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 terminating or repeating decimal." As you dutifully copied down the definition, two things went through your mind:

1. Who comes up with this stuff?
2. 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 stepchildren 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.

Drills

1. Simplify .

The original radicals are both like terms: . So all we have to do is to add the coefficients: 2 + 5 = 7. That gives us (C) as the right answer. Make sure not to add whatever is under the radical.

2. Simplify .

6

When you multiply the numbers under the radicals, you get 12 × 3 under the radical, or . But 36 is a perfect square, and its square root is 6. Technically, we're splitting hairs here, since both (C) and (D) give us the same answer. We really need to get into the habit of simplifying our radicals, so we're sticking with (D) as the answer.

3. Simplify .

8

Multiply the numbers under the radical. If we do, we're left with , which is 8. In fact, this brings up a really cool property of radicals. When we square a square root, we're left with whatever is under the radical sign. Basically, a square root and an exponent of 2 cancel each other out. We think it's cool, anyway.

4. Simplify .

12

First divide the coefficients (deal with rational numbers first). Dividing 12 by 2 gives us 6. Hold that thought. Next, divide the irrationals. We can just take , which is the same as . In other words, . Or, if we want to get even simpler, 2. Take that 6 from the coefficients, the 2 from the radicals, and multiply them together, which gives us 12.

5. Simplify .

None of the above

Okay, cheap shot. But we know we can only add radicals when the numbers under the radical are the same. In this example, they're not. And there's nothing you can do about it. There's no way to simplify that radical 6 so it looks like a radical 3, except in Bizarro World. So the answer to this problem is . It can't be simplified any further.

6. Simplify .

Before we get into subtracting, let's see if we can simplify these bad boys, starting with . We can split 28 up into 2 × 2 × 7, so we can put a 2 outside the radical. That means  becomes (4 × 2), or . Now, we have to subtract that by . Not only is that possible, it's easy. All we have to do is look at the coefficients (which, in the case of , is 1). That means we have 8 – 1 = 7. So our answer is (A).

7. Simplify

This one's a bit trickier, so hold onto your hats. We can treat each number separately when we multiply (rather than addition and subtraction, where coefficients stick to their buddies), so it becomes . In other words, we have . We can multiply the numbers under the radical together, so we get . That's (D), but we can reduce  to  That will give us (C) as the right answer because it's the most simplified, even though (D) technically has the same value. Simplify your radicals, people.

8. Simplify .

As long as everything is being divided or multiplied, we can separate the rational guys from their irrational partners (if only it worked that way in relationships). That means we have , or , multiplied by . We can divide the two radicals so that they reduce to . Then we multiply the rational and (slightly less) irrational numbers back together again and get (D).

9. Which is an example of the sum of a rational number and irrational number being irrational?

First, let's consider what the problem wants from us: a sum. "Sum" means addition. Well, there goes (A). Next, we need a rational number and an irrational number, which eliminates (D). A closer look at (C) will give us the much simpler 4 + 6, since  is 6, a very rational number. That leaves (B), which is a rational number,  or 2, plus an irrational number, .

10. Which is an example of the product of a rational number and an irrational number being irrational?