Rational numbers (“fractions”) have been in use for thousands of years and they, and their rules, have been independently invented at least dozens of times. Constructing the rational numbers and all of their behavior starts with an innocent statement; that “ is the unique number such that “. This one definition is the headwaters for all of the properties of rational numbers that follow. Starting with integers, along with regular addition, subtraction, and multiplication (basic arithmetic tools), and the definition of “ “, you’ll find that all of the weird properties of rational numbers come tumbling out. For example:Even better, we can describe exactly where the rational numbers are!

When you want to outsmart the world, you turn to the facts. And the facts are in the science.