First Article

The numbers we'll be using are the real numbers:


Every curve has a function, or algebraic formula, using a representation in algebraic composition of a parameter.

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 notational option, and substantive equivalent to: $$ \frac{x_1}{x_2} $$ Without further specificity or information on the behavior of holding one constant, the ratio might be summarily posited as a function with two distinguishable parameters.

Formulas of summation

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.

Here are two plots of the identity function $y=x$, showing what a line with slope of unity looks like. It's called the identity function because it takes the parameter $x$ (at every point on the interval) and outputs 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: $[a, b]$.

An introductory plot.

A plot of $y=mx+b$, on the interval $[x_1, x_2]$. There is also a highlight at the point $(x_{\text{hl}},x_{\text{hl}})$.

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 just the identity function $f_{\text{id}}(x)=x$.

The $x$-axis isn't necessarily displayed in the plot, because if the curve doesn't go to zero on the (abscissa) interval of interest, then the lowest $y$ is the lower indicator on the ordinate display.

Basic plot.

A plot of the identity function, on the interval $[1, 3]$. There is also a highlight at the point $y=x=1.9$.

Second plot.

A plot of the identity function, on the interval $[0, 2]$.

The highlight point at $1.9$ appears different in the two plots because the intervals are different. If this was an interesting function, anything other than a line, it would have some curvature.

The ordinate and the abscissa are also known as $y$ and $x$ coordinates, or axes. The ordinate is the vertical axis, and reading a point on the plotted curve (lines in this article) means associating a point (below it) on the abscissa with a point on the ordinate axis, as the pair of numbers at the highlighted point indicate. The ordinate will go from the minimum value of the function (on the given interval) 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 (line in these plots), and the height (function value or magnitude) is as indicated on the left axis.

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

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

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