You're already familiar with arithmetic using positive whole numbers, fractions, and decimals. Now, let's expand our toolkit to include their negative counterparts. Numbers that can be written as a fraction, including positive and negative integers, fractions, and decimals, are called rational numbers.

Thinking about a thermometer is a great way to visualise this. Zero isn't the bottom. Temperatures can drop below freezing, into negative values. Similarly, sea level is a zero point, but divers can go to negative depths, and mountains can reach positive heights. These negative numbers follow specific rules when we add, subtract, multiply, and divide them.

Adding a positive number means moving right on the number line. Adding a negative number means moving left.

For example, to solve $5 + (-8)$, you'd start at 5 and move 8 units to the left, landing on -3.

When it comes to subtraction, the simplest approach is to reframe the problem. Subtracting a number is the same as adding its opposite. This opposite is called the additive inverse.

This trick makes subtraction much simpler. For instance, $-4 - 3$ becomes $-4 + (-3)$, which equals $-7$. A tricky one like $6 - (-2)$ becomes $6 + 2$, which is simply 8. By turning every subtraction into an addition problem, you only need to remember one set of rules.

This concept of opposites is also key to understanding distance on a number line. The distance between two numbers, $a$ and $b$, is the absolute value of their difference: $|a - b|$. Since absolute value is always positive, it doesn't matter if you calculate $|a - b|$ or $|b - a|$; the distance is the same.

The rules for multiplying and dividing rational numbers are straightforward and apply whether you're working with integers, fractions, or decimals. It all comes down to the signs.

A simple way to remember this: if the signs are the same, the result is positive. If the signs are different, the result is negative.

Just as we transformed subtraction into addition, we can transform division into multiplication. Division by a number is the same as multiplying by its reciprocal (or multiplicative inverse).

This is especially useful for fractions. For example, to solve $\frac{2}{3} \div (-\frac{1}{4})$, you would instead calculate $\frac{2}{3} \times (-\frac{4}{1})$. The result is $-\frac{8}{3}$. This technique is also the key to simplifying complex fractions.

These rules aren't just abstract math; they describe how the world works. Let's look at an example.

A submarine is at a depth of -150 metres. It then descends another 80 metres. Its new position is found by adding the change: $-150 + (-80) = -230$ metres.

Later, the submarine needs to rise to one-third of its current depth. To find the new depth, we multiply: $-230 \times \frac{1}{3} \approx -76.67$ metres. The rules for negative numbers give us a precise answer.

Now, let's test your understanding of these concepts.

Mastering these operations is a crucial step that prepares you for more advanced topics in algebra and beyond.