Negative Numbers: Rules for Every Operation

Placing Negative Numbers on the Number Line

You've seen a number line before. Zero in the middle, positives to the right, negatives to the left. Every negative number is the mirror image of a positive: −4 is exactly as far from zero as +4, just pointing the other way. Think of it like floors in a building. Ground floor is zero, the floors above are positive, and the basement levels below are negative.

A couple of things follow from this:

• Numbers get larger as you move right. So −1 is greater than −5, even though 1 is less than 5. Among negatives, the one with the smaller absolute value is the larger number.
• The absolute value of a number is its distance from zero, always expressed as a positive: |−7| = 7, |3| = 3. Absolute value strips away the sign and returns only the magnitude.

Addition means moving right on the number line, subtraction means moving left. Keep that image in your head because it makes everything else with negatives much easier to follow.

Adding Negative Numbers

Adding a negative number moves you left on the number line, which is the same direction as subtraction. So adding a negative produces the same result as subtracting the matching positive.

When you're adding two negatives, add their absolute values and attach a negative sign to the result. They can't cancel each other out. You're moving further left both times.

When you've got opposite signs, subtract the smaller absolute value from the larger one, and keep the sign of whichever number is further from zero.

Real-world version: your account is £30 overdrawn, so the balance is −£30. You deposit £20. New balance: −30 + 20 = −£10. The deposit helped, but you're still in the red.

Subtracting a Negative Number

Subtracting a negative is the one that throws most people off. The rule: subtracting a negative is the same as adding the positive. Two negatives in a row, the subtraction sign and the number's negative sign, flip into an addition.

Think about it this way: subtraction means removing something. If you remove a negative thing, a debt for example, you're better off. Your bank account is −£8. Someone cancels £12 of debt on your behalf. You're now +£4. Taking away a loss is a gain.

On the number line, subtracting a negative reverses your direction. Instead of going left, you go right, which means you end up at a bigger number. Same as adding.

Multiplying Two Negatives

With multiplication, it all comes down to whether the signs match or not.

That last row, negative times negative equals positive, is the one that always raises eyebrows. Here's a pattern that shows why it has to be true.

Look at what happens when you multiply increasing integers by −3:

Each time the left factor goes up by 1, the product increases by 3. So from 0 × (−3) = 0, the next step has to be +3, then +6. The pattern forces it. It's not an arbitrary rule, it's what has to happen to keep arithmetic consistent.

Here's a memory trick: think of "negative" as meaning "opposite direction." One flip gives you a negative. Two flips bring you back to positive. Like turning around twice, you end up facing the same way you started.

Dividing Negatives and Real-World Contexts

Division uses the exact same sign rules as multiplication. Do the division with the absolute values, then figure out the sign based on whether the two numbers match or not.

Not sure what sign to expect? Count the negatives. An even number of negatives in a chain gives a positive result. An odd number of negatives gives a negative. That's all you need.

Negative Numbers in Everyday Life

In every one of those cases, negative isn't a problem, it's information. The sign tells you which side of a reference point you're on: zero degrees, sea level, an empty bank account, even par. Negative just means "on the other side."