Your favorite formula

edited January 14 in Assignments

(5pts)

I assume that you couldn't have made it this far into mathematics without having a favorite formula! So, share yours with the world responding to this post below. Be sure to:

  1. Typeset your formula using AsciiMath or a LaTeX snippet,
  2. say something about where your formula comes from,
  3. include a picture generated by your formula.

For example:


My favorite formula is the computation of the Gaussian integral:
$$\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}.$$

This integral is extremely important in probability and statistics. Geometrically, the formula states the the area under the curve below is %%sqrt(pi)%%.


Here's what I typed in to get this:

My favorite formula is the computation of the [Gaussian integral](https://en.wikipedia.org/wiki/Gaussian_integral):
$$\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}.$$

This integral is *extremely* important in probability and statistics. Geometrically, the formula states the the area under the curve below is %%sqrt(pi)%%.

![](https://marksmath.org/classes/Spring2020CalcIII/MultiCalcTalk/uploads/editor/nb/8d9bi53qdgfj.png "")

That last line was put in automatically by the image uploader. Again, you can generally use either AsciiMath (enclosed with double percentage symbols) or LaTeX. I typed the first formula with LaTeX and the %%sqrt(pi)%% with AsciiMath.

audrey
«1

Comments

  • edited 12:39AM

    My favorite equation in Mathematics comes from the Lorentz Transformations. The transformations help physicists measure properties of an object from different reference frames using Special Relativity. Specifically, the equation I chose is the Lorentz Factor, which is given below.

    %%y=1/(sqrt(1-(v^2/c^2))%%

    %%y=%%the Lorentz Factor
    %%v=%%velocity
    %%c=%%speed of light
    What does this equation do? A number of things, such as measuring how mass, time, and length change in Special Relativity. On the grand scale of the Universe, time, energy, and velocity scales cannot be accurately measured with classical physics, and thus the Lorentz Factor measures how much these variables change from another reference frame (ie: scientists on Earth measuring the mass and velocity of distance objects).

    An example of an application, for instance, is in classical physics. Mass can be given by the variable, %%m%%, and measured by fairly intuitive techniques. However, when an object is in motion, its mass is actually increased. In Special Relativity, the equation becomes %%m=ym0%%

    %%m=%% relativistic mass
    %%y=%%the Lorentz Factor
    %%m0=%% measured initial mass before motion

    From the graph above, you can see for low velocities (velocities in everyday life), the Lorentz Factor will cancel out to approximately one, not affecting the measurement. However, as the velocity of an object increases, so will its weird effect on physical properties, like mass.

    There are more pretty wicked derivations using this factor, such as time dilation, and the length contraction of an object (as the velocity of an object increases, so does its length from an outside reference frame). My favorite is when you take the Classical Physics equations for Kinetic Energy and momentum, and add in the Lorentz Factor. Using this, you can very simply derive the famous %%e=mc^2%% equation.

    mark
  • edited January 14

    My favorite formula is %%x^2+y^2=1%%, which is the equation of a circle.

    It looks like this:

  • edited January 15

    I really like Taylor Series. I don't have a particular favorite, but here's one for %% sin(x) %%

    %% sin(x) = x - (x^3)/(3!) + (x^5)/(5!) - (x^7)/(7!) + O(x^9) %%

    I also really wanted an excuse to use Big O Notation seeing as I'm a Compsci major and it has applications in both Computer Science and Mathematics.

    In this case, it's to approximate an upper bound of the function while in Computer Science it tends to note run-time.

    (Note: sin(x) is denoted in red, the series in blue)

    mark
  • edited January 14

    I have a love/hate relationship with the equation %%intsec(x)dx=ln|sec(x)+tan(x)|+c%%. I've spent many hours forgetting to use this thing in trig subs. May not be groundbreaking or world changing for a lot of people, but it was definitely grade changing for me.
    Below is the graph %%y=ln|(sec(x)+tan(x)|%%

    mark
  • edited January 14

    My favorite formula is:
    %%(x^2+y^2-1)^3-x^2y^3=0%%
    I chose this formula because, when graphed, it creates a heart. Although this isn't terribly mathematically significant, I chose it as my favorite formula because it serves as a great reminder that math doesn't have to be scary and complex all the time. Sometimes, math can be quite fun.

    mark
  • My favorite formula is $$x^{\frac{2}{3}}+y^{\frac{2}{3}}=\text{a}^{\frac{2}{3}}$$
    This formula is known as an astroid or the tetracuspid because of its four cusps. I like this formula because the equation is similar to that of a sphere but it is unique. In the graph shown a is 1.

    mark
  • edited January 14

    Honestly, my favorite equation is the Fundamental Theorem of Calculus $$\int_{a}^{b} f(x) dx = F(b)-F(a)$$

    The moment I understood it and what it meant was the first time I really felt like I understood math on a deeper level than 'what is this this equation and what does it solve for?'


    For this graph I used %%f(x)=1/x%% (red line) and %%F(x)=lnx%% (green line) between the points %%a=1%% and %%b=e%%

    mark
  • edited January 14

    My favorite mathematical formula is probably Euler's Formula:

    %% e^(ix) = cos(x) +isin(x) %%

    cosine(x): real width
    isine(x): imaginary height

    Essentially, this formula is telling us there are two ways to traverse a circle. First is the method using sine and cosine. The second way isto use imaginary growth, which is what %%e^(ix)%% side of the equation is doing. For real growth, numbers are being pulled in the same direction (1,2,3,4....). In imaginary growth, the numbers are always being pulled at 90 degrees. Meaning, that the number isn't being pushed forwards or backwards, it's simply being rotated!

    I love this formula because we are technically dealing with exponential growth, but remain in a circular path. I also love this formula because it was the first time that I had heard of imaginary growth.

    mark
  • My favorite formula is %%y=1/x%% for %%x >= 1%%.

    I like this equation because when the graph is rotated, it produces a solid called Gabriel's Horn. Oddly enough, this solid of revolution has a finite volume, but an infinite surface area.

    mark
  • edited January 14

    My favorite formula is the Pythagorean Theorem:
    %%a^2 + b^2 = c^2%%

    I have always been impressed that this simple formula proves so useful in algebra and geometry.

    There are also numerous interesting proofs of the theorem, my favorite is the rearrangement proof that is illustrated below:

    mark
  • edited January 15

    My favorite function is %%y=- (42-|8x|)^2/120%%, which is similar to McDonald's big M logo

    I like this equation because I like McDonald's nuggets, and if college doesn't work out, I'll probably end up working there

    mark
  • My favorite formula is %%z_(n+1)=z_n^2+c%%, which describes the Mandelbrot set, a well known fractal.
    I have always been fascinated by the Mandelbrot set, even before I had an understanding of what fractals were or how they worked just because of how visually stunning the set is.

    mark
  • edited January 15

    My favorite is the formula:

    %%Z=(Z^2)+C%%

    This is the basis for most fractal formulas and creates a recursive, infinite set that can be used for many applications. Most interesting of which to me is the use of fractals in computer graphics to simulate infinitely small details without requiring excessive computing power.
    In the 1970's and early 80's, Benoit Mandelbrot brought the mathematics of fractals to light in popular culture, programming, and science. Today his breakthroughs are important stepping stones for our current technology and mathematical understanding. Below is a picture of the fractal set he introduced to the world, called the Mandelbrot set.

    mark
  • edited January 15

    My favorite formula is the trig identity %% (sinx)^2+(cosx)^2=1%%.

    This formula states that %%(sin(any x value))^2+ (cos(any x value))^2%% will equal 1. I have found that this formula makes working with trig functions a lot less scary than it first appears.

    Below is a graph of %%(sinx)^2%% (blue), %%(cosx)^2%% (green) and %%(sinx)^2+(cosx)^2 %% (red). Here you can see that this formula equals one at all given values of x.

    mark
  • edited January 15

    My favorite formula is $$\sin(x^2+y^2)=\cos(x*y).$$ I like this formula because it creates an interesting shape when graphed.

    mark
  • My favorite equation is...

    y=mx+b

    This is my favorite equation because it was one of the first math concepts I truly grasped and later learned to love when taking Math 1 in 8th grade. Truth be told, this equation sparked my, otherwise, prior to, none existent, love for math. I find myself still using this equation in all of my natural science classes and I just admire its simplicity.

    When graphed, it looks as so...

    m = 3/4
    b = 3

    mark
  • My favorite formula is the Pythagorean Theorem:

    %%a^2+b^2=c^2%%

    This formula relates the sides of a right triangle to each other by stating that the square of the hypotenuse is equal to the sum of the squares of the other two legs. This has many applications in math such as trigonometry. It can be visualized by the below graphic.

    mark
  • edited January 15

    My favorite formula is the Quadratic Formula:

    $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

    The quadratic formula an extremely useful and helpful tool given in the beginning of high school or even middle school. It's used to solve quadratic equation in a different way. I like this formula because it took me kinda a long time to remember and now I can't forget it. This graph shows a quadratic equation with roots of x=-0.5 and x=1.

    mark
  • edited January 15

    My favorite equation is the one related to Gabriel's horn: %% y=1/x %%. If we just look at the curve on the right and graph it the function goes to infinity as x goes to infinity. But, if you revolve that curve around the x-axis and find the volume of it the volume is actually finite. Infinite length but finite volume. It's neat to think about.

    mark
  • @sara - Whoa!

    I did edit your TeX slightly to the sine and cosine look like $\sin(x)$ and $\cos(x)$, rather than $sin(x)$ and $cos(x)$, and to bring the period inside the displayed equation.

  • My favorite formula is the Law of Sines which combines ratios, which I love, and trigonometry. The formula states that:

    $$\frac{a}{sin A}= \frac{b}{sin B} = \frac{c}{sin C}$$.

    This is extremely useful when you know the angles of a triangle and one side length and are trying to find the other side lengths - especially when adding vectors.

    mark
  • edited January 16

    My favorite equation is the ellipse. Being a physics major, I have many equations that are dear to me. But in the realm of celestial mechanics, the ellipse is very important since all celestial bodies orbit in ellipses.


    where,
    a = 4
    b = 1

    mark
  • My favorite equation is the golden ration:

    $$\varphi = 1+\frac{1}{\varphi}$$.

    This equation is defined in terms of itself and graphing, creates the golden spiral which is found within the Fibonacci sequence and can be rearranged to find the quadratic equation. There are plenty of examples of it within nature such as the sunflower or the nautilus shell.

    mark
  • edited January 16

    I've recently learned about Bayes' Theorem and I think it's super cool!

    %%P(A | B ) = (P(B|A) * P(A))/(P(B))%%

    It is a statistical formula that gives you the conditional probability of some event A being true given that another event B has occurred and that events A and B are somehow dependent on each other. It really is quite simple, but also extremely applicable to various fields including, my favorite, artificial intelligence.


    (Tree representing the conditional probability that a coin is fair given the result of repeated flips)

    As shown in the above picture, this concept can be represented by a tree. There are other visualizations that use areas.

    markarnold
  • edited January 16

    So my favorite "formula" even though it's not really a formula is Euler's Identity.

    $$e^{\pi i}+1=0$$.

    So basically Euler's Identity is considered the most beautiful equation in the world. The reason I like it is because it bring in every type of number; integer, rational, non-real, and even zero.

    Although, Euler does have a formula that goes along with his Identity which is:
    $$ e^{\Theta i} = \cos\Theta + i\sin\Theta$$ .

    mark
  • edited January 16

    My favorite formula is %%r=sin((a/b)(theta))%% which, when graphed in polar coordinates creates a flower-like Mandela that looks very cool:

    mark
  • edited January 16

    My favorite formula is the triangular number series formula. $$T_n=n(n+1)/2$$

    It is also a geometric series. This is my favorite formula because Christmas is my favorite time of the year and the amount of gifts given in the song 12 days of Christmas is an example of a triangular series.

    mark
  • edited January 16

    My favorite formula is pretty simple, however, it has stuck with me all these years. It is Euler's formula:

    %%V-E+F=2%%

    Essentially, if you draw a connected planar graph, the number of dots, or vertices (V), minus the number of lines, or edges (E), plus the number of faces (F) created by the lines will always be equal to two! This is a pretty important equation in graph theory and various fields.

    In the connected planar graph pictured above, there are 7 vertices, 11 edges, and 6 faces and when plugged into the equation above gives you an answer of two.

    mark
  • edited January 16

    My favorite formula is the Law of Cosines.

    $$ a^2 = b^2 + c^2 - 2ab\cos(A)$$

    This equation is useful when you know two side lengths of a triangle and need to find the 3rd side length. It also can be used to calculate one of the angles within a triangle when only side lengths are available.

    mark
  • edited January 16

    My favorite formula is the one that describes Hubble's Law.

    %%v = H times d%%

    Where v is the velocity of the galaxy, H is Hubble's Constant, and d is the distance of the galaxy.

    This equation is used to show how fast galaxies are travelling away from us, and shows that the more distant a galaxy is, the faster it appears to move away from us than closer galaxies.

    This image shows the plot of the apparent velocity of a galaxy compared to it's distance, and the line that travels through all these data points is Hubble's Constant.

    mark
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