−n · Negative Numbers
Below zero, where numbers gained a direction — the negative.
It was 3°C and the temperature dropped 5 degrees. What is it now — what is 3 − 5?
Read position and direction.
The numbers mathematicians once rejected as absurd and impossible.
Open the immersive version →A reading path containing this lesson: Beyond familiar numbers · 1/3
Why does this question matter?
With naturals and fractions alone, 'take 5 from 3' had no answer. Yet the world kept asking — freezing cold, basement floors, debts to repay.
The fix: extend the number line, which used to stop at 0, to the left. Call the number one below zero −1, two below −2 — and 3 − 5 = −2 gives every subtraction an answer.
As a bonus, numbers acquired direction: + to the right (up, gain), − to the left (down, loss). Temperature, altitude, balance — worlds spreading both ways from a zero point fit into one number system.
Pause and try it yourself
Make a prediction first. Change one input at a time and compare what happens with what you expected.
Predict first — it's 3°C and the temperature drops 5 degrees. What is 3 − 5?
Move the thermometer
Starting at 3°C. What happens when you pass below zero?
3℃
current temperature
Read the idea from another angle
The heart of a negative is direction, not size.
−3 means '3, the other way'. Once direction rides on numbers, subtraction becomes 'moving the opposite way' — and the seed of vectors is planted.
Zero is not the end — it's the middle.
Accepting negatives changes zero's status: no longer the left end of the line but its center. Seeing symmetric sides of a reference point is where coordinates and graphs begin.
Now read the definitions with their conditions
What the thermometer showed, in mathematical language.
Defining negatives
The number n below zero is −n. Extend the line leftward and every subtraction gains an answer.
Comparing
Further left is smaller. Among negatives, larger absolute value means a smaller number — minus five is colder than minus two.
Multiplying signs
×(−1) is a direction flip. Two flips return you home — (−)×(−)=+ is the logic of flipping, not memorization.
Make one more distinction
−5 is greater than −2 — because 5 beats 2.
On the number line −5 sits left of −2 — smaller. Minus five degrees is colder than minus two; a $5 debt is poorer than a $2 debt. Among negatives, the bigger the absolute value, the smaller the number.
Negative times negative is positive 'just because' — a rule to memorize.
Multiplying by (−1) is a flip of direction. Flip once: reversed (−). Flip twice: back where you started (+) — like a mirror image of a mirror image. Not a rule but the logic of flipping.
Where else can this idea help?
Weather
−12°C on the winter forecast. Changes too: 'seven degrees colder than yesterday' is written −7.
Bank balances and debt
A balance of −$50 means $50 owed. Accounting's 'in the red' comes from writing negatives in red ink.
Elevators and sea level
Basement 3 = floor −3; 200m under the sea = elevation −200m. Negatives are the language for below-the-reference.
Sports scores
Golf's −7 (under par) means seven strokes fewer than par — in golf, negative is good. A goal difference of −2 uses the same grammar.
Explain it in your own words
Try another question
Q1Which is greater: −3 or −7?Show answer ▾
−3 — it sits right of −7, closer to zero. Just as minus three degrees is warmer than minus seven.
Q2What is (−2) + 5? Think on the thermometer.Show answer ▾
From 2 below zero, rise 5 degrees: 3 above — (−2) + 5 = 3. Adding with negatives never confuses when seen as movement on the line.
Q3Why is (−3) × (−4) equal to +12?Show answer ▾
It's 3 × 4 = 12 with two flips — two factors of (−1) flip the direction twice, landing back at +. Odd flips give −, even flips give +.
Try answering yourself before revealing it — getting it wrong is where learning starts.
What question comes next?
A reading path containing this lesson: Beyond familiar numbers · 1/3
Leads to next
Fractions
Below zero is filled; now the gap between 0 and 1 — fractions fill in the rest of the line.
Go to the Fractions lab →Related lab
Fractions
Below one (fractions) and below zero (negatives) — numbers expanded in two directions.
Go to the Fractions lab →Related lab
Vectors
'Size plus direction' — grow the negative's idea into two dimensions and you get vectors.
Go to the Vectors lab →