• Find the sum or product of two rational numbers and explain why the sum or product is rational
  • Find the sum or product of a rational and a irrational number and explain when the sum or product is irrational

Assignment

  • p9 1–10, 15–23, 25–31 odds, 32–39

Extra Practice

Rational and Irrational Numbers

We’re going to start with talking about two different type of numbers: rational and irrational. Rational numbers are the ones you are used to seeing in math classes. It includes whole numbers, like $3$, fractions like $\frac{2}{5}$, and decimals like $0.7$. They also include repeating decimals, like $0.333\overline{3}$.

Those are all rational numbers because they be written as ratios, or fractions. $3$ is the same as $\frac{3}{1}$, $0.7$ is equal to $\frac{7}{10}$, and $0.333\overline{3}$ is just $\frac{1}{3}$.

Irrational numbers are numbers you likely haven’t run into very often. They include square roots like $\sqrt{2}$, and other never-ending non-repeating decimals. Pi $\pi$ is the most famous of those.

\[3.14159\,26535\,89793\,23846\,26433\dots\]

The exception is are numbers like $\sqrt{9}$, because that simplifies to $3$, so it’s a rational number.

Type Examples
Rational $15, \, \frac{7}{9}, \, 1.14, \, 0.111\overline{4}, \, \sqrt{4}$
Irrational $\sqrt{5},\,1.618033988\dots$

Math with Rationals and Irrationals

There are a few different ways we’re going to work with these numbers. The first is comparing them to see which is larger. The most reliable way to do this is with a calculator. Use division is you are dealing with fractions, and the square root function if it’s a square root, and then compare the decimal results. Start with the left most digit, move right until one is bigger than the other. Just make sure your decimal points are lined up.

The other way we’ll work with them is with addition and multiplication, but we’re not concerned with getting answers at this point (also known as evaluating). We just want to know what kind of a result we’ll get. Will we get a rational or irrational number back out?

This is a lot easier than it sounds. If you are working with just rational numbers, the results will be rational. There is a simple proof of this in the book if you are curious how we can know that without testing an infinite number of numbers.

The other rule is if we have a rational and an irrational, then the result will always be irrational. They tend to muck things up with their irrationality, so once they get involved everything goes irrational.

What about two irrational numbers? Well, then it depends. Most of the time, an irrational number gets spit back out, but a rational number is always a possibility. For example, ${\sqrt{3} \cdot \sqrt{3}}$. That’s the same thing as $\sqrt{3}^2$, which is just $3$.

Numbers involved Result
rational and rational rational
rational and irrational irrational
irrational and irrational ¯\(ツ)/¯