Question

Can you check my answer?
Which of the following is an example of why irrational numbers are 'not' closed under addition?

√4 + √4 = 2 + 2 = 4, and 4 is not irrantonal
1/2 + 1/2 = 1, and 1 is not irrational
√10 + (-√10) = 0, and 0 is not irrational
-3 + 3 = 0, and 0 is not irrational

I was thinking:
–3 + 3 = 0, and 0 is not irrational
because it came up with a different number besides 3.

Answer 1

so does
√10 + (-√10) = 0, and 0 is not irrational

Answer 2

√4 + √4 = 2√4 and is irrational.

Answer 3

watever u picked, observe that -3 and 3 are not irrationals. so they wont output irrationals anyways

Answer 4

But that would still end with a √4 so wouldn't it be one of the last two?
I thought the end had to match what you already had. I thought the 3 and -3 was the answer because of the 10 and 0

Answer 5

i think until you add them with themselves and add the root of square you will see that!

Answer 6

To say that it is "closed under addition" means that anytime you add two irrationals, you must get an irrational.

So to show that is NOT true, you need to add two irrationals, but get a rational number.

Answer 7

√4 is rational, so we may discard that option.

Answer 8

^ Debbie

Answer 9

√10 is 3.66666... and -√10 is 3.1622... so if you add those is would still be irrational but when you add 3 and -3 it would still be rational

Answer 10

Noooo. √10 is √10 . 3.66666 is an approximation for √10 . is it NOT = √10 .

Only √10 is = √10 . :)

Answer 11

(actually typo above: √10=3.1622...)

√10 + (- √10 ) \(\neq\) 3.1622 + (-3.1622)

Because √10 and -√10 are exact. Not a decimal approximation.

Answer 12

so if they are the same does that mean that cant be the asnwer?

Answer 13

I don't understand your question. "if they are the same..." (if what is the same??) "does that mean that they can't be the answer?" huh? sorry.

Answer 14

Really, even with the decimal approximations of \(\sqrt{10}\) and \(-\sqrt{10}\), if you take them to the same number of decimal places, when you add you'll still get 0.

But it is still important, I think, to understand that \(\sqrt{10}\) is the only "exact" representation of \(\sqrt{10}\).

Answer 15

Wait.... do you mean, because \(\sqrt{10}\) and \(-\sqrt{10}\) are "the same"? Is that what you were trying to ask?

Answer 16

Because, well, they aren't the same. They are two different numbers. They have the same square, but they are two, distinct, irrational real numbers. And 0 is as good a rational number as any other rational number.

Again, to show that the irrationals are NOT closed under addition, what you must do is:
Find a sum of 2 irrational numbers that is a rational number.

Any 2 irrational numbers, that sum to any rational number. There are no restrictions, other than that they both be irrationals.

Answer 17

I still don't understand what the difference between the last two were which is what I was asking so I guessed and it was right.

Answer 18

Well, good, but wouldn't it be even better to understand?

√10 + (-√10) = 0, and 0 is not irrational
---- this is a sum of two irrational numbers

-3 + 3 = 0, and 0 is not irrational
---- this is a sum of two RATIONAL numbers

Showing that two RATIONAL numbers sum to a RATIONAL number, doesn't prove anything about IRRATIONAL numbers.

That's the difference.