High-Level Math Sketches

It is often thought that the imaginary unit $i$, and complex numbers by extension, were created by "dreaming up" a solution to the equation $x^2 = -1$, for the purposes of solving quadratics with no real solutions.

Actually, this is not the case, as Noah Snyder comments on Math StackExchange:

I believe the historical context was in solving cubic equations. If you try to solve a cubic equation that has a real solution, sometimes you have to use complex numbers in intermediate steps in the calculation. But then when you get to the end you find a real number that really is a solution to a real cubic equation!

Indeed, people were motivated by Cardano's method for solving cubics. People knew (using tools like the intermediate value theorem or "graphing" without graphing calculators (e.g. Newton's method); nowadays you can just use WolframAlpha) that the real polynomial $P(x) = x^3 - 6x + 2$ (example courtesy of MIT) had all real roots, but Cardano's formula gave gave the roots in terms of the complex cube roots $A = (-1 + \sqrt{-7})^{1/3}$ and $B = (-1 - \sqrt{-7})^{1/3}$ (whatever these mean). For example, using the "nonsensical rule" $\sqrt{-7}^2 = -7$ (from the definition of square root), people checked that $$(A+B)^3 - 6(A+B) + 2 = 0.$$ So somehow, these "nonsensical" expressions $A = (-1 + \sqrt{-7})^{1/3}$ and $B = (-1 - \sqrt{-7})^{1/3}$ (admittedly ambiguous—more on that later) arose as intermediate steps in solving a totally real problem (finding the roots of the real polynomial $P(x) = x^3 - 6x + 2$), and $(-1 + \sqrt{-7})^{1/3} + (-1 - \sqrt{-7})^{1/3}$ is the simplest exact algebraic expression (try putting it in WolframAlpha!) for the irrational (but still "algebraic") real root $A+B = 2.26180224525997\ldots$ (which is just a numerical approximation from algorithms like Newton's method or the bisection method). By the way, as WolframAlpha tells you, we've glossed over the fact that the cube roots $(-1 \pm \sqrt{-7})^{1/3}$ are ambiguous (see here and Euler's formula on Expii), because the fundamental theorem of algebra tells you there are three possible interpretations of a cube root (you can get even more precise information from complex analysis). And the mere existence of a root with an "algebraic" expression (in terms of radicals) like $(-1 + \sqrt{-7})^{1/3} + (-1 - \sqrt{-7})^{1/3}$ has connections with Galois theory (based on the symmetry of roots from Vieta's formulas, and algebraic number theory).

Polynomials

Definition

Division Algorithm

Remainder / Factor Theorem

** Rational Root Theorem**

** Identical Polynomials**

** More Exercises**

Factorizations

Definitions and Methods

Quadratics - Quadratic Formula (factoring linear out)

Cubics and Higher-degree Polynomials (factoring linear out )

Factor Theorem and its uses

Rational Root Theorem and its uses

Vieta's Formulas (related to factorization)

Factoring quadratics or higher out

Factoring Multivariable Polynomials

More Exercises

See Persiflage, Exposition is not underrated. In particular:

I don’t think this issue is really about finding a balance between research and education. It’s about individuals wanting to be rewarded professionally (tenure, salary, etc.) by writing expository articles which neither make a contribution to the corpus of mathematics nor (outside exceptional cases) to the education of the next generation of mathematicians. The language of the debate also seems to ignore the fact that the web is flooded with more expository material than would even have been conceivable thirty years ago, from Wikipedia to MathOverflow to math blogs to online course notes (this is, mostly, a good thing). Many mathematicians have a desire to explain their ideas, and it’s not clear to me that anything profitable can be gained by skewing the incentives. (If you want to argue that teaching should be taken more seriously when it comes to hiring/tenure decisions, that is another debate.) I find myself in agreement with the comment you quote from Dan L: “I also do not believe that your hypothetical “great storytellers of mathematics” really exist, in the sense that I do not believe that a person can achieve that level of mastery over a field without being a successful researcher.”

I view this post as a call to focused action. It means we should think hard about what is likely to be most useful to current and future generations. The context of level and audience are quite important.