Integers — Definition & Examples

Integers

The set of integers is the list $...,-3,-2,-1,0,1,2,3,...$ The integers include the whole numbers (also called the counting numbers) and the natural numbers.

All integers are either positive, negative, or zero. They fall into exactly one of these categories.

We use the symbol $\mathbb{Z}$ to refer to the integers.

The real numbers include all of the other numbers in the diagram. For example, every integer is a real number. All of the rational numbers are also real, and in fact each integer is rational. Over time, you will see more examples of all the types of numbers. This diagram will become easier to remember as a result.

Integers have some really cool properties, like the reflexive, symmetric, and transitive properties. Check them out!!

(Video) Natural Numbers, Whole Numbers, and Integers

by Learn Math Tutorials

This video by Learn Math Tutorials goes over the first three types, or sets, of numbers: natural numbers, whole numbers, and integers.

$\large{\text{Summary}}$

The integers are the numbers we can write without a fractional part. We use the symbol $\mathbb{Z}$ to refer to them. $$\mathbb{Z}=...,-2,-1,0,1,2,...$$

If we were to make a number line to represent the integers, it would look like:

The number line has $0$ in the middle and it spans out to the right and left. The left side has the negative integers which go out to negative infinity, $-\infty$. The right side has the positive integers which go out to infinity, $\infty$.

Integer in a Glance

Integers are numbers that can be obtained by adding a finite number of $1$s or subtracting a finite number of $1$s. From this definition, both $0$ and $1$ must be integers.

What other numbers are integers?

Take $3$ for example. We know $3 = 1+1+1$, so $3$ is an integer. $2 = 1+1$ and $5 = 1+1+1+1+1$, so they are both integers. If we change all the "$+$" signs to "$-$" signs in $3 = 1+1+1$, we find that $-3 = -1-1-1$ is also an integer, since it can be obtained by subtracting $1$s.

What if a number is too large to be expressed as a sum of $1$, for example $100$?

Integers can be defined as a real numbers that contains no fractional parts, i.e. the difference between two successive integers is at least $1$.

If a number does not contain a decimal point or a multiple of $i$, the unit of imaginary numbers, it is an integer. $100$ is an integer, and so is $-2{,}147{,}483{,}647$. $3+4i$ contains no decimal point, but because it has a "$4i$" part, it is not an integer in the usual sense. However, it is an Gaussian Integer.

Is there an integer between $0$ and $1$?

No. One property of integers is that they contain no fractional parts.

However, we cannot prove this property with what we know now. You might say, why, if we keep adding or subtracting ones, we obviously cannot get a fractional part. This is in fact not so "obvious". I will show you some counter-intuitive results later in this article (so keep reading!), but now let's just say what's obvious isn't necessarily right. The problem lies in the distinction of cardinal vs ordinal numbers.

So, here is your challenge... Prove $0.5$ is not an integer!

What is $0.5$? It is a decimal. If you don't know what it is, you most likely don't have to worry about "why there's no integer between $0$ and $1$".

Extension: Why finitely many times?

Here's a peak into some higher level math theory.

When we start talking about an infinite sum, the properties of the sum can change. Try taking the sum $1-1+1-1+1....$ for example.

$1-1+1-1+1....$ = $(1-1)+(1-1)+(1.... = 0 = 1+(-1+1)+(-1+1).... = 1$. $0=1$? What's wrong?

We can do the sum in a slightly different way.

Let $S = 1-1+1-1+1....$.

$1-S = 1- (1-1+1-1+1....) = 1-1+1-1+1.....$

Subtract the second equality from the first. On the left side it's $2S - 1$, and on the right side it's $0$. Therefore $2S=1$, and $S= \frac{1}{2}$

Which do you think is correct?

More on Grandi's series.