🔢 Number Kingdom · Number Sense
Negative Numbers
Extend the number line to the left of zero and compute with signed numbers. Negatives measure debt, cold, and depth below sea level.
In short
- The number line runs both ways; further right is always greater, so -9 < -2.
- Subtracting is adding the opposite, which is why subtracting a negative moves you right.
- For multiplying and dividing: same signs give a positive, different signs give a negative.
- Distance between two integers is always positive, and numbers on opposite sides of zero have their distances added.
The line continues past zero
Counting numbers stop at 0 only because we stop drawing. Extend the number line to the left and you get the negative integers: ... -3, -2, -1, 0, 1, 2, 3 ...
Negative numbers are not strange or imaginary. They measure real things that go below a natural zero:
- temperature below freezing (-8 degrees)
- depth below sea level (-120 m)
- money owed (-45 gold)
- floors below ground (-2)
The zero is a chosen starting point, and the minus sign says "on the other side of it".
Comparing integers
The rule never changes: further right is greater.
That makes -9 less than -2, even though 9 is greater than 2. A debt of 9 gold is worse than a debt of 2 gold; -9 degrees is colder than -2 degrees.
- Every negative number is less than every positive number.
- 0 is greater than every negative and less than every positive.
- Among negatives, the one with the larger digits is the smaller number.
Adding and subtracting
Think of movement along the line. Start at the first number; the second number tells you how far and which way to move.
- Adding a positive moves right: -5 + 3 = -2.
- Adding a negative moves left: -5 + (-3) = -8.
- Subtracting is adding the opposite: a - b = a + (-b).
That last rule explains the one that looks like a trick. 7 - (-4) = 7 + 4 = 11: subtracting a negative moves you right. Removing a debt makes you richer.
When the signs are the same, add the sizes and keep the sign. When the signs differ, subtract the smaller size from the larger and keep the sign of the larger.
Multiplying and dividing
Do the sizes first, then decide the sign separately.
- Same signs give a positive result. 4 x 5 = 20 and (-4) x (-5) = 20.
- Different signs give a negative result. (-4) x 5 = -20 and 4 x (-5) = -20.
The same rule applies to division, because division is multiplication in reverse.
Why does a negative times a negative come out positive? Look at the pattern: (-4) x 3 = -12, (-4) x 2 = -8, (-4) x 1 = -4, (-4) x 0 = 0. Each step up adds 4, so (-4) x (-1) must be 4.
Distance is never negative
The distance between two integers is the gap between them on the line, and a gap is never negative.
From -30 m to 45 m the distance is 45 - (-30) = 75 m, not 15 m. When two positions are on opposite sides of zero, their distances from zero add together.
A quick check: draw a rough line, mark 0, and put both numbers on it. If your answer is smaller than the distance from one of them to 0 alone, something has gone wrong.
Worked examples
Example 1
-14 - (-9) = ?
- Rewrite the subtraction as adding the opposite: -14 + 9.
- The signs differ, so subtract the sizes: 14 - 9 = 5.
- The larger size belongs to -14, so the answer keeps the minus sign.
- -14 - (-9) = -5.
- Check on the number line: from -14, moving 9 to the right lands on -5.
Example 2
At dawn the temperature is -12 degrees. By noon it has risen 19 degrees. What is the temperature at noon?
- Rising means moving up (right) along the thermometer: -12 + 19.
- It takes 12 degrees just to climb from -12 up to 0.
- 19 - 12 = 7 degrees are left after reaching 0.
- The temperature at noon is 7 degrees.
Practice problems, with solutions
Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.
Problem 1
Difficulty 1 of 5Which symbol makes this true? -6 __ -5
- <
- >
- =
Answer: A. <
- Place both numbers on the number line: -6 is further left than -5.
- So -6 < -5.
Problem 2
Difficulty 3 of 5-24 - (-29) = ?
Answer: 5
- -24 - (-29) = -24 + 29 (subtracting is adding the opposite)
- = 5
Problem 3
Difficulty 4 of 5(-24) x (-9) = ?
Answer: 216
- Ignore the signs: 24 x 9 = 216
- The signs are the same, so the answer is positive.
- (-24) x (-9) = 216
Common mistakes
- Ordering negatives by size alone and claiming -9 > -2.
- Getting the size of an answer right but the sign wrong — the classic sign error.
- Treating "minus a minus" as another minus: writing 7 - (-4) = 3.
- Dropping the minus sign from a starting value in a word problem.
What you should be able to do
- Compare and order integers on a number line.
- Add and subtract integers, including subtracting a negative.
- Multiply and divide integers and predict the sign of the result.
- Model temperature, elevation and debt problems with integers.
Where this fits in the curriculum
Common Core
- 6.NS.C.5
Grade 6 — Understand that positive and negative numbers describe quantities with opposite directions.
- 6.NS.C.7
Grade 6 — Order and compare rational numbers, and interpret them on a number line.
- 7.NS.A.1
Grade 7 — Add and subtract rational numbers, including subtracting a negative as adding its opposite.
- 7.NS.A.2
Grade 7 — Multiply and divide rational numbers and know the rules for the sign of the result.
Ontario
- G6.B1.2
Grade 6 — Read, represent, compare and order integers, using a variety of tools including number lines.
- G6.B1.3
Grade 6 — Compare and order integers, decimal numbers and fractions together.
- G8.B2.4
Grade 8 — Add and subtract integers, using appropriate strategies, in various contexts.
- G8.B2.7
Grade 8 — Multiply and divide integers, using appropriate strategies, in various contexts.
SAT
- Heart of Algebra
Signed-number arithmetic inside linear equations and inequalities.
Negative numbers are never the question on the SAT; they are the arithmetic every algebra question is written in.