1)Prove that and are irrational.

3)Suppose that where is an -degree polynomial. Prove that the leading coefficient of is .

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- February 12th 2008, 07:23 PM #1

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- February 13th 2008, 07:13 AM #2
Suppose we let:

Then:

Therefore:

Now we take the real parts:

Using throughout:

We know that:

This is clearly irrational, therefore:

If was rational, would also be rational, which is a contradiction.

Finally:

Using

We get:

If was rational, then would also be rational, and if was rational, then the statement above would also be rational, we have thus arrived at a contradiction:

is irrational.

I have to go to class someone else do or I'll do it later.

- February 13th 2008, 10:27 AM #3
For

Using the math above and taking the imaginary parts:

Using

We know that:

That is clearly irrational.

sin(3) is irrational since sin(15) is irrational.

Finally:

Therefore:

Therefore, since sin(3) is irrational, sin(1) is also irrational.

Using this as logic:

Same with cos(nx)

- February 24th 2008, 12:51 PM #4

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2)We are given that where is a polynomial. Suppose we wish to compute the integral of over the interval . Then . Now there is another way to compute this integral, and that is to use a Riemann sum with equal subdivision point using the right endpoint. In that case we get . This limit needs to be since that is the integral. But this means the polynomial must be degree with leading coefficient of .

In fact, note the following,

.

.

.

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