Showing that a number is rational/irrational

I have a bunch of questions that I don't know how to approach about whether a number is rational or irrational

Could someone give me an idea of how to approach questions like the following:

1 + srt(2) + srt(3/2)

Is this rational or irrational (and why)?

thanks :)

Re: Showing that a number is rational/irrational

Quote:

Originally Posted by

**kinhew93** I have a bunch of questions that I don't know how to approach about whether a number is rational or irrational

Could someone give me an idea of how to approach questions like the following:

1 + srt(2) + srt(3/2)

Is this rational or irrational (and why)?

Can you classify each of as rational or irrational ?

Re: Showing that a number is rational/irrational

Quote:

Originally Posted by

**Plato** Can you classify each of

as rational or irrational ?

Well obviously 1 is rational, srt(2) and srt(3) are irrational, but I don't know about srt(3)/srt(2), or what happens when you sum rational and irrational numbers?

Re: Showing that a number is rational/irrational

Quote:

Originally Posted by

**kinhew93** Well obviously 1 is rational, srt(2) and srt(3) are irrational, but I don't know about srt(3)/srt(2), or what happens when you sum rational and irrational numbers?

If were rational then what would mean ?

Would that mean where is rational ?

Re: Showing that a number is rational/irrational

Quote:

Originally Posted by

**Plato** If

were rational then what would mean ?

Would that mean

where

is rational ?

Yes it would but I don't see the issue with that? I know that the product of a rational and irrational number is always an irrational number...

Re: Showing that a number is rational/irrational

Re: Showing that a number is rational/irrational

Quote:

Originally Posted by

**Plato** Well, if

where

is rational does that not mean that

where

is rational ?

What is wrong with that?

I really don't know. Is it that 3/srt(6) cannot be rational? If it is is still cannot see why

Re: Showing that a number is rational/irrational

Quote:

Originally Posted by

**Plato** Well, if

where

is rational does that not mean that

where

is rational ? What is wrong with that?

Quote:

Originally Posted by

**kinhew93** I really don't know. Is it that 3/srt(6) cannot be rational? If it is is still cannot see why

You have a huge gape in understanding about rational numbers,

If is a **non-square** positive integer then is irrational. Can you prove that?

So if where is rational what in wrong with that?

Re: Showing that a number is rational/irrational

Quote:

Originally Posted by

**Plato** You have a huge gape in understanding about rational numbers,

If

is a

**non-square** positive integer then

is irrational. Can you prove that?

So if

where

is rational what in wrong with that?

Ok gotcha. Ive actually done this question now but I still don't see what I can say about the sum of irrational numbers?

What can be said about

Srt(2) + srt(3) + srt(6)

?

Re: Showing that a number is rational/irrational

Let's go back to your original question.

If is rational, surely r - 1 is rational.

So let's focus on s = r - 1 instead. Again, if s is rational, surely s^{2} is rational as well.

Now: .

If s^{2} is rational, then t = s^{2} - 7/2 must also be rational.

Now , and if t is rational then t/2 = (1/2)t is also surely rational, which means that the square root of 3 is rational.

But this is absurd.

Working backwards, we see that t cannot be rational, which means that s^{2} cannot be rational, which means that s itself cannot be rational.

But this means (finally!) that r cannot be rational, for assuming r rational leads to .

Re: Showing that a number is rational/irrational

The way **I** would do this would be to note that .

Now multiply that by to get

Finally multiply by 4 to get

a fourth degree polynomial with integer coefficients.

Since that was created by multiplying by other terms, is a factor so is a zero of the polynomial.

Now, the "rational root theorem" says that if a polynomial with integer has rational zeros then they must be of the form where the denominator, n, evenly divides the leading coefficient, here "4", and the numerator, m, evenly divides the constant term, here 13.

The only integers that evenly divide 4 are 1, -1, 2, -2, 4, and -4. The only integers that evenly divide 13 are 1, -1, 13, and -13. Thus, the only rational number that could possibly satisfy the equation are 1, -1, 13, -13, , , , , , , and . But putting those into the polynomial show that, in fact, none of them are actually zeros. Therefore no zeros of that polynomial are rational. Since **is** a root, it cannot be rational.