Here’s what **integers** are and **how integers are represented, **solved exercises step by step.

Índice de Contenidos

## What integers are. Set of integers.

Integers are the set of numbers made up of all natural numbers (positive numbers), zero and negative numbers (those that are smaller than zero and have a minus sign in front of them).

- The
**natural numbers**: 1, 2, 3, 4, 5, 6, 7… so to infinity (∞): they are positive numbers, as they could be written as +1, +2, +3, +4, +5,… The positive sign is not usually written. If a number has no sign, it is positive. - The
**zero**: 0 (It is neither positive nor negative, it is neutral) - And the
**negative numbers**: -1, -2, -3, -4, -5, -6, -7…so on down to the least infinite (-∞)

So you’re wondering, are the numbers 1, 2, 3, 4, 5, 6, 7… natural or integers?

It is within the two sets because they belong to natural numbers, but this set is contained in integers, so, in other words, natural numbers are a subset of integers.

The set of integers is represented by the letter Ζ:

Ζ = {…-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6…}

## How integres are represented on the number line

I will now explain how integers are represented on the number line. Here’s how we do it:

1. We draw the line, divided in equal parts and put the 0 in the center:

2. We place the positive numbers to the right of zero, increasing their value by one unit, from left to right:

So far it’s like representing natural numbers.

3. Now we place the negative numbers to the left of zero, with the following particularities:

- Its absolute value (concept explained a little further down) increases from right to left (symmetrically to the positive numbers).
- Everyone has the minus sign in front of them.

Now, after seeing the integer representation, let’s move on to see how they are ordered.

## How integers are ordered

To learn how to order integers among them, it is first necessary to know what the **absolute value** of a number is, a concept that will help us to clear up many doubts.

### Absolute Value

The absolute value of a number is the number that results from removing its sign, positive or negative, from the number. It is represented by enclosing the number and sign between two vertical bars.

The absolute value of a negative number is the number remaining when the minus sign is removed:

In positive numbers or natural numbers, the absolute value coincides with the value of the number. Remember that usually the + sign on positive numbers is not written:

### Comparison of integer numbers

On the one hand we have the positive numbers:

- These numbers, as mentioned above, are represented from left to right and their absolute value increases in this sense as well.
- The direction of the order coincides with that of its representation, that is, from left to right they are ordered from lowest to highest:

With negative numbers, we must be very careful, since their arrangement goes in the opposite direction to that of their representation:

- As the absolute value of a negative number increases, it becomes more negative. This means that the higher the absolute value of a negative number, the smaller it is. Then, the number furthest to the left is less

-8<-2

- Therefore, negative numbers are also ranked from lowest to highest, from left to right (their representation is from right to left).

If we have to compare negative and positive numbers, keep in mind that the sorting always goes from left to right, that is, the smaller the numbers are the more to the left they are and the greater the more to the right they are. Therefore:

- Negative numbers are always less than positive numbers.
- Negative numbers are less than zero.
- Positive numbers are greater than zero.

## Integer exercises

1 – Represents the following integers on the number line: -3, 5, 1 and -6.

2 – Expresses the absolute value of -2, 5 and -4.

3 – Sort the following integers from lowest to highest -5, 0, 5, -1 and 2, using the corresponding symbol (< or >).

**Exercise 1:**

Remember to place the 0 in the center to have a reference point.