(ALL TOO) BRIEF REVIEW OF CALCULUS:

Fundamental Theorem (version 1): $\int_{a}^{b} f'(t)dt = f(b) - f(a)$

Fundamental Theorem (version 2): $d/dx \int_{a}^{x} f(t)dt = f(x)$

REVIEW OF NUMBER SYSTEMS

N = natural numbers = {0,1,2,3,...}
Z = integers = {...,-3,-2,-1,0,1,2,3,...}
Q = rationals = {a/b : a and b integers, b =/= 0}
R = reals = distances along a line = {r.d1d2d3... : r is an integer and
                                      the di are integers from 0 to 9}
C = complex numbers = {x + iy : x and y real} where ii = -1

Each of the sets above is a subset of the one after it. One can think of them as being in a progression resulting from the desire to have solutions (roots) to more equations. Thus in N one can solve 2 - x = 0 but not 2 + x = 0. Passing to Z, one gets a solution to the latter as well, but not to 2x = 1; that solution lies in Q. And xx = 2 has no solution in Q but does in R. Finally, in C there is a solution to xx = -1.

Each of the above sets comes with a natural class of operations, such as addition and multiplication in the case of N, as well as subtraction in case of Z, division (except by 0) in the case of Q, and nth roots in the case of R (except for even-order roots of negative numbers), and nth roots (unrestricted) in the case of C.

There is a special subset of C, the algebraic numbers, which consists of solutions to algebraic equations with integer coefficients (including all the equations shown above). It turns out that very many complex numbers (in fact, very many real numbers) are not solutions to such equations at all; these are "non-algebraic" numbers, usually called "transcendental numbers", and they are much more numerous than the algebraic numbers. The algebraics form a countable set, and the transcendentals an uncountable set. Familiar examples include pi and e.

How many is ``very many'' and what does ``countable'' mean? We will not get deeply into this matter, except to say this: Near the end of the 19th century, Georg Cantor discovered that the notion of infinity, of infinite set, can be made very precise, and that there are different ``sizes'' of infinity. Some sets, such as N and Z and Q, are ``countable'' which means that their elements can be matched up, one to one, with those of N. Some other sets, such as R and C, have too many elements to match up with those of N, so they are ``uncountable''. Moreover, there are sets even much larger, so that their elements cannot be matched up with those of R or C; and so on.

FUNDAMENTAL THEOREM OF ALGEBRA: Every polynomial equation of degree n with (real or) complex coefficients has exactly n roots (counting possible multiple roots, as in $x^2 + 6x + 9 = 0$, which factors into (x+3)(x+3) = 0, giving "two" roots: x = -3 and x = -3).

There also are special finite subsets of Z, sometimes written Z/n or Zn ("Z mod n"): 0,1,2,...,n-1. Z/n has its own special operations, so-called "clock arithmetic", in which one adds (or subtracts) as if the numbers were hours on an n-hour clock, where 0 is at the top and the numbers increase clockwise around the dial up to n-1 just before returning to 0 again. Thus in Z/12, we have 11 + 1 = 0, 11 + 3 = 2, and 2 - 3 = 11. It turns out then for prime n, one can also multiply and divide (except by 0) in Z/n, in a perfectly well-defined way.

Additional systems have been found, continuing the progression above beyond R and C. The most famous of these is the system of quaternions, discovered by William Rowan Hamilton (c 1844). Quaternions have the form x + iy + jz + kw where x,y,z,w are real, ii = jj = kk = -1, ij = k, jk = i, ki = j, and ji = -ij, kj = -jk, ik = -ki. The quaternions VIOLATE a rule held by all the other number systems above: commutativity of multiplication. Quaternions have close connections with matrices and vector algebra. The quaternions include the previous systems N, Z, Q, R, C; for instance, the quaternions of the form x + iy (ie where z = w = 0) are simply the complex numbers.