Numbers, Notation, and Graphics

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.

$e^x:x\in\mathbb{R}$ is a function mapping the reals to the positive reals, and the inverse, $\log{(y)}$, is also defined on a domain of reals, but for a range in the complex, $\log{(-1)}=i\pi: i\in\mathbb{C}$, so it is not closed on the reals.

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 (a notion extended in the reals when describing the irrationals).

In general, every natural number, $n$, is representable 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 distributivity in multiplication over addition (but not the converse), and the set is constructable using nothing more than the Peano Axioms. Commutativity of addition is always true, $a + b = b + a$, but in general for matrices (such as rotation operators), $ab \neq ba$. Associativity is always true for addition and multiplication of general matrices, but as you know it's not the case for subtraction and division of numbers, since $(a/b)/c\neq a/(b/c)$ and $(a-b)-c\neq a-(b-c)$.

$$ a \times (b + c) = ab + ac $$

$$ a + (b \times c) \neq (a + b) \times (a + c)$$

Any of these operations using elements from this set will stay in this set, but when we add a physicality-constraint to a primitive wave composed of the cofunctions, sine and cosine, with arguments in time and space, we necesarily introduce complexity to our picture of “reality”, little by little we'll get there.

The integers, symbolized by $\mathbb{Z}$, are named for their wholeness in the sea of real numbers. They 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 an Equivalence Class, which forms the basis for mapping each integer to a class of such doublets. The equivalence 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 symbol from the term quotient, which is a synonym for ratio, or fraction. $\mathbb{Q}$ is constructed as Equivalence Classes of another type of doublet, this time in the integers, so the equivalence class of one half is, $[1/2]=\{(1,2),(2,4),\dots\}$, or $\mathbb{Q}\simeq \mathbb{Z}\times \mathbb{Z}\setminus{\{0\}}$, where right-side set is integers excluding the single point zero. The equivalency 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 practical use ($9.8 m/s^2$, $3.14/180 rad/deg$)—because to be irrational is to be represented by an infinite series of non-repeating digits, and in particular not representable by any whole number divided by another whole number.

Algebra

Every curve has a function, or set of functions, describing it. The reals are closed under addition, subtraction, multiplication, and division, but not under logarithms. So we have to extend the number system to include these new numbers (complex) which are the results of these significant operations. This is physically motivated because we want to be correct, so we must consider something we thought impossible before it was proven necessary, but required of logarithms, wavefunctions, and special relativity.

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.

$$ \sum^{n-1}_{x=0} r^x = \frac{1 - r^n}{1-r} $$

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 sequence of the kernel. 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 $a$.

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, $f_{\text{id}}$.

Graphs

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, then the lowest $y$ is the lower indicator on the ordinate display, where we draw the abscissa measure over the domain interval. A function, $f$, maps from an interval $X=[x_1, x_2], x_1\lt x_2$, to its range $f(X)=[f_{min}(X), f_{max}(X)]$, 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.

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).