High School: Number and Quantity
The Real Number System HSN-RN.B.3
3. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
At some point in your academic past, you were introduced to the concept of irrational numbers. The teacher probably said something along the lines of, "An irrational number is a number which cannot be written as a non-terminating, non-repeating decimal." As you dutifully copied down the definition, two things went through your mind:
- Who comes up with this stuff?
- When is lunch?
Realistically, we know you really didn't care who came up with this stuff or why, and we're hoping that you got your lunch at some point that day. But we'd like for you to give your students a more realistic (if slightly less mathematically accurate) definition of irrational numbers.
You know when you have an argument with your parents or your siblings or your significant other, and you hear the words, "You are just so irrational!" (Don't lie. We've all been there before.) Well, they love you and all, but they weren't giving you a compliment.
"Irrational" is an ugly word. It implies ugly things.
Irrational numbers are those numbers your students will probably find "ugly." In fact, irrational numbers are so ugly that they need their own symbols, like radical signs or π. They're the evil stepsisters of algebra. (As a math teacher, though, you can probably appreciate the beauty of irrational numbers. Beauty really is in the eye of the beholder.)
Irrational numbers contaminate everything they touch. Add an irrational to a rational, and our answer is irrational. Multiply an irrational by a rational, and our answer is irrational. In fact, the only hope for civilized society is when we multiply or divide irrationals—sometimes the answer will be rational. If we're lucky.
For example, if we multiply by , we get , which is rational since it can (and should, for the sake of everyone's sanity) be simplified to 4.
If we multiply a nice, normal 6 by that ugly irrational , it remains irrational; our answer is .
Basically, here's the rule for multiplying and dividing irrational numbers. Work first with any coefficients (the number before the radical sign) then work with the radicals. If you happen to get something that simplifies to a nice, pretty, rational number, then you really should do so.
To add or subtract, you do the same thing: Simply your radicals so they're "like terms"—all the square root of the same number—if possible, then combine the like terms.
Here is a recap video for rational and irrational numbers.