Positive & Negative Numbers

The forecast says "minus three degrees". The elevator panel shows B2. A game displays your balance as −500. These are numbers smaller than zero: negative numbers.

Numbers used to march one way. With negatives, the number line that place value built can finally stretch to the left of zero — and quantities that move the opposite way, like temperature, floors and debt, finally have somewhere to stand.

Negative numbers in daily life

Negative numbers record quantities heading the opposite way:

Temperature: 5 above zero is $5$ ; 3 below zero is $-3$ ;
Floors: 3 floors up is floor 3; 2 floors down is $-2$ ;
Money: deposit 100 and it is $+100$ ; owe 100 and it is $-100$ .

The sign is a direction sign: first agree on which way counts as positive, and the other way gets a minus.

The number line grows to the left

Extend the line left of 0 and out come $-1, -2, -3, \ldots$ Every positive number has an opposite: $3$ and $-3$ sit the same distance from 0, one on each side.

That distance to 0 is called absolute value: $|-3| = 3$ and $|3| = 3$ . Absolute value answers "how far"; the sign answers "which way".

Comparing: farther from zero is smaller

On the line, right beats left. In negative territory intuition flips: $-3$ sits to the right of $-8$ , so $-3 > -8$ .

Common trap

Since 8 beats 3, you scribble $-8 > -3$ ? Wrong. Compare negatives through absolute value: the bigger the absolute value, the smaller the number, and the negative closest to 0 is the largest. Three below zero is warmer than eight below — same idea.

Adding: walk along the line

Adding a positive moves right; adding a negative moves left.

Example: 5 + (−8)

Start at 5 and walk 8 steps left: 5 steps reach 0, then 3 more land on $-3$ . So $5 + (-8) = -3$ . Subtraction draws the same picture: $2 - 5 = 2 + (-5) = -3$ — subtracting a number equals adding its opposite (another way to see subtraction).

Example: −6 + 4

Start at $-6$ and walk 4 steps right: $-6 \to -5 \to -4 \to -3 \to -2$ . When the signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger: $|-6| = 6 > 4 = |4|$ , so the answer is negative.

Multiplying: minus times minus

The sign rules are tidy:

positive × positive = positive: $3 \times 2 = 6$
positive × negative = negative: $3 \times (-2) = -6$
negative × positive = negative: $(-3) \times 2 = -6$
negative × negative = positive: $(-3) \times (-2) = 6$

"Minus times minus" is not an arbitrary rule — watch the pattern: $3 \times 3 = 9$ , $3 \times 2 = 6$ , $3 \times 1 = 3$ , $3 \times 0 = 0$ ; each time the multiplier drops by 1, the product drops by 3. Keep going: $3 \times (-1) = -3$ , $3 \times (-2) = -6$ . Now a new row: $(-3) \times 3 = -9$ , $(-3) \times 2 = -6$ , $(-3) \times 1 = -3$ , $(-3) \times 0 = 0$ — here each step the product grows by 3, so $(-3) \times (-1) = 3$ and $(-3) \times (-2) = 6$ . The pattern holds all the way: minus times minus is positive.

Check yourself

Quick quiz

1. Which is bigger, −3 or −8?
2. −2 + (−5) = ?
3. (−4) × (−5) = ?