All integers and rational numbers are also real numbers.
The real numbers include "algebraic" numbers, which are the results of finite algebraic problem.
For example, the solution to
x^2 - 2 = 0
is
x = √2 (square root of 2).
It is a real number (exists in geometry) but it is not an integer nor a rational number.
It is an irrational number, and an algebraic number.
pi (the ratio of the circumference to the diameter of a perfect circle) is also a real number (exists in geometry) but is NOT the result of a finite algebraic expression.
But it is an irrational number (cannot be expressed as a ratio) and is also called a "transcendental" number (cannot be expressed as a finite algebraic problem).
There exists algebraic expressions for pi, for example
pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 - 1/13 + 1/15 - 1/17.....
but these expressions go on forever (they are not finite).
Then, you have numbers that were "imagined" as a way to solve algebraic problems that simply had NO solutions:
x^2 + 1 = 0
The Greeks already knew that the square of any number was always positive (except 0, which gives 0). Therefore, it is impossible, with any real number, to add 1 to a square, and get 0.
A few centuries ago, mathematicians "imagined" a number, called "i" (i for "imaginary" -- no kidding) and they defined this number like this:
i is an imaginary number such that
i^2 = -1
("i" itself is not defined, since it falls outside any real number -- it is "imagined")
(but its square does give a real number = -1)
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1 is a natural number (counting?), a whole number, an integer, a rational number (because it can be written as 1/1), a real number and an algebraic number (it is a solution to x^2 - 1 = 0).
1/3 is a rational number, a real number and an algebraic number.
-8 is an integer (but not a natural number, since it is negative) and is also rational, real, algebraic.
9.6 can be written as 96/10 = 48/5 (rational, algebraic, real)
0 Definitely an integer; not a counting (natural) number. Because it is an integer, it is automatically rational and real.
-5 2/5 can be written as -27/5 (rational, algebraic, real)
pi is real (irrational, transcendental)
0.6666666... can be rewritten as 2/3 (rational...)
6 and 721/1 are integers (therefore they are also....)
√3 is algebraic (x^2 - 3 = 0) but not rational. It is therefore irrational (and real)
√16 is an integer (4) and so is √25
square root of -4 does not exist (you cannot take a real number of any kind and multiply it by itself to equal a negative value). Therefore it is an "imaginary" number.