Directed Numbers: Making Sense of Negative Numbers

Imagine it's a bitter winter morning. The thermometer reads 3 °C. Overnight it drops by 7 degrees. What temperature is it now? You might feel that something strange happens when you subtract a bigger number from a smaller one — the answer goes 'below zero'. That's exactly what this lesson is about. Think about where you've seen numbers below zero in real life. A bank account overdrawn by £20? A car park on Level −2? A diver 5 metres below the surface? Jot down one example you can think of.

An integer is any whole number — positive, negative, or zero. No fractions, no decimals. Examples: …−3, −2, −1, 0, 1, 2, 3… A directed number is any number that has both a size and a direction — either above zero (positive) or below zero (negative). All negative integers are directed numbers, but so are positive ones. The sign in front tells you the direction. The number line is your best tool. Zero sits in the middle. Positive integers run to the right, negative integers run to the left. The further left you go, the smaller the number. So −5 is less than −1, even though 5 is bigger than 1 — this surprises many people. Comparing integers: use the symbols < (less than) and > (greater than). On a number line, the number further to the right is always greater. For example, −3 < 2 because −3 is to the left of 2. Adding and subtracting directed numbers — two key rules: 1. Adding a negative number is the same as subtracting its positive value. Example: 4 + (−3) = 4 − 3 = 1. 2. Subtracting a negative number is the same as adding its positive value. Example: 4 − (−3) = 4 + 3 = 7. A helpful way to remember: two negatives written next to each other (like − −) combine to make a positive (+).