Intuitively, Real Numbers are the union of both rational and irrational numbers.
Such numbers can be both positive and negative.

Let's refresh our memories on how the number systems of real numbers, rational numbers and irrational numbers are related to one another.


Most of the numbers we know, and work with, are Real Numbers. The Real Number System (symbol r) 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.
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.

Whole Numbers:
• The whole numbers (symbol W) 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 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, ....}

Rational Numbers:
• The rational numbers (symbol rational) are the set of numbers which can be expressed as a ratio
(a fraction) between two integers (where the denominator is not zero).
• 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..

Irrational Numbers:
• The irrational numbers are the set of number which can NOT be written as a ratio (fraction).
• The symbol for Irrational numbers can be R-Q(meaning Reals minus Rationals), or P)
• Decimals which never end nor repeat are irrational numbers.
• Examples of irrational numbers: rad2 and π


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 (where b is not zero)
complex numbers of the form a + bi, where a and b are real numbers (and b is not zero).

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!


NOTE: The re-posting of materials (in part or whole) from this site to the Internet is copyright violation
and is not considered "fair use" for educators. Please read the "Terms of Use".