Classifying mathematics by its subject matter may be misleading because the kinds of questions being asked may be the same, despite different subject matter.
Nonetheless, classifying by subject matter, we crudely have algebra, geometry and analysis.
Algebra vs. Geometry
It’s insufficient to claim that algebra contains symbols, while arithmetic contains numbers. In advanced mathematics, numbers rarely appear without letters.
Manipulating symbols through rules, e.g. do the same to both sides, is thought of as algebra. Once the concepts are being visualized, then people think geometry.
However, there is more interplay: frequently, one solves an algebraic question by finding a way to visualize it, and one’s visualization helps find useful algebraic methods.
Algebraic Problems Being Solved via Geometric Thinking
Prove that if \(a\) and \(b\) are positive integers, then \(ab = ba\).
With algebra, we’d probably use an inductive argument.
However, imagine a rectangular array that has \(a\) rows with \(b\) objects in each row. The number of items is the same if we consider them as \(a\) lots of \(b\) (counting row by row), or \(b\) lots of \(a\) (counting column by column).
Geometry Problems Being Solved via Algebraic Thinking
What happens if you reflect a circle about a line \(L\) through its center, then rotate it through 40° counter-clockwise and then reflect it about the same line \(L\)?
Think of the circle as a thin piece of wood. Instead of reflecting it about \(L\), turn it upside down. Looked at from below, a 40° CCW rotation is a 40° CW rotation. The final flip puts it right way up. Effectively, it will have gone through a single 40° CW rotation.
The geometric argument, while intuitive, can be tricky to prove. Using transformation matrices and the definition of a unit circle, \(x^2 + y^2 \le 1\), we can prove correctness.
In COS 340, we explored ways in which a geometric argument can lead to subtle mistakes when basing a proof off it.
Algebra vs. Analysis
Crudely, analysis involves limiting processes, e.g. calculus.
It’s insufficient to claim that if we can get to the answer in a finite sequence of steps then we are doing algebra, and otherwise, we’re doing analysis. It’s better to think of analysis as a mathematical technique, just like algebraic thinking, or geometric thinking.
Analytical thinking typically involves finite work to find sufficient conditions for which a certain fairly simple inequality holds.
Prove that \(x^4 - x^2 - 6x + 10 > 0;\ \ \forall x \in \mathbb{R}\)
- \(x \le -1\):
- \( x^4 - x^2 \ge 0 \) and \(10 - 6x > 0\).
- \(-1 \le x \le 1\):
- \(x^4 - x^2 - 6x \le x^4 + x^2 + 6|x| \le 8 \)
- \( \implies x^4 - x^2 - 6x \ge -8 \)
- \( \implies x^4 - x^2 - 6x + 10 \ge 2 \)
- \( 1 \le x \le \frac{3}{2} \):
- \( x^4 \ge x^2 \) and \(10 - 6x \ge 1 \)
- \( \frac{3}{2} \le x \le 2 \):
- \(x^4 - x^2 = x^2(x^2 - 1)\) and since \(x^2 \ge \frac{9}{4}\), \(x^2(x^2 - 1) \ge \frac{9}{4} \cdot \frac{5}{4} = \frac{45}{16} > 2 \)
- \(6x \le 12 \)
- \( \implies x^4 - x^2 - 6x + 10 > 2 - 12 + 10 = 0 \)
- \(x \ge 2\):
- \(x^4 - x^2 = x^2(x^2 - 1) \ge 3x^2 \ge 6x \)
I don’t like this proof. It’s long, and prone to trial and error, e.g. how do I choose \(-1, 1, \frac{3}{2}, 2\) as the boundary points?
The algebraist’s proof would have been:
$$ x^4 - x^2 - 6x + 10 = (x^2 - 1)^2 + (x - 3)^2 > 0 $$
However, deciding whether a polynomial can be written as a sum of squares of other polynomials is a difficult question!
Does it then imply that questions in math can be approached through any technique, but some techniques are simply more efficient than others?