Subsets of Real Numbers

**Natural Numbers** -are used for counting and the set does not include zero.

**Whole Numbers** – include the set of natural numbers and zero.

**Integers** – includes zero, the set of natural numbers, commonly called the positive integers and the opposite of the natural numbers.

**Rational Numbers** – include any number that can be represented by the quotient of two integers. In that quotient the denominator must not be zero.

Given that a and b are integers, then where .

There are all kinds of numbers that fit this description. Any combination of the fractions with the subsets above fit the definition of a rational number.

It is good to remember that all numbers that are termination decimals such as and .

Let’s not for get those decimals that have a pattern that can be identified as repeating. Here are just two examples: and .

**Irrational Numbers** – , , , ,

These are just a few of the examples of irrational numbers. If a number does not fit in as one of the other numbers, it is irrational. There is another set of numbers to, but that is another topic for another day.

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Filed under: Algebra 2, Definitions Tagged: | Algebra 2, Definitions, Subsets of Real Numbers

Guillermo Bautista, on December 9, 2009 at 4:44 am said:If I am not mistaken, in some books, natural numbers include 0.

Mr. Pi, on December 11, 2009 at 11:12 am said:You are 100% correct. There is debate about where 0 should be included. After reading your comment, I opened some older mathematics books that I have and all of them had the natural numbers without zero. Then googled the set of natural numbers and found the conflicting definition of natural numbers that you have mentioned.

Several sites kind of ignored the sets of natural and whole numbers and just defined integers. I defined whole nubmers and natural numbers the way I did, because of the text my students use has it that way.

Brian, on December 16, 2012 at 1:39 am said:I believe the reasoning for including 0 in the natural numbers comes from an axiom of the ZF (Zermelo-Frankel) axiom system, called the Axiom of Infinity. In technical terms, the axiom is (∃N)(∅∈N ∧ ∀x(x∈N⇒℘(x)∈N), where ℘(x) is the power set of x (the set of all subsets of x). This corresponds nicely with the following two axioms from Peano’s axioms for the natural numbers:

1. 0 is a natural number.

2. If n is a natural number, then σ(n), where σ is the successor function, is also a natural number. Furthermore, there does not exist a natural number whose successor is 0.

Axioms 1 corresponds with [∅∈N] since |∅| = 0, and axiom 2 corresponds with [∀x(x∈N⇒℘(x)∈N)], where σ is analogous to ℘, since |℘(∅)| = |{∅}| = 1 = σ(0), etc.

Of course, we can just as easily start natural numbers at 1 in the Peano axioms and still make a one-to-one correspondence between the natural numbers and the elements of the set N defined in the Axiom of Infinity (just shift the correspondence over by 1), but it doesn’t feel as natural since we lose the connection with the size of the elements of N.

jordyn, on December 31, 2012 at 1:25 pm said:that is right…in my algebra book it says natural numbers include 0..

micah, on June 7, 2011 at 10:34 am said:thanks a lot….

god bless…. :)

09019323254367, on June 22, 2012 at 7:04 am said:tnx:)

nikkie garcia, on July 8, 2012 at 6:18 am said:thanks a lot… this could help me for my coming exams….. :D

cherry may bacaoco, on July 20, 2012 at 12:20 am said:thanks u this could help me for my assignment

Hello Kitty, on August 23, 2012 at 5:56 pm said:Thanks I finally found the answer to my problem was searching for hours :)

Lauren Folk, on September 4, 2012 at 6:07 pm said:What is a subset of real numbers?

Mr. Pi, on September 12, 2012 at 6:05 am said:There are two subsets of real numbers: rational and irrational numbers. The rational numbers have other subsets: natural numbers, whole numbers, and integers.

jelsie, on September 13, 2012 at 7:59 am said:thank you for the answer ~_~

ian, on September 22, 2012 at 5:37 am said:thank you

SHERlockedandintheTARDIS, on February 4, 2013 at 7:45 pm said:Wait so would, say -33, be rational as well as an integer?

Mr. Pi, on February 5, 2013 at 8:21 am said:SHERlockedandinTARDIS, -33 is both an integer and a rational number.