God Created the Integers

A path is a function from an interval, like [0,1], into a space of
some sort.  A path is smooth if it's differentiable many times (often,
infinitely many times).

A continuous path is one that's well, continuous.

In a partially ordered set, a `chain' is a subset which is totally
ordered (that is, any two elements are comparable).  An upper bound of
a chain (or any other subset) is an element which is greater than or
equal to everything in the chain.

Zorn's Lemma, which is equivalent to AC, says that any partially
ordered set in which all chains have upper bounds has a maximal element.

Perhaps the `relation' referred to is the partial ordering.

I guess we all know what a function is.

A simple group is one with no nontrivial normal subgroups.  The
classification of all finite simple groups was one of the major recent
achievements of group theory.

In fact, there is precisely one group of order two (up to isomorphism),
and it is finite and simple.

Probably a reference to the identity element of a group.

A tensor is one of those things that's hard to explain... probably the
best concise description is that it's like a matrix, but with
(possibly) more dimensions.  But it's also more general, for instance
it applies to modules over any ring, rather than just vectors.

`WLOG' is often used in a proof to avoid saying similar things over
and over.

A map (of groups, or really anything) is sometimes called faithful if
it is injective (that is, one-to-one), which makes it a bijection onto
its image.  Quotienting by a subobject, such as the image of a map,
means forming a new group in which everything in the subobject is set
to zero.

Two vector bundles are said to be `stably equivalent' if they become
equivalent when both are direct-summed with a large trivial bundle.
The same idea applies to other types of things, like modules over a
ring.

A principal bundle is a bundle whose fibers are the same as the
structure group.

Differential forms are a higher-dimensional vector-like-thing which
generalize vector algebra and vector calculus.  An n-dimensional form
is often called an n-form.  The cross product of vectors generalizes
to the `wedge product' of n-forms, so-called because it is written
as $\wedge$.

Complexifying a real vector space (or bundle, etc) means to allow its
scalars to become complex numbers, rather than only real ones.  In
fancy language, that means `tensoring' with the complex numbers.

A space is `simply connected' if every loop inside of it can be
continuously deformed back to a single point, i.e. if its fundamental
group is trivial at all basepoints.

An open set in R^n is one which contains a small ball around each of
its points.  All topological spaces also come with a notion of open
set.  A dense set is one whose closure is the whole space, so that it
gets arbitrarily close to every point.  For example, the rational
numbers are dense in the real numbers.  There aren't any nontrivial
sets in the reals which are both open and dense, but in some other
spaces there are.

A common construction is to take the `limit' of a sequence of things.
A directed system is a more general thing you can take the limit of;
the objects don't arrange themselves quite in a sequence, but they are
*partially* ordered, and given any two of them, there's another object
which comes after both of them.

The kernel of a map is the set of all things which get mapped to zero.
The rank of a linear map is the dimension of its image.  The dimension
of the kernel and the rank always sum to the dimension of the domain,
so a map of rank one will generally have a large kernel.

An operator generally refers to a sort of linear map.  Operators can
be smooth, and I think analysts talk about different `classes' of
operators, but here I get out of my depth.

Obviously a `mirror pair' has some meaning, but I don't know it.

A functor is a map between categories.  A forgetful functor is one which
just forgets some structure on the objects of its domain.  For example,
the functor from the category of groups to the category of sets which
takes a group to its underlying set of elements is forgetful, because it
just forgets the group operation and identity element.

Associativity refers to the property x+(y+z) = (x+y)+z of addition,
multiplication, and other operations (such as the operation in any
group).

A group action is `free' if no element of the group fixes any point of
the acted-on set.

I believe `purely inseparable' is a term applied to field extensions.
I don't know what it means, though.

Well done.  A few additional comments...

> My heart was open but too dense

An open set in R^n is one which contains a small ball around each of
its points.  All topological spaces also come with a notion of open
set.  A dense set is one whose closure is the whole space, so that it
gets arbitrarily close to every point.  For example, the rational
numbers are dense in the real numbers.  There aren't any nontrivial
sets in the reals which are both open and dense, but in some other
spaces there are.

The complement of any discrete set is both open and dense in the reals.
In particular, the complement of any finite set will be both open and dense.
The Zariski topology on a variety (or scheme) has the property that ANY
open set is dense.

> But we're a mirror pair, me and you,

Obviously a `mirror pair' has some meaning, but I don't know it.

I, also, am out of my depth, but I think this is a reference to mirror symmetry in physics.

> And be a finite simple group, a finite simple group,
> Let's be a finite simple group of order two
> (Oughter: "Why not three?")

It took me a long time to figure out why everyone groaned at the "Why not three?" line.
I was just thinking about the mathematical meaning...

> I've proved my proposition now, as you can see,
> So let's both be associative and free

Associativity refers to the property x+(y+z) = (x+y)+z of addition,
multiplication, and other operations (such as the operation in any
group).

A group action is `free' if no element of the group fixes any point of
the acted-on set.

There are lots of other meanings of free too.  For example, a free group
is one with no relations.   Generally, one can think of a free object with
a set of generators as being one for whom maps from the free object are
determined by where the generators go to.  So, for example, polynomial rings
over a field k are free objects in the category of k-algebras.

> And by corollary, this shows you and I to be
> Purely inseparable. Q. E. D.

I believe `purely inseparable' is a term applied to field extensions.
I don't know what it means, though.

One way to characterize a separable extension L/K is that it is defined by a polynomial f(x) \in K[x]
(irreducible in K[x] because L/K is a field extension) that when you pass to the splitting field of f,
f has no repeated roots.  Alternatively, f and the derivative of f are relatively prime.  This always holds
if K has characteristic 0 or if K is finite.  In fact, it is enough for K to be perfect (all elements have pth roots,
where p is the characteristic of K).  An example of a nonseparable extension is setting k = Z/pZ, K = k[[x]],
L = K[Y]/(Y^p - x).

A purely inseparable extension is one in which the minimal polynomial of every element is of the form (x-a)^n.
By the result I quoted above, K and L must be characteristic p>0, and in fact a condition is that every element u of
L has some power u^(p^m) in K.

In fact, draw all your rotational matrices sideways. Your professors will love it! And then they'll go home and shrink.

e to the pi times i

I have never been totally satisfied by the explanations for why e to the ix gives a sinusoidal wave.

Angular Momentum

With reasonable assumptions about latitude and body shape, how much time might she gain them? Note: whatever the answer, sunrise always comes too soon. (Also, is it worth it if she throws up?)

Riemann-Zeta

The graph is of the magnitude of the function with the real value between 0 and 2 and the imaginary between about 35 and 40. I've misplaced the exact parameters I used.

Moral Relativity

It's science!

Binary Heart

i love you

Pi Equals

My most famous drawing, and one of the first I did for the site

Centrifugal Force

You spin me right round, baby, right round, in a manner depriving me of an inertial reference frame.  Baby.

cited from http://www.xkcd.com/archive/

Proof is one the most important skills required in more advanced math learning, you will encounter them mostly in university, however, Cegep math teachers are using these techniques in their teaching. Moreover, the logic and the way of thinking that ground those techniques can be relevent to any intellectual activity.  

 Proof by Contradiction.

Example:

Theorem. There are infinitely many prime numbers.

Proof. Assume to the contrary that there are only finitely many prime numbers, and all of them are listed as follows: p₁, p₂ ..., p_n. Consider the number q = p₁p₂... p_n + 1.

Proof by contradiction is often used when you wish to prove the impossibility of something. You assume it is possible, and then reach a contradiction. In the examples below we use this idea to prove the impossibility of certain kinds of solutions to some equations.

2.      Mathematical Induction

Example:

Theorem. For any positive integer n, 1 + 2 + ... + n = n(n+1)/2.

Proof. (Proof by Mathematical Induction) Let's let P(n) be the statement "1 + 2 + ... + n = (n (n+1)/2." (The idea is that P(n) should be an assertion that for any n is verifiably either true or false.) The proof will now proceed in two steps: the initial step and the inductive step.

Initial Step. We must verify that P(1) is True. P(1) asserts "1 = 1(2)/2", which is clearly true. So we are done with the initial step.

Inductive Step. Here we must prove the following assertion: "If there is a k such that P(k) is true, then (for this same k) P(k+1) is true." Thus, we assume there is a k such that 1 + 2 + ... + k = k (k+1)/2. (We call this the inductive assumption.) We must prove, for this same k, the formula 1 + 2 + ... + k + (k+1) = (k+1)(k+2)/2.

This is not too hard: 1 + 2 + ... + k + (k+1) = k(k+1)/2 + (k+1) = (k(k+1) + 2 (k+1))/2 = (k+1)(k+2)/2. The first equality is a consequence of the inductive assumption.

3.      Constructive Proof

Example:

Theorem. There is a rational number that lies strictly between the square root of 10¹⁰⁰ and the square root of 10¹⁰⁰+1.

Proof. The square root of 10 ¹⁰⁰ is 10 ⁵⁰. After a little bit of trial and error, we let x = 10 ⁵⁰ + 10 ^-51, which is clearly a rational number bigger than the squre root of 10 ¹⁰⁰. To prove that x is less than the square root of 10¹⁰⁰+1, we compute

x² = (10 ⁵⁰ + 10 ^-51)² = 10 ¹⁰⁰ + (2) 10 ^-1 + 10 ^-102

which is clearly less than 10¹⁰⁰+1.

4.      Case by case

Example:

Theorem. If n is a positive integer then n⁷ - n is divisible by 7.

Proof. First we factor n⁷ - n = n(n⁶ - 1) = n(n³ - 1)(n³ + 1) = n(n-1)(n² + n + 1)(n+1)(n² - n + 1). Now there are 7 cases to consider, depending on n = 7 q + r where r = 0, 1, 2, 3, 4, 5, 6, 7.

Case 1: n = 7q. Then n⁷ - n has the factor n, which is divisible by 7.

Case 2: n = 7q + 1. Then n⁷ - n has the factor n-1 = 7q.

Case 3: n = 7q + 2. Then the factor n² + n + 1 = (7q + 2)² + (7q+2) + 1 = 49 q² + 35 q + 7 is clearly divisible by 7.

Case 4: n = 7q + 3. Then the factor n² - n + 1 = (7q + 3)² - (7q+3) + 1 = 49 q² + 35 q + 7 is clearly divisible by 7.

Case 5: n = 7q + 4. Then the factor n² + n + 1 = (7q + 4)² + (7q+4) + 1 = 49 q² + 63 q + 21 is clearly divisible by 7.

Case 6: n = 7q + 5. Then the factor n² - n + 1 = (7q + 5)² - (7q+5) + 1 = 49 q² + 63 q + 21 is clearly divisible by 7.

Case 7: n = 7q + 6. Then the factor n + 1 = 7q +7 is clearly divisible by 7.

5.      The Pigeon Hole Principle

Example:

Theorem. Among any N positive integers, there exists 2 whose difference is divisible by N-1.

Proof. Let a₁, a₂, ..., a_N be the numbers. For each a_i, let r_i be the remainder that results from dividing a_i by N - 1. (So r_i = a_i mod(N-1) and r_i can take on only the values 0, 1, ..., N-2.) There are N-1 possible values for each r_i, but there are N r_i's. Thus, by the pigeon hole principle, there must be two of the r_i's that are the same, r_j = r_k for some pair j and k But then, the corresponding a_i's have the same remainder when divided by N-1, and so their difference a_j - a_k is evenly divisble by N-1.

Sources:

http://zimmer.csufresno.edu/~larryc/proofs/proofs.html