Math Is Fun Forum

Is it (-7)(x^2) or (-7x)^2?

Using better values for pi, I got:

103993 / 33102 = 3.141592653

On a computer, you can't get more exact with 64bit float values (double).  However, I have long been thinking of making a float class which can represent n bit values by using arrays, maybe I'll start working on that now.

ii. By the ratio test:

Since

Which means that the series converges, although I know of no tests that actually find this number.

And why isn't the latex '<' not working?

Convergence of a solution?  Convergence is normally described for a function, not an equation.  I'm not sure what you mean by convergence in this context.

The quantum physics joke is that the coin lands and stays on it's side, and is both heads and tails at the same time.  Wave/Particle duality rocks!

I would say a lower order is simpler.  But then again, simple is all in context.  A more complex statement is simpler if it allows you to simplify a larger statement that it's in.

Why is a tiny value incorrect if you are trying to get values from 0 to 1?

If you are coding this on a computer or calculator, please, post your code.

What class are you in?  Do you wish to do a formal proof?  Formal proofs just mean you follow a certain structure in writing your proof.  I can help you on writing a formal one if you need it.  But if you're in high school, I doubt that you will.  That being said, the below is not formal:

There are two main ways you prove positive statements (if X, then Y).  One is a direct proof, and if you can do it this way, it is recommended you do so.

In a direct proof, you assum the "if" part, and try to show that this means the "then" part is true.

This means if P(n, m) is true, then Q(n, m) is true.  That's the same meaning as implies, as you wrote.

So we consider P(n, m) to be true.  That is, let n and m be odd integers.

What does this mean about n and m?  What is the definition of odd?  The offical definition is any integer n is odd when n = 2k + 1, where k is any integer.  Lets do the same for m: m = 2l + 1.  Note, you can't use k for both, as that would imply that n and m are both the same odd integers.

n = 2k+ 1, m = 2l + 1

Now we wish to show that nm is an odd integer.

nm = ?
nm = (2k + 1)(2l + 1)
nm = 4kl + 2k + 2l + 1

Is that number odd?  It should be apparent that it is.  This is because we can factor out a 2:

nm = 2(2kl + k + l) + 1

What is 2kl + k + l?  Why, that's an integer!  Let 2kl + k + l = x, where x is an integer.

nm = 2x + 1

Isn't that what we said our defintion of odd was?  So nm must be odd.

You try the second one, it should be easy.  The reason?  All you have to do is work the first one in reverse.  Start from the conclusion, and work your way back up.

For the thrid one, P(n,m) <-> Q(n,m), all you need to show is that P(n, m) -> Q(n, m) and Q(n, m) -> P(n, m).  And that's exactly what the first two are.  So once you've done these, all you have to do is state that the third is therefore true.

i * b = b, a * b = b

i * b = a * b

Let b-¹ be the inverse of b, which must exist since it's a group.

(i * b) * b-¹ = (a * b) * b-¹

The operation is associative so:

i * (b * b-¹) = a * (b * b-¹)

Definition of inverse, b * b-¹ = i (identity must be unique).

i * i = a * i
i * i = i, and a * i = a
i = a

That look good?

I completely understand what you are saying.  But each element of A is two elements.

You have a function: f(x, y).  You can't just plug one number into that.  You have to plug two in.  Because of these, the elements in set A become (n, m), where n and m are from 0-255.