Math Is Fun Forum

I never did buy that proof, but I like this one better.

Consider the distance between .999... and 1.  We will call this distance d.

We know d can't be less than 0, as this would result in a number that is less than .999... 

Let's assume d > 0.  By the Archimedian Principle, there exists an n∈N such that 1/n < d.  But d is the distance from 0.999... to 1.  That is, 0.999... + d = 1.  However, I believe we can all agree (although I'm unsure how to strictly prove) that 0.999... + 1/n ≥ 1.  But 1/n < d.  So 0.999... + d > 1.  Thus, d is not the distance, contradiction.

We shown that d is not greater than or less than 0, so the only thing we are left with is d=0:

0.999... + d = 1
0.999... = 1.

Chemist, nothing is going to work for all functions.  But (x,0) and (x,x) work for a whole lot, and they are easy.

If you plug in (x, mx) and get m, that is enought to say the function is not continuous.

yes, but the normal way to write that is:

y = 0 mod x.

Don't get me wrong, your way is perfectly fine too.  I guess I'm just used to seeing the mod and 0 on the right.

Your notation looks a bit weird to me.  Is that exactly from a question?  What normal notation would be:

Let f:A⇒B. Prove that if f is bijective, so is f-¹

Well, we know that f is a function.  Since f is a map:

For every element a∈A, there must exist a b∈B such that f(a) = b.

Applying an inverse to both sides, we get:

f-¹(f(a)) = f-¹(b)

Or:

a = f-¹(b)

For all a and some b.  Thus, f-¹ must be onto.

Now let f-¹(b1) = f-¹(b2) for any b1, b2 in B.  Since f is 1-1, f(f-¹(b1)) = f(f-¹(b2)).  But f(f-¹(b1)) = b1 and f(f-¹(b2)) = b2.  So b1 = b2.  Thus, f-¹ is 1-1.

Notice how you can go from A->B and then from B->C.  But remember, that f and g are invertible, so you can go from B->A and from C->B.  Use this, and show that if an element is in C, then the element must correspond to some element in A.

Try this, and if you can get it, the thrid should become much easier.

ryos, you'll know soon enough

MathIsFun, how similar is actionscript to java script?  I know basic java script and I plan on learning Flash soon (I got a book, just haven't opened it yet, that's the next step).  If it's pretty much the same, I can write psuedo code that should be easy to put in java script, and thus (hopefully) easy to put in action script.

I have spring break coming up, an entire week off from school with absolutely no plans.  So I'll be working on it then.

Ah, ok, didn't know about that Patrick.

Well Mathisfun?  Aren't you going to put mine on that page as well?

mathyperson, think circle, as in never ending.

That will only make sense if you understand the reason to the sequence.

This sounds more like a science problem than statistics.  There aren't really any outside factors when it comes to statistics, but when it comes to collecting data (science), there are (hypothetically) infinitely many.

To add on what mikau said about crop rotation, they also plant crops which puts nutrients back in the ground, such as peanuts.

But here are the things I would consider:

Enviornment: Could some plants get longer amounts of sunlight?  Or more water?  Is the soil composition mostly homogenous or is it heterogenous?  Could there by varying amounts of humidity?  How about oxygen/carbondioxide concentrations in air?  Did these vary?

People: In any step that includes a person doing something (anything) there is the possibility for human error.  But the specifics would be pretty much be methods for watering and measuring.  Can you find any possible error in these?

Many of these are nit-picks, that is, they probably really don't matter.  But even so, they could matter.

I made it.  Just simpily and you should get:

y/(ab) = x/b + 1/a, which of course is y = ax + b

In Virginia, we use:

Where p is a prime number.