So far the number line has stopped at zero — but the world keeps going below it. In this lesson we extend the line in the other direction and learn to add and subtract these new numbers without ever feeling lost.
The number line keeps going
Picture the counting numbers laid out in a row: \( 0, 1, 2, 3 \) and onward to the right. For a long time, zero felt like the edge of the world — the smallest number there was. But there's nothing stopping the line from continuing to the left of zero. Those numbers to the left are the negative numbers: \( -1, -2, -3 \), and so on, marching away from zero in a mirror image of the positives.
The line now runs in both directions, with \( 0 \) sitting calmly in the middle:
What negatives mean in real life
Negative numbers aren't an abstract trick — they describe everyday situations where something drops below a natural starting point:
- Temperature: a winter morning of \( -5 \) degrees is five degrees colder than freezing.
- Money: if you owe a friend \$8, your balance is \( -8 \) dollars — a debt.
- Buildings: the second basement floor is floor \( -2 \), two levels below the ground floor.
- Elevation: the shore of the Dead Sea sits about \( -430 \) metres — well below sea level.
A number and its opposite
Every number has an opposite: the number the same distance from zero but on the other side. The opposite of \( 3 \) is \( -3 \), and the opposite of \( -3 \) is \( 3 \). They balance each other perfectly, which is why \( 3 + (-3) = 0 \). Think of a \$3 gift cancelling a \$3 debt — you end up exactly where you started.
Adding and subtracting on the line
Here is the one idea that makes everything click. To do arithmetic, you stand on a number and take steps:
- Adding a positive number moves you right. \( \rightarrow \)
- Subtracting a positive number moves you left. \( \leftarrow \)
So \( 2 - 7 \) means: start at \( 2 \), then step \( 7 \) places to the left. You glide past zero and land on \( -5 \). Subtraction can give a negative answer whenever you take away more than you had.
The two rules that trip people up
Two new patterns appear once negatives enter the scene. Both follow from the same stepping idea.
Adding a negative moves left. A negative number points the opposite way, so adding it flips your direction. Starting at \( 2 \) and adding \( -5 \) means stepping \( 5 \) to the left:
\[ 2 + (-5) = -3 \]Adding two negatives just keeps going left from a spot already past zero:
\[ -4 + (-2) = -6 \]Subtracting a negative is the same as adding. Taking away something that pulls you left must push you right instead — the two minus signs cancel:
\[ 4 - (-3) = 4 + 3 = 7 \]In words A plus sign keeps the direction of the number that follows; a minus sign flips it. So \( +(-5) \) ends up pointing left, while \( -(-3) \) ends up pointing right. Read the two signs together and decide which way to step.
- Place yourself at the starting number, \( 2 \).
- The \( -7 \) tells you to step \( 7 \) places to the left.
- Two steps bring you down to \( 0 \): \( 2 \to 1 \to 0 \).
- You still have five steps to take, which carry you below zero: \( 0 \to -1 \to -2 \to -3 \to -4 \to -5 \).
You land on \( 2 - 7 = -5 \). Whenever you subtract a bigger number from a smaller one, the answer drops below zero.
- Start at \( -4 \), four steps to the left of zero.
- You are subtracting a negative, so the two minus signs cancel and become a plus: \( -4 - (-3) = -4 + 3 \).
- Adding \( 3 \) means stepping \( 3 \) places to the right: \( -4 \to -3 \to -2 \to -1 \).
You land on \( -4 - (-3) = -1 \). Removing a debt of \$3 from a \$4 debt leaves you only \$1 in the hole — better off than before.
Tip Subtracting a negative is adding. Picture a debt: if someone cancels the \$3 you owe, your money goes up by \$3 — the same as a \$3 gift. Removing something negative leaves you richer, which is exactly why \( -(-3) \) turns into \( +3 \).
Practice
Try each one yourself, then reveal the full solution.
1. Compute \( 6 + (-9) \).
Adding a negative moves you left, so \( 6 + (-9) \) means start at \( 6 \) and step \( 9 \) places to the left.
Six of those steps bring you to \( 0 \), and the remaining three carry you below zero to \( -3 \).
So \( 6 + (-9) = \mathbf{-3} \).
2. Compute \( 3 - 8 \).
Subtracting a positive moves you left. Start at \( 3 \) and step \( 8 \) places to the left.
Three steps reach \( 0 \) (\( 3 \to 2 \to 1 \to 0 \)), and the remaining five steps continue to \( -5 \).
So \( 3 - 8 = \mathbf{-5} \).
3. Compute \( -2 - (-7) \).
You are subtracting a negative, so the two minus signs cancel: \( -2 - (-7) = -2 + 7 \).
Start at \( -2 \) and step \( 7 \) places to the right. Two steps reach \( 0 \), and five more carry you up to \( 5 \).
So \( -2 - (-7) = \mathbf{5} \).