Most of the numbers we know, and work with, are Real Numbers. The Real Number System includes counting numbers, fractions, terminating decimals, positive numbers, negative numbers, zero, repeating decimals, never ending and non-repeating decimals, numbers that are expressed as radicals, and even pi (π).
The set of Real Numbers contains a series of subsets of other number systems.
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Natural Numbers:
The natural numbers appear within the set of whole numbers.
• The natural numbers (symbol N) are the set of counting numbers {1, 2, 3, 4, 5, 6, ...}
• There are infinitely many numbers in this set of numbers.
• The natural numbers are "closed" under addition and multiplication.
definition
A set is closed (under an operation) if and only if the operation on any two elements of the set produces another element of the same set. If the operation produces even one element outside of the set, the operation is not closed.
 
The addition of two natural numbers creates another natural number.
The multiplication of two natural numbers creates another natural number.
Nclosed under addition and multiplication.
BUT ...
The subtraction of two natural numbers does NOT necessarily create another natural number   (3 - 10 = -7).
The division of two natural numbers does NOT necessarily create another natural number   (1 ÷ 2 = ½).
Nnot closed under subtraction and division.

Whole Numbers:
• The whole numbers are the set of counting numbers (natural numbers) along with zero
      {0, 1, 2, 3, 4, 5, 6, ...}
• There are infinitely many numbers in this set of numbers.
• The set of whole numbers is "closed" under addition and multiplication.

Integers:
• The integers (symbol Z) are the set of all of the natural numbers,
     plus their additive inverses (their negatives), and zero {...-4, -3, -2, -1, 0, 1, 2, 3, 4, ....}
• The integers are "closed" under addition, multiplication and subtraction,
     but NOT under division ( 9 ÷ 2 = 4½).

Rational Numbers:
• The rational numbers are the set of numbers which can be expressed as a ratio
     (a fraction) between two integers.
• Integers are rational numbers since 5 can be written as the fraction 5/1.
• Decimals which terminate are rational numbers. 8.5 = 85/10
• Decimals which have a repeating pattern are rational numbers. 1/3 = 0.3333333...
• The rational numbers are "closed" under addition, subtraction, and multiplication. Under division, we run into the problem of division by 0, which makes the statement that "the rationals are closed under division" false. Some texts state that "the rationals are closed under division as long as the division is not by zero" which is a true statement.

Irrational Numbers:
• The irrational numbers are the set of number which can NOT be written as a ratio (fraction).
• Decimals which never end nor repeat are irrational numbers.
• Irrational numbers are
"not closed" under addition, subtraction, multiplication or division.
    
While a few specific examples may show closure, the closure property does not extend
       to the entire set of irrational numbers.
• Examples of irrational numbers: rad2,
π

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Are there numbers that are not Real Numbers?

complex diagram

Actually, there is a another set of numbers, called Complex Numbers,
which contains the set of Real Numbers and some other interesting numbers.

The numbers that are in the set of Complex Numbers, but are NOT in the set of Real Numbers:
• the imaginary number, i, which is the square root of negative one.
pure imaginary numbers of the form bi.
complex numbers of the form a + bi, where a and b are real numbers.

Imaginary numbers are essential to the study of sciences such as electricity, quantum mechanics, vibration analysis and cartography.)

Do not confuse "imaginary" numbers with "irrational" numbers.
Not the same creatures!

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