.

The absolute value of a number is its distance from zero on the number line. Since distance can't be negative, the absolute value is always positive or zero. We denote it with two vertical bars. For example, $|-7| = 7$ and $|7| = 7$. It simply tells you the magnitude of the number, ignoring its direction (positive or negative).

Now, let's look at a powerful tool for algebra: the distributive property. This rule lets us multiply a number by a sum or difference. It states that multiplying a number by a group of numbers added together is the same as doing each multiplication separately.

This works for all rational numbers—integers, fractions, and decimals. Let's see it in action: $-3(4 + 0.5) = (-3 \times 4) + (-3 \times 0.5) = -12 + (-1.5) = -13.5$

This property is the bedrock of simplifying algebraic expressions.

Let's check your understanding of these concepts.

Mastering these operations is a key step from arithmetic towards the more abstract world of algebra.