Lesson set of real numbers

Historically, the real number system evolved by expanding the notion of "number" as something you could count. These are called "natural numbers".

*NATURAL NUMBERS (N):
1,2,3,4...
(Three dots at the end mean that the list keeps going on and on).
In Figure 1 natural numbers are shown to the right from "zero".


Figure 1

*WHOLE NUMBERS (W):
Whole numbers are natural numbers and "zero":
0,1,2,3,4...
Zero is shown in green color in Figure 1.

*INTEGERS (J):
...-4,-3,-2,-1,0,1,2,3,4...
Here negative numbers are added.
In Figure 1 negative numbers are shown to the left from "zero".

If we add fractions to the set of integers, we get the set of "rational numbers".

*RATIONAL NUMBERS (Q):
all numbers of the form a/b, where a and b are integers (b cannot be zero).
Rational numbers are fractions that can be presented either by terminating decimals (for example, 1.4, 1.3467) or non terminating repeating decimals(for example, 3.2222..., 5.818181...).


Figure 2

All previously defined sets of numbers (N, W and J) are subsets of the rational numbers (Q).

There are numbers that cannot be expressed as fractions, and these numbers are called IRRATIONAL NUMBERS.

*IRRATIONAL NUMBERS (H):
Cannot be expressed as a fraction (for example, square root of 2 or 3: =1.414214...,=1.732051...).
As decimals they never terminate and have no repeating pattern (for example, 1.234567891011...).