Tuesday, November 3, 2015

How to get your mind blown with simple arithmetic

joão pestana

Arithmetics is the oldest of all the branches of mathematics. It's the study of numbers and the basic properties among them such as addition, subtraction, multiplication and division — or simply addition if you read my previous article. Now I know that if you are literate enough to understand the article up to this point, you probably think you are good enough with these basic mathematics operations.

You know how to add, subtract, multiply and probably how to use a division algorithm. Even if you are somehow unable to perform these tasks, I know you can pick up a calculator and use it. The calculator does the math for you, but you still need to know what to input and have some sense of the expected result.

Deep inside of you, there's something working that tells you that you cannot have, as the result of a sum or multiplication, a smaller number than the ones used as inputs. This, I mean, for whole numbers — or, more precisely, the set of all natural numbers `NN`.

This happens because we use a numeral system that's mostly universal and feels natural. We use the decimal system with radix 10 which means we have 10 different symbols to represent 10 different quantities and then we repeat these symbols to represent even greater amounts. It feels natural because we are naturally born with ten fingers.

We think that mathematics is simple if it only consists of these symbols `n in {0,1,2,3,4,5,6,7,8,9}` and is complicated if it involves any other unusual symbol. Even letters from the alphabet, when used in mathematics, seem and feel weird mainly because we do not know what to do with them. Since they're not part of our base system, there's actually nothing we can do but to treat them as alien entities and keep them untouched to be replaced by some more meaningful symbol later — a number. We call these variables.

The decimal digits have very precise meanings as quantities in themselves. An empty park has `0` people in it until a lonely man comes for a walk and it becomes filled with `1` person. Some time later, a couple — that is, `2` different people — may also enter the park and the population is now of `3`. But we would know nothing if we were told that there were `A` people in the park.

There are infinitely many different numerical systems and some have been so influential in the past that they still persist nowadays. Haven't you ever wondered why we measure the degrees and time in multiples of 60? The Babylonians used a numerical system with radix 60. That is, they had 60 different unique symbols to represent quantities and only needed to repeat that for the quantity of 60 just like we do for the amount 10. We always start counting at `0` and then count the number of times we counted until `9` — that's what the `1` in `10` represents.

I could go on for a while about different numerical systems, but I'll move into the mind blowing part — as promised. We take the symbols `0...9` for granted and assume they're the only ones that are mathematicable. Whatever else is included in the mix must be a variable or have some very precise meaning like the symbol `in` which just means something belongs to or is in.

The beautiful thing — now I know I'm not on solid ground here — about mathematics is that it does not need and it does not depend on the numerical system that we choose to use. It always works and yields the same results. We do not have to conform to what everybody is using to and we are free to invent our own symbols to represent whatever quantities we desire. We can even have symbols for fractions, irrational or algebraic numbers — yes, bases like that are possible! Just think of the symbol `pi` that represents nothing more than a number just like `1` does. It sits somewhere close to `3` on the interval from there to `4` in our decimal representation.

For the prestige, I'm going to switch numbers and letters and perform mathematics. I'm going to keep the decimal system in order to keep the mind blowing contained. My ten symbols are `{O,A,B,C,D,E,F,G,H,I}` and everything else is nothing more than an entity which can be used as variables.

A classical example could be that Clara has the recipe for a lemon pie and it requires `A` lemon, `C` eggs, `BOO` grams of sugar and `BEO` grams of flour.  Since she only has `BOO` grams of flour available, what are the remaining required quantities of the other ingredients to keep the proportions correct?

First, we need to get the ratio between the available and the required flour `1=(BOO)/(BEO)=(BO)/(BE)=D/E`. Note that `1` is just my symbol to represent this ratio. I can use anything that's not part of my numerical system. Finally, we simply must multiply every quantity by this value. Clara will need `1xxA=D/E A=D/E` parts of a lemon, `1xxC=D/E C =(AB)/E` eggs and `1xxBOO=D/E BOO=(GOO)/E=AFO` grams of sugar.

I used a fairly intuitive replacement and even `O` to represent the `n u l l` quantity. Everything had a straightforward relation, but imagine a system that's not base 10 and uses symbols that we seldom use or have never seen before! Mathematics will still work, but I'm not sure about our minds.