Question

Which of the following is a true statement?

All square roots are irrational.

All decimals are irrational.

All non-perfect squares have irrational square roots.

All irrational numbers can be written as ratios of two integers.

Answer 1

Let's go through them, but first, do you know the definition of a rational number?

Answer 2

no

Answer 3

A rational number is one that can be expressed as one integer divided by another.

So, 8 is rational, because it can be written as 8/1. \(\pi\) is not, because it cannot be written as a fraction, only approximated.

An irrational number is a number which is not rational.

a) All square roots are irrational. Well, some of them certainly are, such as \(\sqrt{2}\). But all of them? What do you think?

Answer 4

i dont think all of them. just some

Answer 5

Right. For example, \[\sqrt{4} = 2\] and \(2\) is clearly rational, so there are some square roots which are rational numbers, and thus all square roots are irrational is not a true statement.

How about all decimals are irrational?

Answer 6

Let's consider the decimal 0.3 — how do you pronounce that? "three one hundredths", meaning \(\large 3/100\)

Well, that looks like a decimal which is not irrational. Scratch answer choice B from consideration.

Let's skip on to answer choice D: All irrational numbers can be written as ratios of two integers.

Didn't I say that the definition of a rational number is one that can be written as the division of two integers? If that's true, then D is also not a true statement, because it describes rational numbers, not irrational numbers.

Answer 7

Finally, we have answer choice C: All non-perfect squares have irrational square roots.

A perfect square is one that can be illustrated graphically with blocks 1 unit on a side. For example, 1, 4, 9, 16, 25, 36 are all perfect squares, because you can form a square array of blocks 1, 2, 3, 4, 5, and 6 units on a side, giving you 1, 4, 9, 16, 25, 36 blocks.

Any number which cannot be formed in that fashion can be shown to be irrational.

Answer 8

If you want an example of how that is done, see http://mathforum.org/library/drmath/view/57117.html

Answer 9

that is a lot of numbers and information... and im actually understanding! lol

Answer 10

Probably it is sufficient to just take me at my word, however, unless you are taking a number theory class :-) None of the other statements are true, and we are told to choose the one which is true, so this one better be true—the technical term for that is proof by multiple-choice question :-)