This is the beginning of my maths notes. I hope to start from absolutely no knowledge of how numbers work, except for maybe a basic ordering.

Table of Contents

Counting

Let's start with the very beginning, how to count, and what numbers mean. First of all, the following symbols separated by commas are called digits: $0,1,2,3,4,5,6,7,8,9$. To each of these symbols, also called arabic numerals, we associate a certain quantity. While I know this is probably trivial, I am going to list here the symbol and the relative quantity, represented by how many "x" are present in the column.

As it's easy to see, writing 9 is much easier than denoting "xxxxxxxxx". This also does not scale very well, and is relegated to small quantities of objects. The tradeoff here is that numbers can be difficult to visualize or estimate, sometimes lacking a sense of scale.

Speaking of scale, how do we count past 9? While I am not going to fully explain it yet, we use something called base. Rather than using a single symbol/digit for every number, we borrow a similar mechanism to words, and concatenate multiple digits in order to form a numeral (the word, represented by digits instead of letters) which represents a number (the actual meaning of the word, the quantity itself as a concept).

If we were now to represent ten "x" as a "t", we can extend a bit more easily the table.

So 19 means "10 x" and another "9 x" , while 20 means "10 x" and another "10 x". This process means that, in the numeral "35", the "3" in the second column determines the quantity of "t", and the "5" determines the quantity of "x". We can then repeat the procedure with the numeral "534", by defining that "10 x" is "c", we write denote the quantity "5 c, 3 t, 4 x", so, "five hundred, thirty-four".

The exploration of what base means, will be properly defined in the exponents chapter, for now we can give the reasonable assumption that it will have something to do with the number "ten", as we seem to group "ten things" into a higher order each time.

Now we should be able to count any quantity, by making longer and longer numerals, even if we might lack the words to properly describe it. (Things like 100000000000000000000000000000000, which cannot be easily deciphered back into words without losing all sense of meaning, which by the way is "one hundred nonillion" and that sounds made up).

Addition

Now that we can count, we can finally define some operations on numbers. We use symbols called operators to define procedures that involve one or more numbers together, in order to produce a different number.

Addition is defined as "number_A + number_B", where the important bit is "+". How is this procedure defined? We start from the quantity number_A denotes, and we count up as many times as number_B. For example, in the writing "2+3", we start from 2, and count up 3 times. "2 -> 3 -> 4 -> 5". After the third "arrow" or "counting up" that we do, we stop.

Coincidentally, this merges very well with the previous explanation on counting, so that the example "8+4" is both shown as "8 -> 9 -> 10 -> 11 -> 12", but also (8) "x x x x x x x x" and (4) "x x x x", merging together in "x x x x x x x x x x x x", and since we said ten "x" make a "t", we can rewrite it as "t x x", so 12.

Reexplaining the whole procedure with the usual column procedure:

Addition:
   (1) 
       8 +
       4 =
----------
    1  2

And of course, this still extends as usual with multiple columns.

Addition:
(1)  
    3  3  +
 1  7  4  =
-----------
 2  0  7

The extra (1) can mean simply that "3 + 7" made 10 "t"s and those can be grouped up in an extra "c", while the "3+4" did not reach enough "x" to group up in an extra "t".

Also, counting up zero times means not doing it at all, so "a+0" is "a".

Addition tables

Since the cyclical nature of addition is not immediately apparent, here's a quick overview of addition tables, so that a "digit+digit" result can be easily looked up and eventually memorized.

More insights on addition

We said that addition is defined as "a+b", where "a" and "b" are two numbers and we declared that the procedure is to start at number "a" and count up (which is equal to adding 1), "b" amount of times. If we were to do the opposite, and start at "b" instead, counting up "a" times, will we get to the same result? Why? Why not? How do "a" and "b", expressed as "groups of x" factor in the "a+b" result, still expressed in "groups of x"?

Subtraction

We define subtraction by writing the symbol "-", and the writing "a-b" (also called expression, just like "a+b" is also an expression), means that the number "a" counts down "b" times.

Examples are: "9-4" becoming "5" (9 -> 8 -> 7 -> 6 -> 5), or "34-17" becoming "17" (expressed in groups "t t t x x x x" - "t x x x x x x x" , we remove a "t" each and four "x", having a "t t" - "x x x", but we know t is also 10 x so we expand one of those, "t x x x x x x x x x x" - "x x x" and then finally removing 3 "x" each, ending up at "t x x x x x x x").

In columns procedure:

Subtraction:
  3  4  -
  1  7  =
-----------
  2  4  -
     7  = 
----------
   (10) 
  1  4  -
     7  =
---------
  1  7

What happens when we subtract too much?

First of all, subtraction seems to be just the opposite of addition. When we count up in addition, we count down in subtraction. But what happens when we have an expression "5-7"? If we express it in "group of x" notation, we have "x x x x x" - "x x x x x x x", and we can see that if we try to remove 5 x from each, we end up at "" - "x x", and we're still 2 "x"s short. That amount that we're missing, we denote as "-2", and it means "we're two steps below the zero".

In "x"s notation, we define now the number as "-x -x". (Think of it as anti particles, and that they cancel out with positive particles). When an "x" and an "-x" get to be expressed in the same number, the end result is the removal of both elements.

So if in our groups of elements we had "x x x x x x -x -x -x", the resulting number would be "x x x". If we were to group up the "x" and "-x", we would see that we have 6 "x", and 3 "-x".

When adding, we said that we express the entire group as one, "x x" (2) + "x" (1) -> "x x x" (3). Therefore, 6 + (-3) -> 3. We define -number as a group of x where every particle is negative.

This also means that any subtraction we did "7-4" can also be expressed as "7+(-4)", and it means that -"t x x" can mean "-t -x -x".

While this doesn't really change much in terms of expression writing "4-10 -> -6", in terms of philosophy and natural understanding it becomes:

What happens if we swap places?

Just like in addition, what happens when we expand "4-3", and is it the same as "3-4"? How does expanding it in group of xs influence the result? Can you see easily why?

Multiplication

Multiplication is defined as the repeated application of addition. "a*b" means "a+a+a+a+a+...+a" (where there's b amount of "a"s in the expression).

From this point onward, the "group of x" notation starts becoming burdensome, so I will offer a final example of "3*2" where "x x x" * "x x" can be simply seen as

(square brackets for convenience), and therefore

. Also important is that only "x" can be replaced by the second term, so each "t" and "c" must be replaced down to "x"s.

Also, repeating something zero times means not doing it at all, so "a*0" is "0".

Since multiplication is a little bit more involved than addition I am also still going to drop the multiplication tables, which can become common memorization (or you can use just a calculator, there's no need to always do calculation by hand).

What if we want to multiply by 10? Or more? We use multiplication digit by digit.

Multiplication:
  3 4 *
    4 =
--------
1 2   +
  1 6 =
-------
1 3 6

We take each digit of the second term, and multiply it for each digit of the first one, putting it in the column given by summing the columns of the digits involved minus 1. (so, in 34*4, when I do 3*4, the 3 is in the second column, and the 4 is in the first one, so I start writing from the second column (2+1-1)).

Multiplication:
    3 4 *
    4 5 =
--------
1 2      + (3*4, 2+2-1 column)
  1 6    + (4*4, 2+1-1 column)
  1 5    + (5*3, 1+2-1 column)
    2 0  = (5*4, 1+1-1 column)
----------
1 5 3 0

Division

Division is the opposite operation of multiplication. We're going to revisit this more later, as for now, it's not fully defined for every single number we've seen. Just like addition required negative numbers to be fully defined, division will require some more trickery to have a proper result for each number "a" and "b" we plug into "a/b".

For now, we define division as: "If a*b -> c then c/b -> a", that is, it makes a number only if a multiplication of some kind can produce our starting number. More precisely, if we have "a/b", it's only a number if there's a number c, so that b*c produces a.

For the last one, and a bit of a spoiler, is that since 3*2 is 6, and 3*3 is 9 the correct result should be 3*(a number between 2 and 3). What exactly that is, is not really a secret, but it is interesting to know how a simple multiplication and its inverse can expand our thinking of numbers that widely.

Extra Topic

A bit of a personal musing and a review of roman numerals.

Roman numerals are the following:

A number like 8, is written as VIII, so 5+1+1+1. A symbol cannot be repeated more than three times in a row, so a number like 4, isn't IIII, but IV, to signify 5-1. If a unit is before a larger one, it means to subtract it from the bigger one. If we want to represent 4000? We actually cannot, as we are missing bigger and bigger symbols to denote bigger quantities. It is interesting though how this system of I,V,X is very similar to the "group of xs" notation above. But the misuse of the base 10, and no positional meaning to the numerals, meant there was always going to be a "maximum" number it can count to. It's also unwieldy in multiplications, as there's no easy way to column everything together.

Ending notes

And that is it, the very first article/note about very elementary math. This one can probably be skipped, but I added it as a way to fully get all the basics in, and understand math with zero preconceptions or notions beforehand.