Definition of integers

We’ve already learned what natural numbers are and how to perform simple mathematical operations with them, such as addition, subtraction, multiplication and division. We represent the set of natural numbers with $\mathbb{N} = \{ 1, 2, 3, 4 \ldots\}$. These are the numbers we use when we’re counting objects. We can also add zero to the set of natural numbers to represent the absence of quantity (when there is nothing there). Then we mark the set with $\mathbb{N_{0}} = \{ 0, 1, 2, 3, 4 \ldots \}.$

Before we explain what integers are, we would like you to observe the following example:

The air temperature today at noon was 39.2°F and by the evening the air temperature declined by 42.8°F. What was the air temperature in the evening?

We know what natural numbers are and we know how to add or subtract natural numbers. Can we solve example above using just the knowledge we have on natural numbers?

Let’s rewrite the problem using mathematical language:

$39.2-42.8=$

We don’t know how to subtract natural numbers when the subtrahend (second number) is larger than the minuend (first number) so we need to expand the set of natural numbers. To easily understand how we do that, let’s put first few natural numbers on the number line:

All of these numbers are sorted in order from smallest to the greatest. Now, let’s try to do some subtraction with two arbitrarily chosen natural numbers. For example, we know that $\ 4 – 3 = 1$. But what happens if we want to subtract natural numbers when the subtrahend (second number) is larger than the minuend (first number)? Can we do that? Are there numbers that are smaller than zero? Where can we find those numbers on our number line? Well, yes, we can subtract any two arbitrarily chosen natural numbers. Also, yes, there are numbers that are smaller than zero, and, to answer the last question, look at the picture below.

Take a look at this sequence. We have a zero in the middle. Now look at the first number to the right and first one to the left of zero – those are 1 and -1. If we look at the second number to the right and the second one to the left, those are 2 and -2. The third ones are 3 and -3. We can notice the pattern and, to be mathematically correct, we can come up with the following definition: Numbers that are found on opposite sides from zero and are equidistant (or equally distant) with respect to zero are called the opposite numbers or additive inverses.

The set that contains both positive and negative numbers is called the set of whole numbers or the set of integers and we mark the set with: $\mathbb{Z} = \{ \ldots-4, -3, -2, -1, 0, 1, 2, 3, 4 \ldots\}$. It consists of natural numbers and their additive inverses – numbers that, when added to their natural counterparts, result in zero (like -4 and 4, or -16 and 16, etc.). This set is also unbounded from both ends, which means that the smallest and largest integer do not exist! There is always a number that will be smaller than the previous smallest number, and a number that will be larger than the previous largest number.

Every operation that can be done with natural numbers can be done as easily with integers. Addition, subtraction, multiplication, division – you name it! To make things easier and explain them a bit better, we divided these topics into separate lessons, which you can access via the main menu or by clicking on their names a couple of rows above.

That’s it for this lesson! Feel free to explore the website a bit and learn how integers can be used in mathematical operations.