The Real Numbers

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!!

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

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)

What’s the difference? Irrational numbers cannot be expressed as a ratio of integers (a fraction) Example: π or √11 or 0.12112111211112… (non-repeating & non-terminating decimals are irrational)

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

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

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

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

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.

The Path