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Lesson - The Real Number System
Contributed by: Moehle
(Original author: Warren)
There are 5 different classifications of numbersyou need to know: 
The Real Number
        System
natural a.k.a counting numbers
whole numbers
integers
rational numbers
irrational numbers
The natural numbers are also known as the
counting numbers. They begin with 1, 2, 3,...
and they continue indefinitely.
So, the number 254,673,843 is also a 
natural number.
Natural Numbers
The set of whole numbers includes all of the
natural numbers and ONE other number:

                          ZERO!

So, if we wanted to list the whole numbers
in ascending order we would start:
0, 1, 2, 3, 4, 5, 6, 7...
Whole Numbers
The set of numbers called integers (this word
comes from the same root as the word integrity)
include all of the whole numbers (and, thus, allof the natural numbers), as well as,their opposites (negatives).So, we say that integers are all of the positiveand negative whole numbers and zero.
...-3, -2, -1, 0, 1, 2, 3,...
Integers
The root word of rational is ratio, so all rationalnumbers can be written as ratios (a.k.a. fractions).This set of numbers includes all integers, wholenumbers, and natural numbers as well as allnumbers that can be written as a fraction of twointegers, or a terminating (ending) or repeatingdecimal.
Examples:  3/4     -8/2     0.342       -0.75    √64
-15/3, -3.5, -1, 0, 1.2, 6/9, 32/8, 65/6
Rational Numbers
Irrational numbers are a set of numbers thatcannot be written as fractions or decimals thatterminate or repeat. They must be written usingspecial symbols because, if we tried to write a decimal equivalent, we would never be ableto stop writing.
The most commonly known irrational number isπ (pi). All square roots of numbers that are notperfect squares are irrational (i.e. √7, √94, √2).
Irrational Numbers
-56
Rational Numbers
Whole Numbers
-7.5 = -15/2
The Real Number System
Natural
Numbers
0.75 = 3/4
0.6 = 2/3
Integers
0
103
√9
-√4
Irrational Numbers
√6
0.1010010001...
√2
√17
π
√5
√3
e
Let's check what you know:
How would you classify 1,345,752?
Irrational
Rational only
Rational and integer only
Rational, integer, and whole only
Rational, integer, whole, and natural
1,345,752 is rational, integer, whole, and natural
How would you classify 1/3?
Irrational
Rational only
Rational and integer only
Rational, integer, and whole only
Rational, integer, whole, and natural
1/3 = 0.3 and is rational only
How would you classify 8/2?
Irrational
Rational only
Rational and integer only
Rational, integer, and whole only
Rational, integer, whole, and natural
8/2 = 4 is rational, integer, whole, and natural
How would you classify -10?
Irrational
Rational only
Rational and integer only
Rational, integer, and whole only
Rational, integer, whole, and natural
-10 is rational and integer only
How would you classify -2/4?
Irrational
Rational only
Rational and integer only
Rational, integer, and whole only
Rational, integer, whole, and natural
-2/4 = -1/2 = -0.5 and is rational only
How would you classify √11?
Irrational
Rational only
Rational and integer only
Rational, integer, and whole only
Rational, integer, whole, and natural
Irrational
Rational only
Rational and integer only
Rational, integer, and whole only
Rational, integer, whole, and natural
√11 is between 3 and 4 and can only beapproximated and is irrational
How would you classify √81?
√81 = 9 and is rational, integer, whole, and natural
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