The length of a rectangular picture frame has been found to be an irrational number. Dwayne says that because the length is an irrational number, the perimeter and area would always be irrational. Using complete sentences, critique Dwayne’s statement with examples that demonstrate how he is correct or incorrect.

Question

cwrw238 · Answer

If the width is rational then perimeter will be irrational.
If the width is irrational then perimeter will also be irrational

eg 4 + 2sqrt5
or 2sqrt3 + 2sqrt5

For the area  
if width is rational then the area will also be irrational  
eg 2 * sqrt5

if width is also irrational then the area will also be irrational

eg sqrt3 * sqrt7 = sqrt21

the only time you would have a rational area is if its a square 

eg sqrt3 * sqrt3 = 3

anonymous · Answer

Im sorry but I still dont really get it, since I have to put an example I dont want to copy yours

kinggeorge · Answer

@cwrw238 That's not entirely true. For example, you could have a rectangle with width $\sqrt2$ and height$\sqrt8$. Then the area would be 4, but it would not a square.

@kunfuzzled If you don't want to use the example I just used, can you tell me what a very similar example might be (say with $\sqrt3$ instead of $\sqrt2$)?

cwrw238 · Answer

right KingGeorge I missed that !!

anonymous · Answer

@KingGeorge I understand what you're saying, and I guess I will just switch around the numbers

kinggeorge · Answer

Also with the perimeter, you could have length $3-\sqrt2$ and width $\sqrt2$. Then the perimeter would be 6.

anonymous · Answer

So basically, the overall answer would be an irrational plus an irrational does not equal an irrational, and give one of those examples?

kinggeorge · Answer

You need to be a little careful with the wording. Most examples of irrational+irrational are still example irrational. For example $\sqrt2+\sqrt3$ is irrational. But that's not always the case. The examples I provided give ONE example, and don't show that irrational+irrational is always rational.

anonymous · Answer

Ok, thanks for the help!

cwrw238 · Answer

sorry kunfuzzled - I obviously didn't think that one out enough....

anonymous · Answer

@cwrw238  It's fine!

kinggeorge · Answer

You're welcome.

I should also mention that I can't find an example yet of a rectangle with irrational length and width that has both rational perimeter and rational area. I'll keep looking into that additional restriction.

anonymous · Answer

@KingGeorge oh, okay!