Integers: definition, convert to fractions

Por Tips

This article is about whole numbers. You may have already read a short definition of integers in the article Sets of Numbers. In the following sections, you will learn more about integers. After reading this article you will know what integers are and you can test your knowledge with a few practice problems. 😊

Whole numbers definition

The integers extend the set of natural numbers by zero and the negative numbers. It can be written as a set of numbers as follows:

Basically, you can recognize integers by the fact that they all are, i.e. have no comma and no decimal places.

For example, the numbers 5, -13, or 30000 are integers.

Whole numbers can be two different sign have either a positive sign «+» (or no sign at all, because the + is usually not written) or a negative sign «-«.

If you’re not sure what the natural numbers and negative numbers are, you should read the natural numbers and negative numbers articles. Whole numbers are simply all natural numbers with zero and the associated negative numbers. You can find these two articles in the Number Sets chapter.

Whole numbers in everyday life

You encounter whole numbers very often in everyday life. Let’s look at a couple of examples where you encounter whole numbers!

An everyday example is the temperature: you can read 10° Celsius on a thermometer on autumn days

read and on cold winter days -5° Celsius. Both scenarios are integers.

Or when you ride the elevator. You can probably go up to the fourth floor, and all the way down to the basement, where you’ve probably seen a -1 on the elevator display. If you get into the elevator of a high-rise building, you can not only go much further up, but maybe also to the second or third basement, i.e. to floors -2 and -3. These are all examples of integers that you may encounter in everyday life!

Arrangement of whole numbers on the number line

Whole numbers can also be represented on the number line. On the number line, numbers are ordered according to size, the smaller the number the further to the left you have to go and for larger numbers further to the right. The distance between adjacent numbers is always the same.

You will have already learned how to arrange the natural numbers on the number line in elementary school. Now the ray is extended to the left to form a straight line, so it doesn’t stop at zero, but also continues there to infinity, as on the right-hand side. The further to the left a number is, the smaller it is. The further to the right the number is, the larger it is.

Note: The number -40 is less than the number -2. If this isn’t clear to you, you should read the article on negative numbers again!

So the number line for the integers looks like this:

Illustration 1: number line

The number line looks something like a twisted thermometer.

Whole numbers whose absolute value is small are close to zero. Farther away are the numbers whose magnitude is large.

Do you know the difference between a number line and a number line? If not, you can read about that in the article number line and number line.

Formal definition of integers

At this point, the integers should be correctly defined again.

The set of integers includes the set of natural numbers and zero. In the illustration of the most important sets of numbers, it is still relatively far inside. So there are other sets of numbers that contain the whole numbers.

If you are in higher grade and already know the set of rational numbers or the set of real numbers, you should know the following chain:

If you would like to find out more about the rational numbers or the real numbers, we also have an article on this in the chapter sets of numbers.

For an overview you can see the figure of the sets of numbers with highlighting of the whole numbers:

Figure 2: Number sets

Important: Every integer is also a rational number, a real number and can be represented as a complex number. And every natural number is also an integer. But this is not the case the other way around!

Convert whole numbers to fractions

Since every whole number is also a rational number, every whole number can be converted to a fraction. This is quite simple, because you simply write the whole number in the numerator and a 1 in the denominator. The sign of the whole number is simply retained.

Conversely, there are also fractions that are whole numbers.

But this does not apply to all fractions! Only those fractions that can be reduced so much that the denominator is 1 are integers.

Integer subset

Sometimes it makes sense in mathematics to consider subsets of a set of numbers (i.e. only part of the set). Here are a few well-known subsets of the set of integers:

subsets without zero

Calculate whole numbers

For the whole numbers one can the following arithmetic operations use:

  • Addition: 3 + (-8) = -5
  • Subtraction: 10 – 8 = 2
  • Multiplication: 4 * (-5) = -20
  • Division: 8 : (-4) = -2

We have a separate chapter for calculating with whole numbers, you can also find it in the Algebra section. There you will be explained in detail how to calculate with whole numbers and also how to calculate in writing.

Whole number exercises

To deepen your understanding of whole numbers, we have two exercises for you here:

Task 1

Which of the following numbers belongs to the set of integers? Which number is even a natural number?

  1. 3
  3. 0.123456789
  4. -30405
  5. -1.5
  7. 0
  8. 10
  9. 500134
  10. -75.5

solutions

  1. 3 is an integer and in particular a natural number.
  2. is not an integer – it belongs to the set of rational numbers.
  3. 0.123456789 is not an integer because it has decimal places. But it belongs to the set of rational numbers.
  4. -30405 is an integer, but not a natural number.
  5. -1.5 is not an integer. It again belongs to the set of rational numbers.
  6. is an integer because the fraction can be reduced: . This number is therefore even a natural number!
  7. 0 is an integer. It also belongs to the set of natural numbers starting with zero.
  8. 10 is an integer and natural number.
  9. 500134 is also an integer and in particular a natural number.
  10. -75.5 is not a whole number because it has decimal places. It belongs to the rational numbers.

exercise 2

What is the solution to the following arithmetic problems?

  1. 10 – 9 + 8 – 7 + 6 – 5 + 4 – 3 + 2 – 1
  2. 90 : (-9)
  3. 10 (25 – 40)
  4. 10 x 25 – 40
  5. 123 – 456 + 789

solutions

  1. 10 – 9 + 8 – 7 + 6 – 5 + 4 – 3 + 2 – 1 = 5
  2. 90 : (-9) = -10
  3. 10 * (25 – 40) = -150
  4. 10 * 25 – 40 = 210
  5. 123 – 456 + 789 = 456