BetterMath Blog
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Articles, puzzles and things to try: the mathematics behind the course, and how to study it.
Articles
Adding infinitely many numbers
How can a sum that never ends have a finite answer?
Add a half, then a quarter, then an eighth, and keep going for ever. The sum never stops growing, yet it never passes 1. That is the idea behind one of the first results in the course that deals with infinity, and it deserves a proof rather than a formula to memorise.
Read more >Why area and slope are the same calculation
Calculus asks two questions that seem to have nothing to do with each other. The first runs a derivative backwards: given f, find a function F with F′ = f. The second is about area: how much space lies under a curve? The Fundamental Theorem of Calculus says that these are the same question, and it is worth seeing why rather than taking it on trust.
Read more >How to study with BetterMath
AA HL rewards understanding more than memory. The notes are built around that idea, and a few habits will help you get the most from it.
Read more >Try this
Add a half, then a quarter, then an eighth, and keep going.
- Sum so far
- 63/64 = 0.984375
- Gap still left
- 1/64 = 0.015625
Each new term fills exactly half of the gap that is left, so the gap halves but never closes. Add them all, and the sum is exactly 1.
Read why the sum is exactly 1 >The triangle on the cover
Pascal's triangle: each number is the sum of the two above it.
Select any number.
The binomial theorem, Chapter 2 >