Pythagorean Theorem and Hypotenuse Function

The Babylonians (2000 BC) knew the triangle, be it acute, obtuse, or the most useful form, the right triangle.

The three interior angles of a triangle add to 180°, which is the result of work done in Euclid's Elements (300 BC), though much of that work could have been done by students of the Pythagoras school, of the same era. The proof lies in considering not only the interior angles, but also their complimentary angles. So, we construct the Sum of the angles of a triangle with an arrow on the line whose segment is a side, or edge, of the triangle. If a path is drawn around the triangle we know that a vector pointing in the forward direction turns one whole turn upon completing the path, which travels along the three edges, and stops at the three vertices, where it rotates to face the subsequent edge's direction.

φ θ The acute (θ) and obtuse (φ) angles of two crossing lines.

The sum of the outer angles has to add to 360°, and the complementary angles can be summed according to the acute/obtuse diagram, demonstrating that the inner angle sum is always 90° is a good exercise for everyone.

The right triangle has two edges meeting at a 90°-angle, or right-angled vertex. It is known by the three component constraint equation, $$\label{Pythagorean}\tag{1} a^2 + b^2 = c^2 $$ Where $c$ is the hypotenuse, and $a$ & $b$ are the two edges of the right-vertex. The relation for edges of a right triangle is useful anytime there is decomposition of a position into independent components.

A right triangle is one that has a 90° angle, sometimes indicated by a square at such a vertex.

A right triangle with different colored, and dashing, on the edges; the hypotenuse is labelled as c, and has a dot-dash pattern.

The area of a rectangle is the width times the height. The area of a square, with four equal edges, and four right angles, and edge $x$, is $x^2$ ($x$-squared). The right-triangle theorem is geometrically a relationship of areas of the squares.

Depiction of Pythagoras's formula with geometric areas of square polygons.

The area of the right-triangle is half the multiplicative product of the two non-hypotenuse edges. By separating a rectangle into two halves you are left with two identical right-triangles:

A rectangle plus a line drawn diagonally from corner to corner.

which gives you the factor of a half in the formula for the area of a right triangle: $A=ab/2$.

An equivalent statement for $x^2$ is, $xx$ (or, $x$ times $x$). $x^2$, where $x$ is the variable (because it could be $a^2$, $b^2$, or $c^2$), is an example of the exponential notation: $x^n=x*\ldots$, where the "$\ldots$" ellipsis is exactly $n-1$ factors (multiplicands) of $x$. $n$ is an index to an intermediate usage, to demonstrate the exponent position of a variable ($x$).

If you hold one of the edges of the right triangle fixed to a constant value, then you get a function $f$ for the hypotenuse — a single variable function. $$ c=\sqrt{(x^2 + a^2)}=(x^2 + a^2)^\frac{1}{2} $$ where the $x$ was substituted for $b$ in the right-triangle formula, and $a$ is set/fixed as a constant.

It was the year 1620 when Henry Briggs discovered a series solution for this binomial. [1]

When studying functions (algebraically) to find significant features of relationship and behavior, one starts with monomials ($x^n$, $n$ being a positive integer for the Taylor Series), and the next step of complication or sophistication is the binomial, where x is replaced by $(x_1+x_2)$. Plotting monomials is informative, so $x^n$ for $n\in \{1, 2, 3\}$ is plotted here:

Monomials of the first three exponent powers.

Monomials of the first three integer exponent powers: $f_1(x)=x$ is the red one, $f_2(x)=x^2$ is the middle one (on either side of $x=1$, and $f_3(x)=x^3$ is the steepest curve (pale green).

The plotting of monomials is most dramatic around 1, from studying their behavior around 1 one understands the two sides of these single variable functions.

A web search for "history of the square root" has a high quality return (Quora, Mathforum, Stackexchange, Wikipedia), where one can learn that n'th roots were studied and written about in the early 1600s, mostly referred to as radix (since math was written about in Latin) and radix means root, and the related term radical also means related to roots.

Classical physics is often concerned with the slope of a function.

Instead of just calling it the hypotenuse formula (in one variable), one categorizes it by refering to it as $f(x)$, or one could use some freedom in the designation of the subscript on the $f$ to write: $f_a(x)$, where the fixed edge length, $a$, is made an index to which exact $f(x)$, of the infinite values $a$ can represent, particularly if you wanted to compare the slopes of these two functions.

Between the two equations Pythagorean and hypotenuse, there is an exponent applied to both sides of the equals sign, and on the left side we are left with $$ ((c)^2)^\frac{1}{2}= (c)^{(2)(1/2)} = (c)^1 = c $$ Which leaves the hypotenuse on the left side of the equation, and we call that formula for the hypotenuse (right side of the equals sign) a function of $x$.

Any given number is silently "raised" to the exponent one, or I should say there is an exponent position on every set of parentheses.

The hypotenuse function, conventionally and alternatively referred to as $f(x)$, for the formula in subject being a function of any single variable, can be illustrated from $x$ equals zero to any constant (measuring along an imaginary tape measure that goes as far or as microscopic as is classically imaginable); the interval of interest can be categorized as to when $f(x)$ is small, about one, and large. Not all single variable functions stay finite for finite $x$, one-over-x ($1/x$) can't be plotted for $x$ close to and including zero, because infinity can't be plotted. To plot it we consider the values of this function at every point along the interval of interest in $x$, which is the set of values from $x_A$ to $x_B$.

For this function, $x=4$ has imperceptable variation (a fraction of a pixel) from the asymptote with variable "y-intercept" and slope of one. A single variable function is understandable because you can plot it, and to understand a multivariable function's behavior, we will hold all but one fixed to constants to isolate it's behavior, but we'll stay with the hypotenuse function long enough to get some function handling experience and to appreciate the darkness of science subsequent century after century of nothing advancing The Theorem.

Plot of the hypotenuse function, and dashed straight line with slope one to signify asymptote.

Plot of the hypotenuse function, $c$, with edge $b$ measured on the abscissa, and fixed edge $a=1$.

The dashed line is a straight segment, described by the familiar formula $y=mx+b$, and it is drawn in order to compare the shape of the hypotenuse curve, near $x=a=1$, that is around where both edges off the right angle are about equal, with the distant behavior of the curve. This $f(x)$ curve always has a small amount of curvature, never a straight line, but the difference between it and the straight line drawn, $f-y$, is decreasing up until the point of their intersection.

So there is not really a line with slope one (ie for any given $b$, $y$-intercept) which won't be crossed at some point. Which just means the swinging door needs a little space around it, no matter how thin you make the door.

So, for $x\gg a$ (much greater than) $a$, the slope of the curve, in graphical display of the value of the function, which is $c$, the hypotenuse, the slope is increasing from close to a constant (horizontal line, independent of $x$), approaching the slope of the asymptote.

The function is plotted over the domain [0,3] (which is to say 0 through 3, including the two endpoints), and the horizontal axis ( the abscissa) is labeled in evenly separated measurements (three of them here). The ordinate of the plot is the curve of values of $f(x)$ as they are calculated for "each point" above a given value of x, labeled to the left of the vertical axis in three places, starting at the value $2$ ($a=2$ in hypotenuse formula), which was an arbitrary value for the fixed edge of a right triangle. The values of 2 for the fixed edge and 3 for the non-hypotenuse edge are kind of determined by an important facet of this function--having to do with convergance of the binomial series around the value of $(x/a)=1$, but it also makes a good looking plot using those two plot program inputs.

Plot of the hypotenuse function, and dashed straight line with slope one to signify asymptote.

Plot of the hypotenuse function, with fixed edge $a=1$, and variable edge on the interval [2,5]

From $x=2$ on up, plotting the hypotenuse as a function of one variable triangle-edge pretty much looks the same, with the slope getting closer and closer, but never equal to one.

The function can be rewritten an endless number of ways (because you can multiply by 1 or add zero and make another equals sign), but if it is divided through by $a$ and leaving an $a$ out front, and knowing a few rules of algebra such as the following: $$ (ab)^\frac{1}{2}= (a)^\frac{1}{2} (b)^\frac{1}{2} $$ Letting $c^2=a$ in the previous equation, we have: $$ ((c)^2b)^\frac{1}{2}= ((c)^2)^\frac{1}{2} (b)^\frac{1}{2} = c (b)^\frac{1}{2} $$

We can look at an alternate form of the hypotenuse function, in particular we will separate the function into asymptote (on the left of the composite) and the other (compound) part on the right. In this constructed way we can look at the hypotenuse function over the [1, ∞] part like so: $$ \label{f_plus} \tag{2} f_{1+}(x)=\Big(x\Big)\left(1 + \left(\frac{a}{x}\right)^2\right)^{1/2} $$ As you can see the $x$ is on the left, and the normalized binomial function is on the right. This is the correct form for $x$ greater than $a$ because the behavior of the left and right composites are well defined both at $x=a$ and for $x=\infty$, as opposed to the $f_{1-}$ factoring, which is ill behaved on that interval. A composite number is one that can be factored (multiplication) into two whole parts (where a part can itself be composite), whereas a compound number is one that is represented as the addition of two parts. We'll explore the complicated part by next looking at the inverted monomial (one over $x$). Inversion, generally, is not simply swapping the numerator and denominator, it's a formulation in which the combination of the inverse and the original forms the identity function.

$f_{1+}$ is a product of x and another function of $x$, $$ \label{BinomialForm} \tag{3} (1 + y)^{1/2} $$ where $y=(a/x)^2$, because that's a constant ($a^2$) times one-over $x^2$. Understanding the single term functions gives understanding for how a compound, or composite function looks at different intervals. Monomials (a function with a single term) are expressed by a variable, like $x$, with an exponent. The exponent can be a fraction, whole number, positive, or negative, making the term a monomial. So the hypotenuse function is a composite function, and $f_{1+}$ is a monomial times a composite. So we look at the components in the composite part and see that we have nice behavior in the monomial component $(a/x)^2$, in the interval which exludes $x$ smaller than 1 (the interval of 1, on up).

A monomial multiplied by a constant has the same behavior as the monomial scaled by unity. For any formula of a single variable $g(x)=af(x)$, then we can talk about the difference between $f$ and $g$ as being scale. So note that the $y$ substitution is of a similar form because putting $x$ or $x^2$ in the denominator makes the function go small for large $x$.

The right side of the formula for $f_{1+}$ has a similar term in it, $$ x^{-2} = \frac{1}{x^{2}} $$ which puts the $x$ variable in the bottom (denominator, being the dividing variable) of the ratio (fraction). If we wanted to add a column to the table for $f(x)=1/x^2$, we could just square the $1/x$ column. $$ x^{-2} = x^{(-1)(2)} = (x^{-1})^2 = \left(\frac{1}{x}\right)^2 $$ In this way, we see the relationship between the two functions of $x$. Monomials are monotonic over the domain [0,∞], which means either never increasing, or never decreasing.

From the table, and by inspecting the function, inversion replaces the starting point (zero or infinity) with the other one. One can see what inversion does to the $x$-monomial, which is pivots the function around $1$, but with a sort of "compression" to get the same range you have for $f(x)$ on [1, ∞] to "fit" in the range on [0, 1].

For the partial domain of [0, 1] (zero to one), the function is labelled with the subscript $1-$: $$ \label{f_minus} \tag{4} f_{1-}(x)=a\left(1 + \left(\frac{x}{a}\right)^2\right)^{1/2} $$ For [0, 1], the asymptote is $a$, on the left side of the rewritten function, and multiplying on the right side is a function deviating from $1$. Now with $y=(x/a)^2$, with which to use the binomial series on. Both forms for f(x), above in equations (\ref{f_plus}) and (\ref{f_minus}), are the same as f(x), algebraically you can put the asymptote part of the function back into the outer power of 1/2, and you get the hypotenuse function. $$ f(x) = f_{1+}(x) = f_{1-}(x) $$

It took 1,800 years to get from the Pythagorean theorem to the generalized binomial series, so it is informative to Binomials were studied for a very long time, with a global history of what the formula is for $(a+b)^n$ for arbitrary positive integer n. To use the binomial series, to better understand the plot of the hypotenuse function, we need two series: one for the domain interval from [0,1] and the other function expansion for over 1, because we need a convergant series expansion.

The plot also explains a feature of the swinging door — that you only have to make the clearance in its frame a small fraction of the thickness of the door. If you had a very thick door, and didn't want to leave a big space around the closed door and its frame, you would have to radius the door, or bevel it so that the inner edge of the door didn't obstruct the opening motion. So it is instructive to demonstrate the narrow gap around 1.5" thick, household doors corresponds to a linear, one-to-one asymptote over the domain of $x\gg$(a=1.5) by looking at the hypotenuse function over the configuration of a swinging door in a tight frame, or about $x=33 \textrm{in.}$ :

The hypotenuse function at x=32 inches to x=34 inches, for a fixed edge of 1.5 inches, is 32.04 and 32.03 inches respectively, and has very subtle curvature.

This plot has curvature that is imperceptable to the eye, but means something to someone who ever wondered how swinging doors didn't hit the frame, which we get to elucidating with what physicists call convergance. And also, by looking at a function modification, considering the variance from x, $$ g(x)=f(x)-y $$

A convergent function is when some complicated function looks like a monomial, at certain parts of the domain (tending towards infinity). We measure the difference between the binomial hypotenuse function and the monomial, at the width-to-thickness ratio of doors, and we see a frame minimal clearance of much less than the thickness.

The hypotenuse function looks like a. The asymptotic monomial, written explicitly with the exponent, as $x^1$ (x to the first power), not to be confused with how many numbers there are inside the outermost exponent (that's what the mono and bi refer to in the nomials). The linear asymptote with slope one ($f(x)\approx x$) as $x>>a$ ($x$ is much larger than $a$, the other edge of the triangle), where I changed the variable $x=b$ (from $b$ to $x$) because $a$ and $b$ are associated with constants, in equation (\ref{Pythagorean}).

f(x, a=1.5)xg(x)

So by inspection of this table we can see that the variance of $f(x)$ from $x$ (the asymptote) is still decreasing, but it's in the third decimal place (thousandths, not very much). If we take the difference between $f(x)$ and $x$, you can see that difference is decreasing, which means that the wider we make the door (distance from the hinge axle), the closer the hypotenuse is to the breadth, so the tighter we can make the frame.

In this plot of the hypotenuse as a function of x, for real ratios of door width to door thickness, we see that the variance of the hypotenuse, at x= 32" and x=34" (normal door widths, as measured left to right standing behind one) is very close to an f(x) value (ie, the "y=mx+b" slope is getting closer to 1 with increasing x). It should also be noted that the three points in the abscissa and ordinate axes correspond to linearly increasing numbers, signifying that the axes are Cartesian, or like a grid with rolling tape measures going as far as we need them--this is by far the most common type of coordinates to plot with, but you always need to check that half way between two points is actually the average.

One of the most important changes to all this as you progress in your physics studies is that the hypotenuse function becomes the radius, which is decomposed into orthogonal components of position measurement. You already know orthogonal, it's what the two non-hypotenuse edges of a right triangle are to eachother -- two lines (along which we measure position) intersecting at a 90° angle.

If you're wondering what the next step is to analyze the nature of this single variable function, then you're in the same boat anyone studying the works of Pythagoras and Euclid was in--for hundreds of years, in fact there wasn't another breakthrough in physics until 1665 ( The hypotenuse function f(x) is called a binomial function with power of one half.

Another notational characteristic of physics is that all constants and variables are single character in length (could be uppercase or lowercase, Greek or Latin, script or bold, variants) -- there is some convention as to where character variants are used in the physics equations, but the significance of a formula is summed up as being a "function" of some variable ($x$ as in the area of a square with edge $x$, or any other single variable function), or the formula could be a function of more than one variable, as in the function of $a, b$ for the hypotenuse length of a right triangle.

The Pythagorean theorem lets us solve problems like calculating how much time it takes to “motor” straight across a river with uniform flow, and parallel banks. We go straight across a river by knowing the speed of our boat in still water, then angling it so that the centerline from the point of the bow through the middle of the stern is along the line of a hypotenuse, with one edge parallel to the bank, and the other parallel with the line going straight across.

There is a Physics Stackexchange problem on a river and a "motor boat", asking about the path across a river where the motor boat is always pointed at a fixed point on the bank, directly across the river from the starting point, but the question has not dismissed because it is an uncommon exercise: swimmer with a focal direction. The idea is reminiscent of the Newtonian "centripetal force", which was addressing a readership of very different paradigm, one that was trying to explain the solar system using the refracting telescope, developed in 1608. Now, if you make the point on the river bank a focus, the force always acts towards the point, not that motion is constrained in that way, which is similar to Newton's concept of centripetal force.

[1] D. T. Whiteside. Newton's Discovery of the General Binomial Theorem. The Mathematical Gazette, vol. 45, no. 353, 1961, pp. 175-180. JSTOR, . Accessed 24 Oct. 2020.

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