Recent content by gunch

I'm not bothering to show the base cases as they are easy. I'm leaving out some detail here and just giving a proof sketch, but the general approach is as follows: Strengthen the induction hypothesis slightly by claiming that f(y) \in \{0,1,2\} in addition to f(y) \equiv y \pmod 3. The key...

I'm pretty sure it naturally occurred to me. I was raised by Christian parents who didn't impose their views on me, but if asked they would simply state that they believed in a god. I think it was in 4th grade that a teacher first mentioned evolution in a class in reference to how animals...

Remember that these are competitions so the problems are made sufficiently hard that many good students won't be able to solve them. Therefore you shouldn't feel bad about not being able to solve them. This is not to say that you shouldn't try though. Recreational/competition math is kind of fun...

How about simply letting the induction hypothesis being: P(n)=there exist positive odd integers x,y such that 2^n=7x^2+y^2? You have shown P(3) and you are pretty close to showing that P(n) implies P(n+1). If you have shown that exactly (x+y)/2 or |x-y|/2 is odd you can chose one of those to...

Considering the whole thing mod 121 quickly gets messy, and we see that 11|33 so we should instead try to see what we get if we consider it modulo 11 instead. From your factorization you should be able to see that, 11|(n+7)(n-4)+33 What can you say about n modulo 11 from this?

I respectfully disagree. For a real, well-designed physics test this may be the case, but due to the standard format of the SATII and the fairly limited amount of stuff covered and the shallow way in which it's covered it's extremely easy to cram for it. If this is simply for SATII, not...

I depends on how you define decision problem, Turing computation, etc. Take any book on computation theory, start with the definition of complexity classes given by AUMathTutor and work backwards by substituting each concept by its definition and eventually you should end up with an ugly...

Assuming you mean the factorial of 0, then factorial is usually defined recursively by, 0! = 1 n! = (n-1)! * n for n > 0 So it's true by the definition of the factorial. If you mean why 0 doesn't equal 1 then you have to state explicitly some formal properties of the integers. For instance a...

As you said yourself the rule only applies to a nonsingular system of linear equations, not to arbitrary systems of equations (this is one of the most basic facts in linear algebra). Consider, a^2 + b^2 = 0 with a,b unknown real variables. We have two unknowns, one equation and one unique...

This is incorrect. It yields, d = (c-10)(c+12) which is prime if c=11. So h=2, c=11, d =23 is a solution (I assume this is the one HallsofIvy meant).

By the definition given in your book, then no they aren't equal and that is the ordinary definition. I have never seen them defined in terms of ranges instead of codomains. Properties such as surjectiveness would stop making sense. Consider the floor function; with the real numbers as the...

This is quite surprising to me because I have a TI-89 Titanium and it can easily solve inequalities. Writing your original query I got: -1<x<1. I was under the impression that the CAS was identical, but apparently they upgraded it.

How exactly do you mean think? Do you want the definition usually given, or the intuitive concept of subtraction. Normally we take it as an axiom that every x has an additive inverse y such that x+y=0, and we then think of subtraction as the addition of the additive inverse. Or in more ordinary...

I don't really see why those are issues. Wikipedia is not a math textbook and should therefore not adhere to the same standards. Under definition, it defines pi as, "In Euclidean plane geometry, pi is defined as the ratio of a circle's circumference to its diameter" So it specifically mentions...

I don't have any experience with the books you are considering so I can't say whether they are right for you, but I have tried other "for dummies" books and in my experience they only give you a superficial understanding of the subject matter (I know others who swear by them, so your experience...