You might have heard of the following problem.
As I was going to St. Ives, I met a man with seven wives. Every wife had seven sacks, every sack had seven cats, every cat had seven kitts. Kitts, cats, sacks, wives, how many were going to St. Ives?
This is a rather good problem and a rather old problem. In fact some of the hieroglyphics found on papyrus more than 3000 years old are believed to represent a very similar problem!
I think this problem, while amusing, is a little big ambiguous and slightly alarming. Is the toothed, clawed, furry and wild-haired army en route to St.Ives like the narrator or is the narrator passing them while they are in their their home? Can we
assume the man meets the army head on as they are returning from St.Ives? Is the man travelling with his entourage or is just meeting them on his way to St.Ives? What kind of person carries 7 cats and 49 kittens in a sack? What...
This journey is heavily inspired by the youtube mathematician Vi Hart, whose videos describing mathematical concepts through doodling in a notebook were the inspiration for much of my mathematical journeys series. I'll put a link to her video on this
topic at the end of the journey, and I highly encourage everyone to go check her out.
Let's talk exponents.
But to do that, first we should talk about multiplication. Multiplication is a shortcut for adding a bunch of the same number together. If I gave you:
5 + 5 + 5 + 5 + 5 + 5 = ?
You could just add them normally, treating each of those 5's as a size-5 step along the number line. But since each of these addition steps is the same size, a faster way to figure out the result would be to determine two things: the size of the step, and how
many steps we have. Then we can multiply the size of step (in this case, 5) by the number of steps. In this case, we have a total of 6 size-5...
Thanks, Vihart of YouTube!
In this video, elementary algebra is built up from counting to positive numbers to negative numbers to multiplication and division to exponents and logarithms. All these concepts are tied together under the common theme of "fancy counting." That is, each
of these operations is tied directly to the fundamental operation of "+1".
All of this in just over 9 minutes. This is how experts think of algebra. Novices, new to the world of algebra, see the mechanical steps to be taken, the shuffling of symbols, the confusing mass of numbers, letters, and symbols and get lost. Experts think
of algebra as chunks of value that are being operated upon by a (relatively) small set of operations, made smaller by lumping an operation with its inverse (addition/subtraction, multiplication/division, exponents/logarithms) and realizing they are two sides
of the same coin.
"How I Feel About Logarithms,...