Real Numbers vs Integers: Difference and Comparison

Numbers can be of two types, real and imaginary. The actual 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 whole numbers and their negatives. Real numbers are a set of natural numbers and zero.

Key Takeaways

1. Real numbers are a broad category of numbers that include all rational and irrational numbers, such as integers, fractions, and decimals.
2. Integers are a subset of real numbers, consisting of whole numbers and their opposites, such as -3, -2, -1, 0, 1, 2, 3, and so on.
3. Both real numbers and integers are categories of numbers. Still, real numbers encompass all rational and irrational numbers, while integers are a specific subset of real numbers containing whole numbers and their opposites.

Real Numbers vs Integers

Real numbers are an extensive category of numbers, which include different types like decimals, fractions, integers, and rational and irrational numbers. Integers are subsets or types of real numbers which consist of all whole numbers, both positive and negative, on the number line.

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

Parameter of Comparison	Real Numbers	Integers
Classification	Integers, rational, irrational, natural, 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 a fractional or a decimal number.
Representation on the Number Line	Any point on the number line is an actual number.	Whole numbers and their negatives on the number line are 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	René Descartes coined the term “real” 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 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”.

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What are Real Numbers?

Real numbers are an integral part of 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 actual.

Be it positive, negative, fractional, irrational or even 0.

An actual number, and therefore its subsets (integers, rational numbers, irrational numbers, natural numbers and whole numbers), can be represented on a natural number line.

To distinguish them from imaginary numbers, Descartes coined the term “real” to describe a polynomial’s roots.

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, an unlimited number of real numbers exist between the two points.

The symbols “R” or “ℝ” 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 untrue.

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 makes this system unique and valuable. Real numbers have a fascinating history behind their origin.

In 1563, Arbermouth Holst was conducting an experiment involving bunnies and elephants.

To help him with this experiment, he invented 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 natural number line, say 0 and 1, there are no integers between the two points.

The letter “Z” represents the set of all integers.

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Main Differences Between Real numbers and Integers

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

References

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