Since we were children playing with alphabet blocks we've had some degree of physical, observational learning about composition of symbols, the stability of the untouched block, trying to remember what's on the obverse, and of course taking away and removal to put the hand's grasp upon the block. But why not formalize your concepts and notions about it with a little mathematical logic; so, We study the logic of math to help our understanding of what is logical and what is mystery to ourself. That this helps with physics is, supported by the , where the path is going along in a logical way motivated by the impact of mathematical research on physics like studying its geometry, analytically eliminating unecessary vagueness with method and exercises, and then a few math textbooks later (real analysis, linear algebra, abstract algebra, differential topology), you are ready to understand the the complex manifolds of the most recent physics. This is because only when the particulate mechanics equation (Schrödinger) is generalized to the quadratic form of energy and momentum, connecting them to the most correct theory of physics in history, General Relativity, do we get the formula for the charge density distribution, and the wavelength of the particle which is ten orders or magnitude off from the Compton kinetic wavelength.
Classical wave mechanics is at the heart of all wave mechanics, so we must study the sine and cosine, and their identities, which is codeveloped with calculus. Sequences and series are also fundamental to our evaluation of curves.
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 kind of being the zero element of sets), $\{0\}\neq\emptyset=\{\}$, 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 name 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 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 $y=x$, in pursuit of logical consistency, including being a closed set under the operations of the system, including multiplication and addition.
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 (dollars), 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 the natural numbers, $\mathbb{N}$.
$\mathbb{N}$, the natural numbers is a set of numbers equipped with commutativity, associativity, and transitivity in both multiplication and addition, constructable using nothing more than the Peano Axioms. In particular, any of these operations using elements from this set will stay in this set (closure under the operation, in the set). $\mathbb{Z}$, the integers are constructed by first structuring the target (non-negatives + negatives) as a Cartesian product 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 an integer, so the convention is to subtract the second number from the first to obtain the value. The difference between $a$ and $b$ is representable in what is known as a congruancy 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, $[0]=\{(0,0),(1,1),(2,2),...\}$.
$\mathbb{Q}$, 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 $\mathbb{Q}\simeq \mathbb{Z}\times \mathbb{Z}\\0$. 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.
Every curve has a function, or algebraic formula, using a representation in algebraic composition of a parameter (abscissa), because the reals.
Formulas use the parentheses in two ways. The first being to group nomials, or terms, and the second is to indicate the parameter(s) 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 needs a parameter.
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 $x_1/x_2$, which is a substantive equivalent to: $$ \frac{x_1}{x_2} $$ 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.
Formulas can be expressed as a compact kernel inside a summation symbol (a kernel is a compact way to describe a multi-term function) and always a combination of a variable, commonly $x$, as the abscissal parameter, and constants of significance and convention, as well as those of logical particularity.
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: $[a, b]$.
A plot of $y=mx+b$, on the interval $[0, 2]$. There is also a highlight at the point $(1.9,3.9)$.
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 graph of the $f=x+2$ function along with $f_{id}$, on the domain interval $[1, 3]$.
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 $y$ coordinates, or axes, but these are parallels as the axes only intersect at the origin (first plot, 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.