classification of the real number system. real rational - any number that can be written as the...
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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 Irrational
Not 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
Rational
Integers
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
Rational
IntegersIrrational
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
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