Number Theory and Sets

Early BirdI wanted to include the diagram of number sets from today’s class on the website along with clarification of what numbers are included in various sets. Don’t forget to ask if you need clarification. Questions during class usually help everyone learn!

This chart shows how various sets of numbers are related and how some sets contain other sets.
This chart shows how various sets of numbers are related and how some sets contain other sets.

To review some older number theory, remember that you first learned your counting numbers:

\left\{ 1,2,3,4,5... \right\}

Then, you added zero, and got whole numbers:

\left\{ 0,1,2,3,4,5... \right\}

Eventually, you added the complexity of negative numbers and got integers:

\left\{ ...,-3,-2,-1,0,1,2,3,... \right\}

Rational numbers include any number that can be written as the ratio of two integers such as:

\left\{ 0,-1,37,\frac{14}{3},\frac{3}{2} \right\}

Rational numbers when written in decimal form will always have decimals that either terminate or repeat.

\begin{array}{l}\frac{1}{4}=0.25\\\frac{1}{6}=0.1\bar{6}\\\frac{1}{9}=0.\bar{1}\end{array}

Irrational numbers cannot be written as a ratio of two integers. A rational number can’t be irrational, and an irrational can’t be rational. Two famous irrational numbers are \pi and e. Irrationals are also found when taking the square root of numbers that are not perfect squares like:

\left\{ \sqrt{27},-\sqrt{31} \right\}

Note that it’s perfectly ok to take the negative of the square root of 37, but if you try to find the square root of negative 37, you encounter a pegasus unicorn, or an Imaginary number. You’ll learn more about those in future coursework.

Real numbers (often shown using the symbol: \mathbb{R}) include all of the irrational, rational, integers, whole, and counting numbers.