Negative numbers & the number line
The idea
Zero is not a wall — the number line keeps going to the left, and the numbers there carry a minus sign: −1, −2, −3, and everything between. You already know addition, subtraction, and how to place numbers on a line, and negatives reuse all of it. They show up wherever a quantity has two natural directions: temperatures below freezing, floors below ground level, money owed instead of owned, yards lost instead of gained.
Read every calculation as motion: adding moves you right, subtracting moves you left, and a negative sign flips the direction of the move. The widespread trap is ranking negatives by their digits, deciding −10 must beat −3 because 10 beats 3. It is the reverse: −10 sits farther left, so −10 < −3 — a debt of $10 leaves you worse off than a debt of $3. When comparing, picture positions, not digits.
Worked example
At dawn the temperature is −7 °C. By mid-afternoon it has risen 12 °C, and after sunset it falls 15 °C from the afternoon reading. What is the night temperature, and is it colder than dawn?
- Start at −7 on the number line and move 12 to the right for the afternoon rise: −7 + 12 = 5 °C.
- From 5, the evening fall moves 15 to the left: 5 − 15 = −10 °C, crossing zero along the way.
- Compare the two below-zero readings by position: −10 lies farther left than −7, so −10 < −7 and the night is colder than dawn.
- Sanity-check the day as one trip: the net change was +12 − 15 = −3, and −7 − 3 = −10 °C, which agrees.
Answer. The night temperature is −10 °C, which is 3 degrees colder than the −7 °C dawn.
Check your understanding
- Why is −10 smaller than −3 even though 10 is bigger than 3, and how would you convince a skeptical friend?
- What real situations would lose meaning if negative numbers did not exist — what would we have to say instead?
- How does picturing motion on a number line predict whether an answer lands above or below zero?
- What is the relationship between a number and its opposite, and where do the two always sit on the line?
Build the foundations first
Negative numbers & the number line builds on these concepts. If any feel shaky, start there.