Calculus
Calculus is a mathematical discipline focused on limits, continuity, derivatives, integrals, and infinite series.
- Differentiation and Integration are the 2 main pillars of calculus. They are inverses of one another.
"Transitioning from something approximate to something precise is pretty subtle and it cuts deep to what calculus is all about."
"The emergence of calculus in the 17th century signalled the birth of modern mathematics and was the key to the successful applications of mathematics in the sciences." It was an insight of classical Greek mathematics (Eudoxus and Archimedes) that it doesn't really matter if you're measuring a circle or a polygon with very many very short sides, since the areas of each scenario will be so similar that they will be close enough for any practical purpose that can be thought of.
Newton's insight was that there is nothing special about a perfect circle, because if we zoom in close enough at any part of the circle, it appears as a straight line, just at different angles. Unlike classical Greek mathematicians however, Newton wasn't satisfied with increasingly smaller lines; his insight was to reduce the field of view until the lines were infinitisemal— so small that it's smaller than any size you can name, but not zero.
- What was previously almost a line has become an actual line, the slope of which is referred to as the derivative.
Imagine you were swinging a ball attached to a string above your head and suddenly released it. That ball would fly in a straight line away from you.
- spec: this kind of insight demonstrates that a circle really is made up of infinitely small and infinitely many lines. The moment we release the string, the ball goes in a straight line in accordance with the infinitely small line that it was a part of when it was released.
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