1. The Real Numbers

2. What makes a number REAL?
Any number that can be shown on a number line is REAL!!
If a number cannot be shown on a number line it is not real. It is
IMAGINARY!!

3. Rational or Irrational?
All real numbers must be either
Rational or Irrational

4. What’s the difference?
Rational numbers can be expressed as a ratio of two integers (as a fraction)
Example: 2 is rational b/c it can be expressed as 2/1.
(repeating and terminating decimals are rational)

5. What’s the difference?
Irrational numbers cannot be expressed as a ratio of integers (a fraction)
Example: π or √11
or …
(non-repeating & non-terminating decimals are irrational)

6. Rational Numbers
If a number is rational it MAY also fit into other sub-categories.
INTEGERS
WHOLE NUMBERS
NATURAL NUMBERS

7. INTEGERS Rational numbers that are whole numbers and their opposites.
Examples: 0, 3, -3, 49, -49, etc.
NO DECIMALS

8. Whole numbers Rational numbers that are natural numbers and zero.
Examples: 0, 1, 2, 3, etc.
No negatives and no decimals.

9. Natural Numbers All “counting” numbers 1 to infinity.
Example: 1, 2, 3, 4,… to infinity
No decimals, no negatives, and no 0 !!!

10. And so……
In order for a number to be a natural number, it must be a whole number AND an integer.
In order for a number to be a whole number, it must be an integer.
In order for a number to be an integer, it must be rational.

11. The Path