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Kindle Notes & Highlights
by
Kalid Azad
Read between
December 3 - December 19, 2019
Did you know that negative numbers were only created in the 1700s, and were considered absurd? That imaginary numbers had the same fight when introduced?
Interesting math facts
The essays highlight my favorite learning method: get the context of an idea, formulate analogies, and cover examples using those analogies.
Learning method of the book author
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Newsletter of this e-book
Feel free to adjust the settings; I find the smaller sizes work best. The diagrams are in high-contrast color to balance readability on black & white and color screens.
Ideally read this book in smaller font, and if you want to better understand the diagrams, check out the PDF/online Kindle version
We’re left with arcane formulas (DNA) but little understanding of what the idea is.
Way we're usually taught math with no intuition behind it
The points (x,y) in the equation x2 + y2 = r2 (analytic version of the geometric definition above)
One of the countless definitions of a circle
math is about ideas — formulas are just a way to express them.
Step 1: Find the central theme of a math concept.
When trying to understand math, come to the roots of the problem for understanding its application
Step 2: Explain a property/fact using the theme.
Later, translate formulas into their real meanings
Step 3: Explore related properties using the same theme.
Lastly, Find where else this statement applies
Looking at e’s history, it seems it has something to do with growth or interest rates. e was discovered when performing business calculations (not abstract mathematical conjectures) so “interest” (growth) is a possible theme.
Roots of e
Definition 1: Define e as 100% compound growth at the smallest increment possible
Money makes money, which makes money, which...
by always growing it means you are always calculating interest – it’s another way of describing continuously compound interest!
Continuously compound interest
e is the number where you’re always growing by exactly your current amount (100%), not 1% or 200%.
e
the natural log (ln) as shorthand for this “time to grow” computation.
Natural log(ln)
ln(a) is simply the time to grow from 1 to a.
ln(a)
e is the amount of growth after waiting exactly 1 unit of time!
e
Once we have the core idea (“e is about 100% continuous growth”), the crazy equations snap into place — it’s possible to translate calculus into English. Math is about ideas!
It's possible to translate calculus into English
If it doesn’t click, come at it from different angles. There’s another book, another article, another person who explains it in a way that makes sense to you.
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
In square it's a side, in circle it's a radius
Area = Factor ·(line segment)2
Universal equation
given area formula works for all similar shapes, where “similar” means “zoomed versions of each other”. For example: All squares are similar (area always s2) All circles are similar,
All triangles are not similar: Some are fat and others skinny – every type of triangle has its own area factor based on the line segment you are using.
Triangles differ
when you zoom (scale) a shape, you’re changing the absolute size but not the relative ratios within the shape.
When zooming
Area can be be found from any line segment squared, not just the side or radius Each line segment has a different “area factor” The same area equation works for similar shapes
Summarising
Any right triangle can be split into two similar right triangles.
Circle of radius 5 = Circle of radius 4 + Circle of radius 3.
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
Universality of the Pythagorean Theorem
Metcalfe’s Law (if you believe it) says the value of a network is about n2 (the number of relationships).
Metcalfe's Law
In terms of processing time: 50 inputs = 40 inputs + 30 inputs Pretty interesting. 70 elements spread among two groups can be sorted as fast as 50 items in one group.
Assuming the boats are similarly shaped, the paint needed to coat one 50 foot yacht could instead paint a 40 and 30-footer. Yowza.
Energy at 500 mph = Energy at 400 mph + Energy at 300 mph With the energy used to accelerate one bullet to 500 mph, we could accelerate two others to 400 and 300 mph.
Kinetic energy = 1/2 mv^2
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.
Pythagorean Theorem - real use
As you can guess, the Pythagorean Theorem generalizes to any number of dimensions.
It appears humans can’t tell the difference between colors only 4 units apart; heck, even a distance of 30 units looks pretty similar to me:
In the system of 255 values
You can even unscramble certain blurred images by cleverly applying color distance.
If you can quantify it, you can compare it using the the Pythagorean Theorem.
Core idea
the Pythagorean Theorem: Works for any shape, not just triangles (like circles) Works for any equation with squares (like ½ mv2) Generalizes to any number of dimensions (a2 + b2 + c2 + ...) Measures any type of distance (i.e. between colors or movie preferences)
Summary of the Pythagorean Theorem
Before numbers and language we had the stars. Ancient civilizations used astronomy to mark the seasons, predict the future, and appease the gods
Before we used stars
isn’t it strange that a circle has 360 degrees and a year has 365 days?.
Constellations make a circle every day. If you look at them the same time every day (midnight), they will make a circle throughout the year.
Constellations as the time measurement
Degrees measure angles by how far we tilted our heads. Radians measure angles by distance traveled.
Degrees & Radians
Radian = distance traveled / radius
Getting a normalised angle
radian is about 360/2π or 57.3 degrees.
Think about it — “Hey Bill, can you run 90 degrees for me? What’s that? Oh, yeah, that’d be π/2 miles from your point of view.” The strangeness goes both ways.
Degrees to radians
Strictly speaking, radians are a ratio (length divided by another length) and don’t have a dimension.
think of radians as “distance traveled on a unit circle”.
With degrees, you’re comparing your height on a circle (sin(x)) with how far some observer tilted their head (x degrees), and it gets ugly fast.
