Helpful Tools for Calculus, Chapter 0: Numbers, Notations and Notes
Previous Section: 0.7 Geometry
“Mathematics is the science of patterns and the art of engaging the meaning of those patterns.” – Dr. Francis Su, Mathematics for Human Flourishing (P44)
In all levels of mathematics abstraction plays a major role in improving methods, comprehension, and productivity. Without identifying patterns, each problem has no meaning in the grand scheme of things. By identifying how problems, functions, strategies and ideas relate to each other we are able to connect and intertwine them in a web on knowledge. By seeing how exponents and finance are related we can create general formulas to define the value in a bank account overtime. By connecting physics and calculus we can delve into a whole new world of math and science with differential equations to provide the depth and clarity that basic algebra formulas don’t provide.
Let’s take a look at some pictures.
We might notice that in the first picture we have a square. Perhaps its total area is 1 square unit, or at least we could say that it is 100% of the square. In the next picture it is cut in half and each side represents half a square unit. In the next the same thing happens and now you have a half and a fourth. By repeating this we have the following image.
Hidden in this image is a pattern, and formula, and a really interesting claim. Let us connect this picture to the fractional parts we have mentioned:
1 square=1/2 square+1/4 square+1/8 square+1/16 square+⋯
We are claiming that by cutting each part of the square in halves, infinitely that it still adds up to the whole square, which in and of itself isn’t surprising. What is surprising is the hidden detail that 1/2+1/4+1/8+1/16+⋯ =1.
There’s a few parts to this, but let’s ignore the fact that we just claimed an infinite number of parts added together is not equal to infinity, but rather 1. I think the picture is evidence enough of that for now. Let’s focus on connecting this idea to mathematics.
1/2+1/4+1/8+1/16+⋯=1
1/21 +1/22 +1/23 +1/24 +⋯1/2∞ =1
So, the sum of the powers of 1/2 add to 1.
Let’s look at some more pictures and see if we can make more conclusions:
1/3+1/9+1/27+1/81+⋯=1/2
1/x1 +1/x2 +1/x3 +1/34 +⋯1/3∞ =1/2
And one more time…
1/4+1/16+1/64+1/256+⋯=1/3
1/41 +1/42 +1/43 +1/44 +⋯1/4∞ =1/3
Based on this you can probably take a guess about what happens in general:
1/x1 +1/x2 +1/x3 +1/x4 +⋯1/x∞ =1/?
Give it a try, then draw a picture to convince yourself!