Classification of the Real Number System

Real

Rational - any number that can be written as the ratio of two integers, which consequently can be expressed as a terminating or repeating decimal.

Irrational - numbers that cannot be written as a ratio of two integers.

Rational IrrationalNot Real

Integers are positive and negative whole numbers and zero such as … -4, -3, -2, -1, 0, 1, 2, 3, 4 and so on.

Integers do not have any fractional parts. So numbers such a ½, .3, 2 ¼ , 25% etc are not integers because they involve fractional parts.

Important Tip

RealRational IrrationalIntegers

When determining if a number is rational the number must be able to be written in such a way that the numerator and denominator is a positive or negative whole number.

Also …

The numerator can be zero but not the denominator.

Additionally …

Number Ratio of twointegers

Terminating decimal or repeating decimal

5.000 terminating

.250 terminating

20% = .20 terminating

repeating

-8 -8.0 terminating

-2.5 -2 = - -2.50 terminating

-6.0 terminating

0 0.0 terminating

Examples of Rational Numbers

• The set of rational numbers has subsets• Some common subsets of rational

numbers are• Natural/counting numbers• Whole numbers• Integers

• Some numbers fall into more than one category

Real

Natural

Natural/counting numbers (N) are positive whole numbers beginning with 1. A way to remember natural / counting numbers is to think about what number you begin counting with --- 1. So natural / counting numbers are numbers such as 1, 2, 3, 4, etc.

Real

Whole

Natural

Whole numbers (W) include ALL counting numbers and 0. So whole numbers are 0, 1, 2, 3, 4, etc.

Real

Integers

Whole

Natural

Integers (Z) were explained previously but to recall they include all natural/counting numbers and whole numbers. They are positive and negative whole numbers and 0 such as … -4, -3, -2, -1, 0, 1, 2, 3, 4 …

Real

RationalIntegers

Whole

Natural

Rational Numbers (Q) recall that they are zero and all positive and negative numbers that can be expressed as a ratio of two integers (with no zero in the denominator), including integers, whole numbers, and natural/counting numbers.

Real

RationalIntegers

Irrational

Whole

Natural

Irrational Numbers (I) recall that they are real numbers that are not rational and cannot be written as a ratio of integers.

Examples of Irrational Numbers

√20 Pi 𝞹 √32

Irrational numbers are considered real numbers.

The real number system can be divided into two categories – rational and irrational. Many students tend to think that irrational numbers are not real.

This is not true. Irrational numbers ARE real but just are expressed differently than rational numbers.

3.1415926535897932384626433832795… (and more) 4.47213594…

0.8660254…

0-1-2-3-4-5-6 1 2 3 4 5 6

Basically in order to determine if a number is real, ask yourself if the numbers can be placed on a number line. If the number can be placed on a number line or be ordered, then the number is real.

0-1-2-3-4-5-6 1 2 3 4 5 6

−√𝟑

-6 -4.2−√𝟑 2.5 √𝟏𝟔√𝟑𝟓

0-1-2-3-4-5-6 1 2 3 4 5 6

√𝟑𝟓−√𝟑

-6 -4.2−√𝟑 2.5 √𝟏𝟔√𝟑𝟓

0-1-2-3-4-5-6 1 2 3 4 5 6

√𝟑𝟓2.5−√𝟑

-6 -4.2−√𝟑 2.5 √𝟏𝟔√𝟑𝟓

0-1-2-3-4-5-6 1 2 3 4 5 6

√𝟑𝟓2.5−√𝟑-6

-6 -4.2−√𝟑 2.5 √𝟏𝟔√𝟑𝟓

0-1-2-3-4-5-6 1 2 3 4 5 6

√𝟑𝟓2.5−√𝟑-4.2-6

-6 -4.2−√𝟑 2.5 √𝟏𝟔√𝟑𝟓

0-1-2-3-4-5-6 1 2 3 4 5 6

√𝟑𝟓2.5−√𝟑-4.2-6 √𝟏𝟔

-6 -4.2−√𝟑 2.5 √𝟏𝟔√𝟑𝟓

Numbers Not Considered Real

𝒙𝟐=−𝟏

𝒙=√−𝟏The square root of any negative number are numbers not considered real.

𝟑𝟎−𝟕 .𝟑𝟎

√𝟏𝟖𝟎

These numbers are undefined because zero is in the denominator and cannot be considered a real number. They are not numbers at all.

Rational

Integers

Whole

Natural/Counting

Irrational

Numbers Not Considered Real

-5 7018%

− 23

82 √202

−2.73

2613

020

√1213

√−25-5

-5

−√67

Rational

Integers

Whole

Natural/Counting

Irrational

Numbers Not Considered Real

-5 7018%

− 23

82 √202

−2.73

2613

020

√1213

√−25-5

-5 𝟕𝟎

−√67

Rational

Integers

Whole

Natural/Counting

Irrational

Numbers Not Considered Real

-5 7018%

− 23

82 √202

−2.73

2613

020

√1213

√−25-5

-5 7018%

−√67

Rational

Integers

Whole

Natural/Counting

Irrational

Numbers Not Considered Real

-5 7018%

− 23

82 √202

−2.73

2613

020

√1213

√−25-5

-5 7018%

82

82

82

82

−√67

Rational

Integers

Whole

Natural/Counting

Irrational

Numbers Not Considered Real

-5 7018%

− 23

82 √202

−2.73

2613

020

√1213

√−25-5

-5 7018%

82

82

82

82

26

26

26

26

−√67

Rational

Integers

Whole

Natural/Counting

Irrational

Numbers Not Considered Real

-5 7018%

− 23

82 √202

−2.73

2613

020

√1213

√−25-5

-5 7018%

82

82

82

82

26

26

26

26√1213

−√67

Rational

Integers

Whole

Natural/Counting

Irrational

Numbers Not Considered Real

-5 7018%

− 23

82 √202

−2.73

2613

020

√1213

√−25-5

-5 7018%

82

82

82

82

26

26

26

26√1213 √−25

−√67

Rational

Integers

Whole

Natural/Counting

Irrational

Numbers Not Considered Real

-5 7018%

− 23

82 √202

−2.73

2613

020

√1213

√−25-5

-5 7018%

82

82

82

82

26

26

26

26√1213 √−25

−√67

−√67

Rational

Integers

Whole

Natural/Counting

Irrational

Numbers Not Considered Real

-5 7018%

− 23

82 √202

−2.73

2613

020

√1213

√−25-5

-5 7018%

82

82

82

82

26

26

26

26√1213 √−25

−√67

−√67

−𝟐𝟑

Rational

Integers

Whole

Natural/Counting

Irrational

Numbers Not Considered Real

-5 7018%

− 23

82 √202

−2.73

2613

020

√1213

√−25-5

-5 7018%

82

82

82

82

26

26

26

26√1213 √−25

−√67

−√67

− 23−2.73

Rational

Integers

Whole

Natural/Counting

Irrational

Numbers Not Considered Real

-5 7018%

− 23

82 √202

−2.73

2613

020

√1213

√−25-5

-5 7018%

82

82

82

82

26

26

26

26√1213 √−25

−√67

−√67

− 23−2.73

√𝟐𝟎𝟐

Rational

Integers

Whole

Natural/Counting

Irrational

Numbers Not Considered Real

-5 7018%

− 23

82 √202

−2.73

2613

020

√1213

√−25-5

-5 7018%

82

82

82

82

26

26

26

26√1213 √−25

−√67

−√67

− 23−2.73

√202

𝟏𝟑

Rational

Integers

Whole

Natural/Counting

Irrational

Numbers Not Considered Real

-5 7018%

− 23

82 √202

−2.73

2613

020

√1213

√−25-5

-5 7018%

82

82

82

82

26

26

26

26√1213 √−25

−√67

−√67

− 23−2.73

√202

13𝟎

𝟐𝟎

𝟎𝟐𝟎

𝟎𝟐𝟎