Math From First Principles
Short derivations rather than formulas to memorize. Where each result comes from, so you can rebuild it when you forget it, which you will.
7 articles · about 36 min in total
Start with Why Pi Is 3.14A formula you have memorized is a formula you will eventually misapply. A formula you have watched fall out of something simpler is one you can rebuild from scratch, and that is a different kind of knowing.
So every step here is a derivation. Pi is not 3.14 because a textbook says so. It is 3.14 because of what happens when you compare the edge of a circle to its width, and seeing that once beats seeing the digits a hundred times.
The order runs from geometry you can see to rules you have to trust. Pi and the Pythagorean theorem can both be shown with shapes on a page. By the time you reach the quotient rule, the argument is algebraic, and by then that should feel like the same activity rather than a new one.
These are short on purpose. They are written for someone who needs the answer today and would rather understand it while they are here.
Key takeaways
- Pi is the ratio of a circle circumference to its diameter, which is why it is the same number for every circle regardless of size.
- The Pythagorean theorem can be derived by rearranging four copies of one right triangle, with no algebra needed to see why it holds.
- A factorial counts distinct orderings of a set, which is why zero factorial equals one: an empty set has exactly one arrangement.
- The law of sines and the law of cosines solve different triangle cases, so the information you already have decides which one applies.
Step 1: Why Pi Is 3.14
Pi is the ratio of a circle’s circumference to its diameter. It’s 3.14159... because that’s how circles work, and the value is the same in every universe with circles.
Nov 18, 2025 · 5 min read
Step 2: Deriving the Pythagorean Theorem
The Pythagorean theorem is the result every middle-schooler memorizes and most adults never see proven. The classic proof using rearranged squares is one of the cleanest mathematical arguments ever made.
Oct 28, 2025 · 5 min read
Step 3: What Is a Factorial
A factorial is the simplest fast-growing function in math. 5 factorial is 120. 10 factorial is 3.6 million. 70 factorial is bigger than the number of atoms in the observable universe.
Nov 25, 2025 · 5 min read
Step 4: The Law of Sines, Explained
The Law of Sines is the cleanest result in triangle trigonometry. The ratio of any side to the sine of its opposite angle is the same for all three sides. The proof is one line, and the consequences are surprising.
Nov 11, 2025 · 5 min read
Step 5: The Law of Cosines, Explained
The Pythagorean theorem only works on right triangles. The Law of Cosines is the generalization that works on any triangle, with one extra term that captures the angle.
Nov 4, 2025 · 5 min read
Step 6: What Is the Quotient Rule (Calculus)?
The quotient rule is the calculus shortcut for taking the derivative of a fraction where both the top and the bottom depend on x. It's mostly memorization, but understanding why the formula has the form it does makes the rule much harder to forget.
Sep 18, 2020 · 4 min read
Step 7: How to Get Standard Deviation in Google Sheets
Standard deviation is one of those statistics that gets used a lot more than it gets understood. Google Sheets has six functions for it, and which one you use depends on whether you're describing a sample or a whole population. Worth getting right.
Aug 29, 2020 · 7 min read
Frequently asked questions
- Why is pi equal to 3.14?
- Because dividing any circle circumference by its diameter always gives the same number, roughly 3.14159. The ratio is fixed by the geometry of a circle, so it does not depend on size, and it is irrational, meaning the decimal never terminates or repeats.
- Why is zero factorial equal to one?
- Because a factorial counts arrangements and there is exactly one way to arrange nothing. It also keeps the recursive definition consistent, since n factorial equals n times (n minus 1) factorial only works at n equals 1 if zero factorial is one.
- When do I use the law of sines instead of the law of cosines?
- Use the law of sines when you know two angles and any side, or two sides and a non-included angle. Use the law of cosines when you know two sides and the angle between them, or all three sides. The information you have decides it.
- What is the quotient rule?
- It gives the derivative of one function divided by another: the bottom times the derivative of the top, minus the top times the derivative of the bottom, all over the bottom squared. It follows directly from applying the product rule to a reciprocal.