The integers are the whole numbers together with their negatives. Before we let letters into mathematics, we make sure every signed number behaves — including the one rule that surprises everyone.
You have already met negative numbers back in Stage 1, so the line that runs below zero is not new. What this lesson does is tighten those ideas into a single tidy family — the integers — and then add the piece that finally opens the door to algebra: the sign rules for multiplying and dividing. Get these solid now, and the letters that arrive later will feel like old friends.
The integers, in order
The integers are the counting numbers, their opposites, and zero — written out, they march off in both directions forever:
\[ \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots \]That is the whole story: the whole numbers and their opposites, with no fractions in between. There is no integer sitting between \( 1 \) and \( 2 \); the family skips straight from one whole step to the next. The number line is what puts them in order — anything to the left is smaller, anything to the right is larger. So, reading along the line,
\[ -5 < -2 < 0 < 3. \]Drag the marker below along the line and watch the value change — notice how moving left always lowers the number, even once you are below zero.
Absolute value: distance from zero
Sometimes we care only about how far a number is from zero, not which side it lies on. That measurement is its absolute value, written with two upright bars: \( |x| \) is how far \( x \) sits from zero, ignoring direction. Because it is a distance, it is always zero or positive — you can never be a negative number of steps away from home.
For example, \( |{-5}| = 5 \) and \( |3| = 3 \). The first counts five steps to the left of zero, the second three steps to the right, and absolute value reports each as a plain positive distance. A neat consequence: two opposite numbers share the same absolute value, since they sit the same distance from zero on either side.
In words \( |x| \) is a distance, so it can never be negative. That is why \( |{-9}| \) and \( |9| \) are both \( 9 \) — they live the same number of steps from zero, just on opposite sides.
Adding and subtracting (a quick recap)
You met these moves with negative numbers already, so here is the short version, all in one place. When the two numbers have the same sign, add their sizes and keep that shared sign:
\[ -3 + (-4) = -7. \]When they have different signs, subtract the smaller size from the larger and take the sign of the larger:
\[ -3 + 7 = 4. \]And subtraction never needs its own rulebook — subtracting is just adding the opposite:
\[ 5 - 8 = 5 + (-8) = -3. \]The sign rules for multiplying and dividing
Here is the genuinely new idea of the lesson. When you multiply or divide two integers, the sizes behave exactly as you would expect — what is new is what happens to the sign. The rule is wonderfully short:
- Same signs give a positive result.
- Different signs give a negative result.
Watch it work across all four combinations:
\[ (-3)(4) = -12, \qquad (-3)(-4) = 12, \] \[ -12 \div 4 = -3, \qquad -12 \div -3 = 4. \]Why should two negatives make a positive? Think of what \( -1 \) does on the number line: multiplying by \( -1 \) flips a number across zero to its opposite. So multiplying by \( -3 \) flips the direction and stretches by \( 3 \). Multiply by a negative twice and you flip twice — the second flip undoes the first and brings you back to the positive side. Two flips return you home; that is the whole reason behind the rule.
Comparing and ordering
The line settles every comparison for you: farther right means greater, farther left means smaller. With positives this matches everyday instinct, but negatives quietly reverse it. We feel that \( 8 \) is "bigger" than \( 1 \), yet
\[ -8 < -1, \]because \( -8 \) sits further to the left of zero — a deeper debt, a colder temperature. The size grows as you move out, but the value shrinks. When ordering signed numbers, trust the line, not the gut feeling about how big the digit looks.
Tip To fix the sign of a product or quotient, just count the minus signs. An even number of them gives a positive answer; an odd number gives a negative one. Work out the sizes separately, then stamp the sign on at the end.
- The sizes multiply as usual: \( 6 \times 2 = 12 \).
- Both factors are negative — two minus signs, an even count — so the signs match.
- Same signs give a positive result.
So \( (-6)(-2) = \mathbf{12} \). Two flips across zero bring you right back to the positive side.
- Place each number on the line: \( -7 \) sits farthest left, then \( -3 \), then \( 0 \), then \( 2 \) on the right.
- Least to greatest just means reading the line from left to right.
So the order is \( \mathbf{-7, \ -3, \ 0, \ 2} \). The number with the largest digit, \( -7 \), turns out to be the smallest of all.
Practice
Try each one yourself, then reveal the full solution.
1. Compute \( -8 + 5 \).
The two numbers have different signs, so subtract the smaller size from the larger: \( 8 - 5 = 3 \).
The larger size belongs to \( -8 \), which is negative, so the answer keeps that sign.
So \( -8 + 5 = \mathbf{-3} \).
2. Compute \( (-4)(3) \).
The sizes multiply to \( 4 \times 3 = 12 \).
One factor is negative and one is positive — different signs — so the result is negative.
So \( (-4)(3) = \mathbf{-12} \).
3. Compute \( |{-9}| - |4| \).
Take each absolute value first: \( |{-9}| = 9 \) and \( |4| = 4 \).
Then subtract: \( 9 - 4 = 5 \).
So \( |{-9}| - |4| = \mathbf{5} \).