Since we were once children playing with alphabet blocks we've each had some physical, observational learning about composition of symbols, the stability of the untouched block, one's memory of the obverse, and of course taking away and removal operations. So, why not formalize concepts about it with some foundations of mathematics, review of trigonometry, and analysis, towards a more rigorous understanding? We study the logic of math to help our understanding of what is new information, and what is implicit by physical laws—all the contributors to theory have a good background in mathematics.

Classical wave mechanics is at the heart of all wave mechanics, so we must study the sine and cosine, and their identities, and Fourier analysis, and calculus. Sequences and series are also fundamental to our evaluation of curves as it gives us the logarithm as integral of the hyperbola.

Physics is based on the mathematical field of real numbers, but we have to build the most fundamental analytical set of real numbers from the cardinal natural numbers. In the early 20th century, mathematician Giuseppe Peano reduced the rules of natural numbers to what is known as Peano Arithmetic Axioms. Counting requires notation, and for this there is the representational convention we use known as the (infinite) set of natural, or non-negative integers, $\{0,1,2,…\}$, being a composite of the zero element (not an empty set with it alone, because of the (any) element versus no element difference in sets, with the empty set, itself being the zero element of sets), $\{0\}\neq\{\}=\emptyset=0$, or, the set which includes just zero being not equal to the empty set, along with the positive integers, $\{x\in{}\mathbb{Z}: x\gt{}0\}$ (an appeal to the common experience in place of using more rigorous notation, by the way, $\mathbb{Z}$, the set of integer numbers gets its symbol from the German word for numbers, zahlen), read as, all numbers candidate for the value of a variable $x$ being in the integer set, of those the ones which are greater than zero. The natural numbers are constructable from two elements, zero and one, with every number being a unique quantity of applications of the successor function, being to add such ab-initio non-zero element (one) to the number in argument, $S(n)=n+1$. Further axioms, or posited and not derived rules, state that no application of the successor will result in the zero element, and that if $x=y$, then $y=x$ being assignment symmetric, and if $y=z$ then $x=z$ which is called transitivity, and in pursuit of logical viability, include closure under the operations of the system, for our most common and everyday sets of numbers, the natural numbers, integers, rationals, and reals, which are all closed under both multiplication and addition. But we'll see that there are some functions, being empirically motivated, which reveal even more complex sets of numbers, being the complex numbers.

We are inately equipped with the concept of order and comparative value, as you have already in your imaginative memory the notions of miles to a destination (GPS naviagation), date/age (years), monetary (cents), and street number comparison. These sets of discretely stepping numbers are equipped with several properties, the most important of which is order, or the discernability and comparability of every position in any subset of these natural numbers, $\mathbb{N}$. The notion of zero is independent of the representation of the natural numbers, and it is the basis for foundational mathematics of the 19th century onward, set theory. The child with blocks knows when they don't see any blocks, or when there are various numbers of blocks, and can compare the numbers of blocks in two different piles—so this is a physical basis for the natural numbers being fundamental to everything numerical. As each type of number is constructed from a more fundamental type of number, we will see that the properties of the more fundamental set are inherited by the more complex sets. A more abstract method of construction of the natural numbers is to use set theory, where each number is represented by the set of all smaller numbers. For example, the number 3 can be represented as the set $\{0,1,2\}$, and in general, the number $n$ is represented as the set $\{0,1,2,\ldots,n-1\}$. So, $0$ is $\emptyset=\{\}$ the empty set, $1$ is $\{\{\}\}$ (the set containing the zero element), $2$ is $\{0,1\}=\{\{\},\{\{\}\}\}$, and so on. This construction is known as the von Neumann construction of the natural numbers.

The natural numbers are a set equipped with commutativity, associativity, and transitivity in both multiplication and addition, and the set is constructable using nothing more than the Peano Axioms. Any of these operations using elements from this set will stay in this set (closure under the operation is staying in the set, as opposed to any sort of localizing operation, which might take any (large) number and map it to a local neighborhood). $\mathbb{Z}$, the integers are constructed by first structuring the target (non-negatives + negatives) as a Cartesian product (ordered pair, from chart notation) of one set of the natural numbers with another identical set, $\mathbb{N}\times{}\mathbb{N}\to{}\mathbb{Z}$, represented as two numbers, or a doublet, $(a,b)$. This doublet is representing a single integer, where the convention is to subtract the second number from the first to obtain the value, introducing the hyphen as negativity indicator. The difference between $a$ and $b$ is representable in what is known as a congruency class, which forms the basis for mapping each integer to a class of such doublets. The congruency class of zero in this doublet notation is, $[0]=\{(0,0),(1,1),(2,2),...\}$, one is $[1]=\{(1,0),(2,1),(3,2),...\}$, and minus one would be $[-1]=\{(0,1),(1,2),(2,3),...\}$.

$\mathbb{Q}$, the rational numbers get their name from the term quotient, which is a synonym for ratio, or fraction. $\mathbb{Q}$ is constructed as congruency classes of another type of doublet, this time in the integers, so $[1/2]=\{(1,2),(2,4),\dots\}$, or $\mathbb{Q}\simeq \mathbb{Z}\times \mathbb{Z}\setminus{0}$. The congruency class of zero in the rationals $0\in \mathbb{Q}$ is, $[0]=\{(0,1),(0,2),\dots\}$, and one is $[1]=\{(1,1),(2,2),\dots\}$. This doublet is representing a rational number, so the structured value is interpretted as the first place being divided by the second number, hence the set exclusion of zero from the second place.

The Number System

Anything expressible as a decimal, with a most-significant digit at the leftmost position, and the least-significant digit at the rightmost position, which is an integer to the left of the decimal point, and a fraction of unity (one) to the right of the decimal point--which is always rational in our use!—because to be irrational is to be represented by an infinite series of decimals.

Negative integers and rationals, by structuring all such numbers as having a negative sign before them or not, these can be thought of as a second set of $\mathbb{N}$, or $\mathbb{Q}$, in a doublet structure, so that if the number is positive, it is in the first position of the doublet, and if it be negative then it goes in the second position. Then, you can define the algebra for addition and multiplication, subtraction and division. Because in physics you're not just measuring from the tip of the tape-measure, you also want to talk about other positions, to designate the distance to the mirror-opposite direction (left, for the x-axis) as negative, in one to three spatial dimensions.

Algebra

Every curve has a function, or algebraic formula, using a representation in algebraic composition of a parameter (abscissa), but the reals aren't closed under all mappings, since something happens when you square a number, or raise it to the argument of an exponential function, or take its logarithm. The reals are closed under addition, subtraction, multiplication, and division (except by zero), but not under roots, exponentials, and logarithms. So we have to extend the number system to include these new numbers which are the results of these operations. And this is physically motivated, because we want to be able to measure and calculate things like distances, areas, volumes, growth rates, and decay rates, which often involve these operations.

Formulas use the parentheses in two ways. The first being to group terms in a tuple, and the second is to indicate the parameters to a function. If a function is designated with a letter, then whether the parameter has been indicated in parentheses following the letter or if the parameter had been stated clearly verbally doesn't affect the construction of the function because a function always takes an argument implicitly.

The difference of two quantities, $x_1$ and $x_2$, is expressed as $x_1 - x_2$ or its additive inverse, $x_2-x_1$. The ratio of two quantities is expressed as $x_1/x_2$ or $\frac{x_1}{x_2}$. So, as a function in $x_1$, $f(x)=x/x_2$ is basically the identity function, with a scale factor of $1/x_2$; and as a function in $x_2$, $f(x)=x_1/x$ is the hyperbola of $x_1$ scale.

Formulas of summation

Formulas are always a composition of a variable, commonly $x$ or $y$, and sometimes the abscissal parameter can be expressed as a compact kernel inside a summation symbol, such as in the expression for the geometric series. Formulas of summation have a pattern in that there is a summation symbol, and there are notations below and above the summation symbol signifying the limits bounding the summation of the kernel formula. The two types of summation which will be used here are discrete ($\sum\limits_a^b$) and with calculus, the infinitesimal ($\int\limits_a^b$), where the first term in the discrete sum is the (absent here) kernel-function, evaluated first at $a$, commonly zero.

One is the multiplicative identity, as the product of multiplying one by any number is identical to the other number. Likewise, the additive identity is zero, because zero along with any other number is the same as the quantity which is the non-zero summand. The functional identity is the one that produces a quantity identical to the number in argument, $f_{\text{id}}$.

Here are two plots of the identity function $y=x$ compared with the same with a non-zero y-intercept, $f(0)=2$, showing what the reference basis of slope unity is. It's called the identity function because it takes the parameter $x$ (at every point on the interval) and maps to the same number (identically). An interval here means two different real numbers, symbolically $a$ and $b$, along with everything in between, using square brackets it's written like so: $[a, b]$.

The $y=mx+b$ line equation is a constant-slope line in a two-dimensional space with position measured from some origin so that $b$ is obtainable algebraically and we use the familar slope as a ratio of rise-over-run, $m=(y_2-y_1)/(x_2-x_1)$. To get the given $y-$intercept, one applies the formula with known variables for $m$, $x_1$, and $y_1$ to get $b=(y_1-mx_1)=((y_1)-(m)(x_1))=((x_1)-(1)(x_1))=0$. So $b$, as a constant for the formula of $y$ on $x$, is a function of any two given points on any line in a plane, not the identity function $f_{\text{id}}(x)=x$, alone.

The $x$-axis isn't necessarily displayed in the plot, because if the curve doesn't go to zero on the ordinate interval range, or function image, of interest, then the lowest $y$ is the lower indicator on the ordinate display, where we must draw the abscissa metric over the domain interval (an x-axis-parallel measuring line). A function, $f$, maps from an interval $X=[x_1, x_2], x_1\lt x_2$, to its image $f(X)=[f_{min}(X), f_{max}(X)]$, where the minimum of $f_{min}(X)$ is the lowest point on the curve within the interval, likewise for $f_{max}(X)$, which are in general different than $f(x_1)$ or $f(x_2)$.

The highlight point at $1.9$ appears different in the two plots because the domain intervals are different.

The plots on this site are all 2-dimensional planar, so far, and the rectangular area presented can be thought of in terms of the graph set notation, $(X,f(X))$, where $X$ is the bounded interval of the real numbers, $X=\{x_0\lt{}x\lt{}x_1:x_0,x_1\in\mathbb{R}\}$, and $f(X)$ is called the image of $f$, being the whole plotted ordinate range.

The plot is a visualization tool, not an exact or complete representation of the mathematical formula—because a curve has no mathematical width, it is simply a sequence of points, but since the eye can't percieve that, curves are width-stroked with colors to make them visible.

The abscissa and ordinate are also known as $x$ and $y$ axes, or coordinates. The axes only intersect at the origin (as in the first plot below, with domain $[0,2]$), which isn't in every graph, $G$. The ordinate is the vertical axis, and reading a point on the plotted curve means associating a point below it (in the positive-definite plots of this article) on the abscissa with a point on the ordinate axis, as the pair of numbers at the highlighted point indicate in the $G$-doublet, $(x, (y=f(x)))$. The ordinate will go from the minimum value of the function to the maximum of the curve, indicated as the upper and lower numbers on the vertical axis, so for the identity function the range of the ordinate is the same as the interval of the abscissa. In the plots the function value as indicated by going straight up from a point on the abscissa (an $x$ value in the interval of interest) to the curve (lines in these two plots), and the height (function value or magnitude) is as indicated on the left axis by drawing a line parallel to the abscissa at the height of the point (positive or negative, $\pm$) and seeing where it intersects the axis-parallel metric ordinate.

A bit on typography, if ever you want to see what the $\LaTeX{}$ source code looks like for a particular equation or expression, you can get it from the interactive tools of the MathJax widgets, just by right-clicking on the equation and selecting "Show Math As" > "TeX Commands". To get you started with reading the math mode TeX, the superscript notation is written as `^{}`, and the subscript notation is written as `_{}`, where the curly brackets are used to group terms as the argument to the command, and of course these two commands imply a leftside element to the expression (like $2$, $x$, or $e$, though only the second one will exhibit usage of notation in both positions, of course).