Negative Numbers
Negative numbers are one of those topics where the rules feel arbitrary until they suddenly click. Once you see the logic behind them , especially why a negative times a negative is positive , everything gets a lot easier.
Negative numbers are less than zero. On a number line, they're to the left of zero. Adding a negative is the same as subtracting. Subtracting a negative is the same as adding. Multiplying or dividing two negatives gives a positive. One negative and one positive gives a negative.
The number line and what negative actually means
Numbers extend in both directions from zero. To the right, they get bigger. To the left, they get smaller. Negative numbers are just numbers on the left side of zero.
-7 is further from zero than -3. So -7 is actually less than -3, even though the digit 7 is larger than 3. This trips people up. The further left a number is, the smaller it is. Think of temperature: -20 degrees is colder (smaller) than -5 degrees.
Or think of debt. If you owe $7, you're in a worse financial position than if you owe $3. Your "score" is -7 vs -3. Lower number, worse situation.
Absolute value is the distance from zero, ignoring direction. |−7| = 7 and |7| = 7. Both are 7 steps from zero, just in opposite directions.
Adding and subtracting negatives
The confusion here usually comes from seeing expressions like 8 − (−3) and not knowing what to do with the two minus signs. The trick is to just treat them as operations.
Adding a negative: moving left on the number line. 5 + (−3) = 5 − 3 = 2. You're adding something negative, which is the same as subtracting a positive.
Subtracting a negative: two negatives become a positive. 5 − (−3) = 5 + 3 = 8. Taking away a negative is like a double negative in grammar , it reverses the direction.
Multiplying and dividing: the sign rules
For multiplication and division, the sign of the answer depends only on whether you have an even or odd number of negatives. Two negatives: positive. One negative: negative. Three negatives: negative. Four negatives: positive.
The rules:
Same rules apply to division.
Why does negative times negative equal positive? Here's an intuitive way to see it. Multiplying by −1 reverses direction. So −1 x 5 = −5 (5 gets reversed to −5). Now −1 x (−5): you're reversing −5, which means pointing back in the positive direction. −1 x (−5) = 5. Two reversals bring you back to positive.
Where negative numbers show up in real life
Temperature below zero is the obvious one. In the US, weather rarely dips below −20°F. In Canada or Northern Europe, −40 is a real possibility.
Bank balances go negative when you overdraft. Being $200 overdrawn is a balance of −$200. Depositing $350 brings you to −200 + 350 = $150 positive.
Elevation: Death Valley in California sits at about −86 meters , 86 meters below sea level. The difference in elevation between Death Valley and a nearby mountain at 3,368 meters is 3,368 − (−86) = 3,454 meters. Subtracting a negative adds it.
Profit and loss: a quarterly business loss of $12,000 is −$12,000. A following quarter gain of $18,000 gives a net of −12,000 + 18,000 = +$6,000.
The mistakes people make
Not applying the subtraction to the entire number. If you have −x − 7 = 12 and you add 7 to both sides, you get −x = 19, not x = 19. The negative in front of x is still there , you need one more step, dividing by −1, to get x = −19.
Confusing the sign of a number with the operation of subtraction. The 7 in −7 is a negative number. The minus in 10 − 7 is an operation. They look the same but mean different things. Context usually makes it clear, but pay attention.
Forgetting that in inequalities, multiplying or dividing by a negative flips the inequality sign. This doesn't apply to equations , −3x = 12 just gives x = −4 with no flipping. But −3x > 12 gives x < −4. Different rules for equations and inequalities on this point.