Math geeks only
 

- OR -

Yes, I am geeky enough to be kept awake at night by number theory


I need some formal proof help. Never my strong suit. I grasp the general concept, but when it comes to being properly formal and pedantic, I get a bit fuzzy.

What I'm struggling with now is whether a certain kind of deductive conclusion is permissable, or if I need more rigor.

What I'm trying to prove is that for any integer value of n > 2, n*7 < 10^(n-1).

It's certainly a true statement, but I'm trying to formalize it. The best I've come up with is to know that for the value n=3 it is true and that the expression on the left grows linearly while the expression on the right grown exponentially.

Is that sufficient proof? That given any 2 functions f and f` such that f(n) < f`(n) AND f grows linearly while f` grows exponentially, f(x) < f`(x) for all values > n?

Not a math geek, but is 2 the absolute top of the non-qualifying values for n in this? I mean, short of testing it myself, is there any value of n=2.x that might also be disqualified?

I specified integer values only.

But to answer your question, yes. There is a non-integer value (that I'm ill equipped to work out at the moment) of n, between 2 and 3, for which n*7 = 10^(n-1). Below that value, the result is > 10^(n-1), above it, it's > 10^(n-1).

Ah, that would be me forgetting that integer meant a whole number

(Proof that I'm not a math geek... now ask me the integer value that represents how many times I had to take algebra II in order to pass it. Hint: n>2)

Doesn't it also work for zero?

Just because this IS a number theory thread, I should point out that in most mathematic circles "whole numbers" consist of the positive integers (or the very similar "non negative integers" which also includes 0). So while whole numbers are indeed integers, an integer is not necessarily a whole number.

But in this particular case, we do happen to be dealing with only whole numbers.

That wouldn't nullify the equation though. He's stating for all integer values greater than 2, but not addressing values 2 and under. Right?

Yes it does work for zero (7*0 = 0, 10^-1 = .1). However, I'm not concerned with any of the infinite other values of n may or may not work. All I'm interested in proving is that for intergers > 3, it's true. Anything else is irrelevant to the overall problem.

ETA: Thank you Mori, precisely.

How about proving it with a curve? Does that do anything for you? I mean, demonstrating that one increases linearly while the other increases exponentially... a curve would demonstrate that, yes?

(ETA: I can't believe I have even this many posts in a math thread... did I mention n>2?)

Then how much of a number theory problem is it?

Is "For all C > -40: 9/5 * C + 32 > C" a number theory problem?

Yes, the graphs seem to be visual proof. And my rather wordy "proof" above was essentially saying what the graph would say. But that feels a little circumstantial to me and may or may not be solid enough proof to include it as a rather pivotal step in a larger proof (the crux of number theory being to avoid any ambiguity where you just never know, somewhere waaaaay out on the number line you might run across a weird hiccough in the procedings).

A little bit of hand waving is acceptable in proofs. You don't have to go back and prove that 2>1 every time you start a proof. And I know this one is pretty trivial. I'm just not well versed in the theory enough to know if it's trivial enough for me to get away with leaning on some higher level concepts (exponential and linear growth) rather than formally proving the nuts and bolts.

Yes, it is. A very striaght forward one, but a provable theory none the less.

Like I said, I'm aware that it's a fairly self-evident truth. But in terms of mathematic rigor, would more detail be required (the answer may very well be no)?

Is it warm in here, or is it just me... anyone got a thermometer? ;)

Mori's getting all moist in the math thread!

</invasion of math thread by non-geek>

Solving for y1=x*7 at x=3; y=21.
Solving for y2=10^(x-1) at x=3; y=100.

Slope:
Derivative of y1: y1'=7 at x=3 (and all values of x)
Derivative of y2: y2'=10^(x-1); at x=3, y2'=100
Second derivative of y2: y2''=10^(x-1)

y1<y2 at x=3
y1'<y2' at x=3
y2'' is positive for all values of x indicating that y2' increases as x increases
therefore y1<y2 for all integers where x>2

That's way more complicated than just a pretty picture showing the plot of each equation on one graph, but I think it satisfies the burden of proof.

And also a slightly more formal way of saying "linear growth vs. exponential growth," got it. I think that's good enough for me.

But where do you get y2'=y2? Isn't the derivative of 10^x=10^x ln (10)? (too brain fried to figure out how the "x-1" affects that)

I have to ask.

Where in real life did this come from?

Nevermind, ln not log.

My warped mind and a cold that kept me awake.

A puzzle from NPR's Sunday Puzzle got me off on a tangent.

Good grief.

If you're that bored I have some math homework you can do ;)

A group of ducks is crossing a bridge. There's a duck in front of two ducks, a duck between two ducks, and a duck behind two ducks. How many ducks are on the bridge?

*MouseWife just passing through*
:p

Three

You're jumping to conclusions.

Which bridge?
Suspension bridge?
Golden Gate Bridge?
London Bridge?
Terabithia?

Led Zeppelin's "Confounded bridge"

Or could it be the bridge to the song Disco Duck?

Wait...it's Huey, Dewey and Louie.

What was the question again?

Conclusions jumped: That all of the ducks are walking single file. That all of the ducks are headed in the same direction. That the conditions of all of the ducks, vis a vis the other ducks, are described. That cricket is not involved.

Correcting for those conclusions I am going to say the answer is 15, one of which is a cricket batsman who scored a duck.

Honestly, I had forgotten how to do the derivative of 10^x so I looked it up. And the reference I used said it's 10^x log(10). Since log(10) is 1 I simplified. So I guess it's wrong.

Yes, it was a more convoluted way to say exponential growth, but it essentially proves that it is exponential growth.

Why did the ducks cross the bridge? Are they European or African ducks?

DUCKS!



Oh come on, you couldn't think that was going to go unmentioned.

What do you mean? An African or European duck?