What Are Integers? Positive and Negative Numbers

Numbers that go both ways

Counting numbers only go one direction. Integers go both:

$$\ldots, \; -3, \; -2, \; -1, \; 0, \; 1, \; 2, \; 3, \; \ldots$$

An integer is a whole number or the negative of one. $-3$, $0$ and $17$ are all integers. $1.5$ and $\tfrac{2}{3}$ are not, though they are still numbers with a place on the line.

What a negative number describes

A negative number is a quantity in the opposite direction from a positive one. Which direction counts as positive is a choice, and once it is made the other direction is negative.

Situation	Positive means	Negative means
temperature	above zero	below zero
elevation	above sea level	below sea level
money	money you have	money you owe
a lift	floors above ground	floors below ground
time	after launch	before launch

So $-30$ metres is thirty metres below sea level, and $-40$ dollars is forty dollars owed. The minus sign is not saying the quantity is small. It is saying which way it points.

Opposites

Every number has an opposite: the number the same distance from zero on the other side.

$$\text{opposite of } 4 = -4 \qquad \text{opposite of } -7 = 7$$

Taking the opposite twice returns you to the start:

$$-(-7) = 7$$

That is worth reading carefully. The minus sign means “the opposite of”, so “the opposite of $-7$” is $7$ — you have crossed zero and come back.

Why zero has no sign

Zero is the point everything else is measured from. It is neither above nor below, neither owed nor held.

It is also its own opposite: the number zero steps from zero on the other side is zero again. Since a sign records which side of zero a number is on, and zero is on neither side, it needs no sign.

Why negatives were worth inventing

Without negative numbers, $3 - 8$ has no answer. You can say “you cannot take $8$ from $3$”, and for a while school mathematics does exactly that.

But the question keeps coming up in situations where an answer plainly exists. A thermometer at $3$ degrees that falls $8$ degrees reads something. An account with $3$ dollars that pays out $8$ owes something. The measurement is real, so the arithmetic ought to produce it.

$$3 - 8 = -5$$

Extending the number line leftward makes subtraction always work. That is the pattern behind every extension of the number system: fractions make division always work, and negatives make subtraction always work. Each one removes a case where a sensible question had no answer.

It also means one number line can hold a whole measurement. Temperature does not need one scale for warm and another for cold — it needs one line through zero.

Numbers between the integers

The line does not only hold integers. Fractions and decimals sit between them, and negatives of those sit to the left:

Any number that can be written as a fraction of two integers is a rational number. That includes every integer, since $5$ is $\tfrac{5}{1}$.

Worked examples

Common mistakes

Practice problems