Negative Numbers and the Number Line

Why we need numbers below zero

You could write the balance out in words, as “a debt of five dollars”. A phrase like that is awkward to calculate with and awkward to compare. One symbol, $-5$, does the whole job: the $5$ gives the amount, and the minus sign records the owing side of zero.

What a negative number is

A negative number is a number less than zero, and a positive number is a number greater than zero. We write a negative number by putting a minus sign in front of a positive one, like $-3$, $-17$ or $-2.5$. We read $-3$ aloud as “negative three.”

The counting numbers $1, 2, 3, \ldots$ are all positive, and so is a number like $2.5$. A positive number can be written with a plus sign, like $+3$. That plus sign is almost always left off, so a bare $3$ is understood to be positive.

Every negative number is the partner of a positive one. The negative $-3$ records the same size as $3$ (three units), but on the opposite side of zero. If $+3$ means three steps in one direction, then $-3$ means three steps in the other.

Zero sits apart from both groups. Zero is neither positive nor negative. It is the single dividing point between the two sides, the reference from which “above” and “below” are measured. Writing it as $-0$ means nothing different; $-0$ is just $0$.

The number line, extended to the left

A number line is a straight line with zero marked on it. The whole numbers are zero together with the counting numbers. Zero sits at that mark, and the counting numbers are spaced out along the line at equal steps to its right. To make room for the negatives, we simply keep the same even spacing going to the left of zero. Then we label those ticks $-1, -2, -3,$ and so on:

The positive numbers run to the right of zero and the negative numbers run to the left. Each step along the line is the same length as every other. The line has no last tick on either end: the three dots mean it continues forever in both directions. What matters most is that every number now has a fixed home, a single position on this line.

The integers

The whole numbers, together with all of their negatives, form a named set: the integers.

The curly braces just mean “the collection of.” The three dots on each end signal that the list runs on without stopping in both directions. The integers are exactly the numbers you land on by starting at zero and stepping by ones, in either direction. Stepping right gives the positives and stepping left gives the negatives. The integers split into three kinds: the positive integers $1, 2, 3, \ldots$, the negative integers $-1, -2, -3, \ldots$, and zero, which belongs to neither side.

Not every number is an integer. A number like $\tfrac{1}{2}$ or $2.7$ lands between two neighbouring integer ticks, so it is not an integer. Even so, it still has a perfectly good position on the line.

Opposites

Two numbers are opposites if folding the line at zero lands each one on the other, so they sit the same distance from zero. The opposite of $5$ is $-5$, and the opposite of $-5$ is $5$. Fold the line at zero and the point five ticks to the right lands on the point five ticks to the left. Folding carries a point across zero rather than along its own side. So for every number except zero, the two land on opposite sides of zero. On the line, a number and its opposite are mirror images across zero.

Taking the opposite is a kind of flip across zero, and flipping twice returns you to the start. So the opposite of the opposite of $5$ is $5$ again. Putting a minus sign in front of a negative number is how we write “the opposite of.” So the opposite of $-5$ is written $-(-5)$, and that equals $5$.

One mark is doing more than one job, and its position tells you which. In front of a bare numeral, as in $-5$, it marks a negative number. In front of a number already written down, as in $-(-5)$, it is an instruction: take the opposite of that. Between two numbers, as in $7 - 5$, it is the subtraction you have used since your first arithmetic. Read the position first and the job is never in doubt.

One number is its own opposite. Zero sits at the fold, so reflecting it across zero leaves it exactly where it was. The opposite of $0$ is $0$, and zero is the only number for which that happens.

Comparing integers by position

For positive numbers you have always known that $5$ is greater than $2$. Negatives need a rule that still works once you cross to the left side, and the number line supplies one that never fails.

Look at where $5$ and $2$ sit: $5$ is to the right of $2$, and $5$ is the greater. Now compare $-5$ and $-2$. The point $-5$ is five steps left of zero, while $-2$ is only two steps left. So $-5$ lies farther left, which means it is the smaller:

Read that as “negative five is less than negative two.” It can feel backward, because $5 > 2$ for the positives, but the order genuinely flips when you cross zero. In both pairs, though, it was the number sitting farther right that came out greater, and that holds everywhere on the line:

Of any two numbers on the number line, the one farther to the right is the greater. Of the same two, the one farther to the left is the smaller.

The comparison symbols are the familiar $<$ (“is less than”), $>$ (“is greater than”), and $=$ (“is equal to”). A reliable way to read them is that the wide open end always faces the larger number and the narrow point faces the smaller. So the statement $-5 < -2$ has its point toward $-5$, the smaller number.

Two consequences follow straight from the picture and are worth keeping at hand. First, every negative number is less than zero, since every negative sits to the left of zero. Second, every negative number is less than every positive number, since the whole left side lies to the left of the whole right side. You never have to compute anything to know that $-1000 < 4$.

Why does position decide it, and not the digits? Because on the number line, “less than” is a statement about position: one number is less than another when its point lies to the left.

Plotting and reading points

To plot an integer is to mark its position on the line. Start at zero and count equal ticks: to the right for a positive number, to the left for a negative one. To plot $-4$, count four ticks left of zero and mark that point. To plot $3$, count three ticks right.

Reading a marked point reverses the process. Count how many ticks the point sits from zero, and note which side it is on. A point to the left of zero is a negative number, and a point to the right of zero is positive. The count gives the size and the side gives the sign.

Counting from a point other than zero

Zero is the natural place to count from, but any point will do. The distance between two points is the number of one-unit steps that take you from one to the other. From $-10$ to $-6$ is four steps, and from $-6$ to $-2$ is four steps again. So $-10$ and $-2$ sit the same distance from $-6$, one on each side. That is the same pattern as $-5$ and $5$ sitting either side of zero.

When the two points fall on opposite sides of zero, count the two pieces and add them. From $-3$ to $4$ is three steps up to zero and then four more, which is seven steps in all.

The same counting finds the point halfway between two others. Count the steps from one to the other, halve that count, then count that far in from either end. From $-10$ to $-2$ is eight steps, and half of eight is four, so four steps in from either end lands on $-6$. The figure above shows both counts.

The word between asks a related question. Which integers are greater than $-8$ and less than $-4$? Walk right from $-8$ and keep only the numbers strictly inside: $-7$, $-6$ and $-5$. The two ends are left out, because “greater than $-8$” does not include $-8$ itself.

Signed quantities in the real world

A sign is a compact way to record which side of a reference point a quantity falls on, or which direction it points. The reference (the thing that counts as zero) changes with the situation, but the same number line models them all:

• Temperature. Take zero degrees as the reference. Then $-8^\circ$ means eight degrees below zero and $+8^\circ$ means eight degrees above. A colder temperature is a smaller number, farther left.
• Money. Take a zero balance as the reference. A balance of $-40$ dollars means you owe forty dollars, and $+40$ means you have forty.
• Elevation. Take sea level as the reference. An elevation of $-30$ meters means thirty meters below sea level, and $+30$ means thirty meters above.
• Time. In a launch countdown, zero is liftoff, so $-3$ seconds means three seconds before launch.

Because all of these share one number line, comparing signed quantities is nothing more than comparing positions. A temperature of $-8^\circ$ is colder than $-3^\circ$ for the same reason $-8 < -3$: the point $-8$ lies farther left.

Common mistakes