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Kindle Notes & Highlights
by
Kalid Azad
Read between
December 28, 2018 - January 23, 2019
Unfortunately, math understanding seems to follow the DNA pattern. We’re taught the modern, rigorous definition and not the insights that led up to it. We’re left with arcane formulas (DNA) but little understanding of what the idea is.
Missing the big picture drives me crazy: math is about ideas — formulas are just a way to express them.
Math becomes difficult and discouraging when we focus on definitions over understanding.
The Pythagorean Theorem can be used with any shape and for any formula that squares a number.
The area of any shape can be computed from any line segment squared
The Pythagorean Theorem applies to any equation that has a squared term. The triangle-splitting means you can split any amount (c2) into two smaller amounts (a2 + b2) based on the sides of a right triangle. In reality, the “length” of a side can be distance, energy, work, time, or even people in a social network.
Given this relationship, it makes sense to partition elements into separate groups and then sort the subgroups.
the directions are independent (the geek term is orthogonal).
The Pythagorean Theorem lets you find the shortest path distance between orthogonal directions. So it’s not really about right “triangles” — it’s about comparing “things” moving at right angles.
If you can quantify it, you can compare it using the the Pythagorean Theorem.
Degrees measure angles by how far we tilted our heads. Radians measure angles by distance traveled.
Radian = distance traveled / radius
Negatives were considered absurd, something that “darkened the very whole doctrines of the equations” (Francis Maseres, 1759).
“imaginary numbers” are as normal as every other number (or just as fake): they’re a tool to describe the world.
A complex number is the fancy name for numbers with both real and imaginary parts.
Magnitude measures a complex number’s “distance from zero”, just like absolute value measures a negative number’s “distance from zero”.
The conjugate is a way to “undo” a rotation.
Pi is the ratio between circumference and diameter shared by all circles.
e is the base amount of growth shared by all continually growing processes.
e is not an obscure, seemingly random number. e represents the idea that all continually growing systems are scaled versions of a common rate.
e is like a speed limit (like c, the speed of light) saying how fast you can possibly grow using a continuous process.
The natural log gives you the time needed to reach a certain level of growth.
Nature doesn’t wait for a human year before changing.
Unfortunately, calculus can epitomize what’s wrong with math education.
We’ve created complex mechanical constructs to “rigorously” prove calculus, but have lost our intuition in the process.
