Number Theory Help!...

What would these fit into.

{1, 1/3, -8, 9.6, 0, -5 2/5, Pi, 0.666..., 6, 721/1, Square root of 3, square root of 16, square root of -4, - square root of 25}

Whole Numbers
Integers
Rational
Real
Natural
Irrational
Imaginary

A Natural number, also called a counting number, is a non-negative integer that does not include zero (and the Greeks used to debate whether 1 should be included, because if you have "one" of something, you do not need to count it).

An integer is a whole number, positive or negative (includes zero). It is a number without any fractional part (hence the name "whole number")

A rational number is a number that expresses a ratio of two integers. Its value can be expressed as a fraction (a ratio) of two integers, normally called p and q
(the rational number p/q).
Integers are included in the rational numbers, as they are "ratios over unity" (meaning: q = 1)

The integer 7 can be expressed as the ratio 7/1; therefore the integer 7 is included in the rational numbers. So is zero.

If a number of any shape, size or form CAN be expressed as a ratio (even if it is presented differently), then it is counted as a rational number.

For example, the number
7.142857142857142857142857... (going on forever)
can be rewritten as 50/7
Because 50/7 is a rational number (ratio of two integers), then the number
7.142857142857142857142857... (going on forever)
is also a rational number.

Numbers that can be represented as the solution to geometrical problems are called real numbers.
For example, the square root of 2 (the hypotenuse of a right-angle triangle with both small sided equal to 1) is definitely NOT an integer. It is also a number that CANNOT be expressed as a ratio of integers (the Greeks had proven this over 2000 years ago). But it is a number that can be expressed as the solution to a geometry problem. It is therefore a real number. But because it is not a rational number, it is called an "irrational" number.