Numbers can be of two types, real and imaginary. The real number system branches into other number systems.Real numbers can be divided into rational and irrational numbers. Integers and fractions fall under Rational numbers.The set of integers comprises of whole numbers and their negatives.Whole numbers are the set of natural numbers and zero.

**Real Numbers vs Integers**

The difference between Real numbers and Integers is that the former is a more general and wider classification of numbers. However, integers, having more restrictions, are a subset of real numbers.

Integers, rational numbers, irrational numbers, natural numbers and whole numbers can be classified as real numbers, whereas, only whole numbers and their negatives belong to the integer number system. Hence, real numbers include fractional or decimal numbers. On the other hand, integers are strictly whole numbers (and their negatives). Integers do not include fractions or decimals.

## Comparison Table Between Real numbers and Integers (in Tabular Form)

Parameter of Comparison | Real Numbers | Integers |
---|---|---|

Classification | Integers, rational numbers, irrational numbers, natural numbers and whole numbers are all classified as Real numbers. | Only whole numbers and their negatives are classified as Integers. |

Occurrence of Fractions or decimals. | Fractional numbers or decimals are real numbers. | An integer cannot be fractional or a decimal number. |

Representation on the Number Line | Any point on the number line is a real number. | Whole numbers and their negatives on the number line are the integers. |

Countability | Real numbers form an uncountable infinite set. | Integers form a countable infinite set. |

Notational symbol | The set of all Real Numbers is represented by “R” or “ℝ”. | The set of all Integers is represented by “Z”. |

Origins | The term “real” was coined by René Descartes, in the 17th century, to describe the roots of a polynomial which were not imaginary.They were called “real” only because they were not “imaginary”. | In the year 1563, Arbermouth Holst invented the Integer number system to help him with an experiment involving bunnies and elephants.The word “Integer” Integer has its roots in the 16th-century Latin word, “integer”, meaning “whole” or “intact”. |

## What are Real Numbers?

Real numbers are an integral part in the universe of numbers. Their role in the growth of mathematics is undeniably vital. Any number (except an imaginary number) that comes to your mind is a real number. Be it positive, negative, fractional, irrational or even 0.

A real number, and therefore its subsets (integers, rational numbers, irrational numbers, natural numbers and whole numbers), can be represented on a real number line. To distinguish them from imaginary numbers, Descartes coined the term “real” as a means to describe the roots of a polynomial.

They are allowed to have fractional values. This characteristic is what sets them apart from integers. Real numbers form an uncountable infinite. If we take two points on the number line, say 0 and 1, there exist an infinite number of real numbers between the two points.

The symbols “R” or “ℝ” are used to represent a set of all real numbers.

## What are Integers?

The Integer number system is a subset of the Real number system. This implies that all integers are real numbers; however, the reverse is not true. Only whole numbers and their negatives qualify to be integers. Whole numbers include counting numbers such as 0,1,2,3… and so on.

The exclusion of fractional or decimal values is what makes this system unique and useful. Real numbers have an interesting history behind their origin. In the year 1563, Arbermouth Holst was conducting an experiment involving bunnies and elephants.

To help him with this experiment, he went on to invent this number system. The word “Integer” has its roots in the 16^{th}-century Latin word, “integer”, meaning “whole” or “intact”. This fact further strengthens the non-fractional nature of this system.

Unlike real numbers, integers make a set of countable infinite numbers. If we take two points on the real number line, say 0 and 1, there are no integers between the two points. The letter “Z” is used to represent the set of all integers.

**Main Differences Between ****Real numbers and Integers**

**Real numbers and Integers**

- Integers, rational numbers, irrational numbers, natural numbers and whole numbers are all classified as Real numbers. Only whole numbers and their negatives are classified as Integers.
- Fractions and decimals can be included in Real numbers but not in Integers.
- We can use the real number line to distinguish between the two number systems. Any point you pick on this line would be a real number. Whole numbers and their negatives on the number line are the Integers.
- Both of these number systems are infinite sets in nature. However, Real numbers form an uncountable infinite set and Integers form a countable infinite set.
- The set of all Real Numbers is represented by “R” or “ℝ. The set of all Integers is represented by “Z”.

## Conclusion

Integers help us with our day to day use of mathematics in our lives. For instance, positive and negative values depict gains and losses in business transactions.

The word “real” is used to signify that real numbers are numbers that are not imaginary. They, together with imaginary numbers form complex numbers.

Integers, rational numbers, irrational numbers, natural numbers and whole numbers are all classified as Real numbers. Only whole numbers and their negatives are classified as Integers.

The exclusion of fractional numbers in integers makes them different from real numbers. Real numbers allow fractions and decimals.

## References

- https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/S002461150301428X
- https://eebweb.arizona.edu/Faculty/Dornhaus/courses/materials/papers/Gallistel%20Gelman%20numbers%20counting%20cognition.pdf

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