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## Subsets of Real Numbers

Subsets of Real Numbers

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

$1, 2, 3, 4, 5, ...$

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

$0, 1, 2, 3, 4, 5, ...$

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

$... -2, -1, 0, 1, 2 ...$

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 $\frac{a}{b}$ where $b \neq 0$.

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 $0.25 =\frac{25}{100}= \frac{1}{4}$ and $0.4 =\frac{4}{10}= \frac{2}{5}$.

Let’s not for get those decimals that have a pattern that can be identified as repeating. Here are just two examples: $0.\overline{6} = \frac{2}{3}$ and $0.\overline{4} = \frac{4}{9}$.

Irrational Numbers$\pi$, $\sqrt{2}$, $\sqrt{5}$, $\sqrt{\frac{6}{7}}$, $1.021502160217...$

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.

### 15 Responses

1. If I am not mistaken, in some books, natural numbers include 0.

• 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.

• 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.

• that is right…in my algebra book it says natural numbers include 0..

2. thanks a lot….
god bless…. :)

• tnx:)

3. thanks a lot… this could help me for my coming exams….. :D

4. thanks u this could help me for my assignment

5. Thanks I finally found the answer to my problem was searching for hours :)

6. What is a subset of real numbers?

• There are two subsets of real numbers: rational and irrational numbers. The rational numbers have other subsets: natural numbers, whole numbers, and integers.

7. thank you for the answer ~_~

8. thank you

9. Wait so would, say -33, be rational as well as an integer?

• SHERlockedandinTARDIS, -33 is both an integer and a rational number.