The last group of numbers that we are going to discuss are irrational numbers. Remember that rational numbers are numbers that can be represented as a ration of two numbers. Irrational numbers are numbers that can't be represented as a ration of two numbers. Irrational numbers fill in the spaces between rational numbers on the number line.
The Greek Pythagorean Society studied numbers and their effect in the universe. They believed all numbers were rational. When one of them asked what the square root of 2 was, they were so upset by the resulting irrational number that they took out in a boat, tied him up and threw him into the sea! Mathematicians are more receptive to new ideas these days.
Rational numbers are numbers that are either terminating or repeating and nonterminating. Irrational numbers are nonrepeating and nonterminating. Many numbers fall in this category. The roots of prime numbers are all nonrepeating and nonterminating. Roots of prime numbers are all irrational. An ancient special number is irrational. Pi is the ratio of the diameter of a circle to its circumference. This ratio has been known for thousands of years, but everyone in antiquity used a rational approximation to pi. Today, we recognize that pi can't be represented as a a rational number so we simply use the Greek letter π to represent it. Another special number is used to find an exponential growth like with rabbits or bacteria. It's called e and it's irrational also.
With the natural numbers, the integers, the rational numbers and the irrational numbers our alphabet is essentially complete. These numbers together are called the real numbers.
With our alphabet complete, we can begin talking about all topics in mathematics.
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