The sometimes 'Beauty' of mathematics??? - Page 1
Forum section: electronics / general

author: GlennSprigg
author_rank: Super Contributor
timestamp: August 04, 2021, 01:48:17 pm
content: I don't know how many people have seen my 'Tag' (?) thing, at the bottom of my posts/replies?Diagonal of 1x1 square = Root-2. Ok.Diagonal of 1x1x1 cube = Root-3 !!! Beautiful !!Doesn't that not strike people as mathematically amazing! that the internal diagonal of a 'Cube' (1x1x1) =Root-3!!!I love the natural relationships with certain numbers, that can easily be proven/shown, and be so simple!GREAT!! Good-ol basic maths can show/prove it all !!!...Then I found out recently, that there isNO KNOWNcorrect formula, to calculate theCircumferenceof an 'Ellipse'!!You don't believe it... then look it up!! Sigh... Just lost faith in Maths again...

author: TimFox
author_rank: Super Contributor
timestamp: August 04, 2021, 01:55:46 pm
content: The "elliptic integrals" that appear when calculating the circumference of an ellipse are not "elementary functions" such as combinations of sines and cosines, but are tabulated functions that can be found in math-table books, entered into the books from numerical computations. There are other non-elementary functions that occur in science and engineering, such as Bessel functions, that are solutions of practical math problems and are also tabulated in such books.That being said, in school where a problem could be reduced to elliptic integrals, we left it at that. One such problem I remember (senior undergraduate classical mechanics) was how long does it take for a car door to slam shut if it is open at the start of the car's acceleration.

author: tggzzz
author_rank: Super Contributor
timestamp: August 04, 2021, 02:11:00 pm
content: Euler's identity shows how five fundamental numbers are related in a single equation: \( e^{i \pi} + 1 = 0 \)

author: mawyatt
author_rank: Super Contributor
timestamp: August 04, 2021, 02:16:08 pm
content: If you like this sort of thing, then consider a string wrapped around the earth's equator (assuming the earth as a perfect sphere) and tied off at the ends. Now raise the string above the surface by some small amount all the way around, say 1 meter. How much longer is the string when raised and tied at the ends? Do same for the Moon, Sun, Jupiter, the solar system and the entire universe assuming all are spheres. Then redo for a beachball, basketball, baseball, pingpong ball, solder ball and an atom (assume all are spheres). Now compare the results, amazing indeedIf you like mathematics in electronics, look into analog filters and how the transfer functions of Butterworth, Bessel, Cauer, Chebyshev, Gauss, & Legendre interact in the frequency and time domains. Fun stuff indeed for those interestedBest,Edit: Signal processing, both analog and digital, is another fascinating mathematical area

author: daqq
author_rank: Super Contributor
timestamp: August 04, 2021, 02:19:04 pm
content: For delightful visualisations of mathematical concepts see:https://www.youtube.com/c/3blue1brown/videoshttps://www.youtube.com/c/PrimerLearning/videos

author: Brumby
author_rank: Unknown
timestamp: August 04, 2021, 02:30:58 pm
content: Quote from: mawyatt on August 04, 2021, 02:16:08 pmThat constraint is more limiting than the problem requires. You simply need to assume the earth's equator is a perfect circle. The same applies to the other bodies.The question, however, is a simple one - and some very basic algebra will yield the answer.

author: mawyatt
author_rank: Super Contributor
timestamp: August 04, 2021, 02:51:19 pm
content: Quote from: Brumby on August 04, 2021, 02:30:58 pmYes, but 3D bodies seem more useful in visualizationAnd yes the answer is very simple, but astoundingBest,

author: magic
author_rank: Super Contributor
timestamp: August 04, 2021, 02:57:29 pm
content: I have heard a different version:Imagine a circle lying on the Earth's equator.Then increase its circumference by 1 meter.Is it enough for a mouse to pass under it?

author: TimFox
author_rank: Super Contributor
timestamp: August 04, 2021, 03:38:38 pm
content: Another story from my school days, roughly 50 years ago. The senior (and very distinguished) professor in the mathematics department “crossed the aisle” to the physics department colloquium to tell us about recent work in mathematics that might be of interest to us, including work on generalizing the concept of “function”.He started with an anecdote about his technical work during WW II, where his security clearance allowed him to read classified technical reports. One report computed a function relevant to aircraft design and tabulated it asy=f(x). The tabulated values were all close to 1. He and a buddy went back to the original equations and found they could be reduced tof(x) = sin2(x) + cos2(x).This is an example of mathematical beauty: even though you need recourse to tabulated values for trig functions of arbitrary values, it is straightforward to prove relations between them (what we call “trig identities”) with (in this case) Euclidean geometry. Draw a circle with radius equal to the hypotenuse of a right triangle and apply the Pythagorean theorem to the other two sides that represent sine and cosine.

author: armandine2
author_rank: Frequent Contributor
timestamp: August 04, 2021, 05:38:40 pm
content: sometimes the numbers seem to be well chosen - almost like someone wastrying to get you to remember the formula!

author: rstofer
author_rank: Super Contributor
timestamp: August 04, 2021, 06:21:15 pm
content: Let's say a person has an BS is Applied Mathematics and an MS in Mathematics. Where do they find a job? What kind of industries hire theoretical mathematicians?

author: TimFox
author_rank: Super Contributor
timestamp: August 04, 2021, 06:23:12 pm
content: Quote from: armandine2 on August 04, 2021, 05:38:40 pmIn primary-grade arithmetic (say, fourth grade), the numbers in the homework problems always "came out even", so that the clever students knew that it was probably wrong when the process did not come out even.

author: mawyatt
author_rank: Super Contributor
timestamp: August 04, 2021, 07:02:07 pm
content: Quote from: rstofer on August 04, 2021, 06:21:15 pmWe had a whole staff of mathematicians including my colleagues wife a Fulbright Scholar, she studied at a French university (I can't remember the details. They were mostly working on Kalman filters for SOTA Inertial Nav Systems based upon ESG and RLG Gyros, but this was ~45 years ago. Also had a colleague professor when I was an adjunct that studied under Nash at Princeton, he was brilliant and authored a grad level book on Complex Variables!! However, today I'm not sure where one would look outside a university for work specifically in advanced mathematics.Best,

author: mawyatt
author_rank: Super Contributor
timestamp: August 04, 2021, 07:09:32 pm
content: Quote from: TimFox on August 04, 2021, 06:23:12 pmRecall from undergrad calculus a test problem with integration by parts, you kept coming up with the same integral over and over that you started with, and another term. The clever student, not merealized the answer was the other term divided by 2Best,

author: TimFox
author_rank: Super Contributor
timestamp: August 04, 2021, 07:14:30 pm
content: I’m retired, and not an authority on the current job market, but at my employer we had high-ranking employees with advanced degrees in mathematics working on image processing and similar topics.

author: daqq
author_rank: Super Contributor
timestamp: August 04, 2021, 07:34:11 pm
content: Quote from: rstofer on August 04, 2021, 06:21:15 pmIt depends - if you are actually good there's any number of serious engineering companies, whether they are software, hardware, civil engineering... There are real world applications even (or maybe especially) for the really advanced stuff.If you are not particularly good, no idea.

author: rstofer
author_rank: Super Contributor
timestamp: August 04, 2021, 08:55:41 pm
content: Quote from: TimFox on August 04, 2021, 07:14:30 pmI would think the hot topics are cryptology or machine learning and other branches of AI. I know Google is into AI bigtime and so is Microsoft plus, perhaps, government agencies. In addition to math skills, I suspect programming skills are absolutely required. MATLAB, Octave, wxMaxima, Python (and all of the libraries, especially AI libraries) and so on. I'm really old so I would throw in a dash of Fortran as well.But I'm just guessing...

author: TimFox
author_rank: Super Contributor
timestamp: August 04, 2021, 09:21:32 pm
content: By the time I retired, the younger employees doing the analytical work mainly used Matlab, but I found it easier to refer the relevant problems to them rather than learn it myself.The nice thing about Matlab is that my employer could purchase a license (for a quite reasonable price) that allowed our guys to develop "stand-alone" applications that could be used internally by the non-Matlab folks and even sold to our customers.

author: EPAIII
author_rank: Super Contributor
timestamp: August 05, 2021, 09:23:15 am
content: You would be surprised.Start with the folks who drill for oil and gas.Quote from: rstofer on August 04, 2021, 06:21:15 pm

author: Brumby
author_rank: Unknown
timestamp: August 05, 2021, 09:29:26 am
content: Quote from: magic on August 04, 2021, 02:57:29 pmI like that variation.

author: rstofer
author_rank: Super Contributor
timestamp: August 05, 2021, 03:08:36 pm
content: I like "Gabriel's Horn" better. Consider the curve 1/x for x between 1 and infinity - it's a really long horn...Compute the volume of revolution and you will find that it takes PI units of paint to fill it completely.Then compute the surface area and you will find that it takes an infinite amount of paint to cover the interior surface.That's just weird!Hints:Volume = integral from 1 to infinity of pi * radius squared dx but r = 1/x so integral of pi/x2dx = piSurface Area = integral from 1 to infinity of 2 * pi * r which is the integral of 2 * pi / x dx. Remember, integral of 1/x = ln(x) and the result diverges (runs off to infinity)

author: magic
author_rank: Super Contributor
timestamp: August 05, 2021, 09:06:20 pm
content: You may run out of paint on the exterior, but the interior cannot be painted with a layer of any thickness because most of it is too narrowIf you want to see some seriously fucked up consequences of playing with all that "infinity" stuff, look up the Banach-Tarski paradox(not going to pretend I know the proof of that)

author: TimFox
author_rank: Super Contributor
timestamp: August 05, 2021, 10:13:22 pm
content: Another similar paradox applied to natural objects is to compute the coastline of Norway. Obviously, the area of Norway (with mountains projected back onto the globe) is finite, since you can draw a closed curve around it, of finite area, that encloses it. If you do the actual experiment of computing the physical coastline as you go to finer and finer map scales, the value (in meters) increases without bound. (Please don't quibble about atomic dimensions. This is math.)An ideal geometric figure that does the same: the Koch snowflake, with finite area but infinite perimeter.Start with an equilateral triangle, of side 1. Total perimeter = 3.Iteration 1, on each of the three sides of that triangle erect another equilateral triangle whose side is 1/3 of the first. This gives you a Star of David hexagram. Total perimeter = 4Iteration 2, on each line segment (12 for this step) of that figure erect another triangle of length 1/3.etc.There exists a circle that encloses all of the resulting figures, but the perimeter increases without bound. The length afterNiterations is 3 x (4/3)NThe asymptotic area (after an infinite number of iterations) is (8/5) x the area of the original triangle.(I read this discussion many years ago in Mandelbrot's popular introduction to fractal geometry.)

author: rstofer
author_rank: Super Contributor
timestamp: August 05, 2021, 10:24:54 pm
content: Quote from: magic on August 05, 2021, 09:06:20 pmIf I can have quantifiably small internal diameters, I should be able to have really tiny paint molecules.My math skills are too weak for either of the last two paradoxes. I just try to balance my checkbook and that isn't easy as it sometimes requires imaginary numbers.

author: TimFox
author_rank: Super Contributor
timestamp: August 05, 2021, 10:33:51 pm
content: The interiorsurface areamight be infinite, but thevolumeof paint required to paint the interior is finite due to decreasing clearance, even if the thickness goes to subatomic dimensions. The volume of paint must be less than the volume of the interior (equal to pi according to the post, I haven't checked), which must be less than an external volume that contains the horn.Paint at Home Depot is sold by volume.

author: The Electrician
author_rank: Frequent Contributor
timestamp: August 05, 2021, 11:17:07 pm
content: Quote from: GlennSprigg on August 04, 2021, 01:48:17 pmOf course there is a correct formula. The relevant observation is found on this page:http://www.numericana.com/answer/ellipse.htm#elliptic"There is no simple exact formula: There are simple formulas but they are not exact, and there are exact formulas but they are not simple."Also here:https://wj32.org/wp/2012/12/15/formula-for-the-circumference-of-an-ellipse/

author: Circlotron
author_rank: Super Contributor
timestamp: August 05, 2021, 11:21:52 pm
content: The one I like is phi, 1.6180339.....In particularphi^2=phi+11/phi=phi-1

author: vk6zgo
author_rank: Super Contributor
timestamp: August 05, 2021, 11:56:40 pm
content: Quote from: rstofer on August 04, 2021, 06:21:15 pmQ: What does a maths graduate say?A: "You want fries with that?"

author: vk6zgo
author_rank: Super Contributor
timestamp: August 06, 2021, 12:04:00 am
content: Quote from: GlennSprigg on August 04, 2021, 01:48:17 pm"Is this not strange & wonderful?"Quote

author: free_electron
author_rank: Super Contributor
timestamp: August 06, 2021, 12:49:49 am
content: 3x+1

author: RJSV
author_rank: Super Contributor
timestamp: August 06, 2021, 05:06:56 am
content: First year, college CALCULUS:That instructor...Well, he had done some 'colorfull' career stuff previous (to teaching). Yeah, he was a PRO race-car driver, in national competition.Had an usual zest for living, and a very dynamic speaking style, presenting thinking theory behind the Calculus. The 'Infinitesimal' : what it is, and why useit ?Anyway, oh and one other thing: The professionalracing had cost him a LEG so there was that. Us students never asked about that, directly.===========================================After writing lots of CODE and using binary / hex number notations, I have noticed some coincidences in the number translations.For example decimal '64' is a kind of top end, when counting using 6 bits, and if you reverse, to 64hex, that translates to decimal '100'.For quick 'Rule of Thumb', I use a casual rule of tens;Meaning that with 10 bits, you have 1k.With 20 bits: you get 1 Meg,With 30 bits, you've got 1 giga (roughly).Makes for quick estimating.

author: tggzzz
author_rank: Super Contributor
timestamp: August 06, 2021, 09:09:08 am
content: Quote from: RJHayward on August 06, 2021, 05:06:56 amEverybody at my state school did calculus for external exams at 16yo. Great fun. differentiation and integration of polynomials except1/x

author: GlennSprigg
author_rank: Super Contributor
timestamp: August 06, 2021, 11:39:48 am
content: O.P. here... MY simplistic 'brain', at thecurrentmoment anyway...), tried to solve this by Reverse thinking!!!Or at least from a totally different 'Angle'... (excuse the pun, butNOTactually a 'pun')Consider a perfect circle. Ok, the calculation of it's 'circumference' is OBVIOUSLYC = 2 Pi r.Now rotate that circle 90-deg until it 'appears' as a straight line!! Now the APPARENT (from the viewers perspective!), is just2 x DSo to ME!, all the 'InBetween' apparent 'Circumferences,from one's view point!, while observing such an apparent Circumference, MUSTsimply be not only between those 2 above limits, but MUST be directly related mathematically as a result of thatspecific anglewe are viewingit from? 'One' needs to let go of thoughts about a 'physical' Ellipse per-se, and one that is 'apparent' from the angle we are viewing it!?Surely then, the actual 'Maths' is simple!! Without those complex theorems/styles being applied?, or 'trying' to apply them ??

author: RJSV
author_rank: Super Contributor
timestamp: August 06, 2021, 07:18:00 pm
content: But but but... Hey folks, don't keep me in suspense:Increase circumference by 1 meter:π X D = CThen what is C + 1 ?(π X D) + 1 = C + 1Solve for new D value, (don't have immediate clue)...

author: RJSV
author_rank: Super Contributor
timestamp: August 06, 2021, 07:26:50 pm
content: (continued).... Getting schetchy here, but my inclination is to set up, maybe TWO 'simultaneous' equations. Having a diameter D1 and a later diameter D2.You already have C1 and C2 (C2 is just 1 meter larger).At this point, realize that we need, one of those MATH types, here.

author: mathsquid
author_rank: Frequent Contributor
timestamp: August 08, 2021, 03:31:54 pm
content: Let \$r\$ be the radius of the earth. Put a rope around the earth at the equator to get a circle. Its circumference is \$2\pi r\$. Now take another rope that is one meter longer, and use it to make a circle that is \$a\$ meters above the earth.Since the longer rope is one meter longer, the circumference of the bigger circle is \$ 2 \pi r + 1\$.The radius of the bigger circle is \$r + a\$, thus the circumference of the bigger circle is \$2\pi (r+a) \$Together we have$$2 \pi r + 1 = 2\pi (r+a)$$$$2 \pi r + 1 = 2\pi r + 2\pi a$$$$ 1 = 2\pi a$$$$ a = \frac{1}{2 \pi}$$

author: mawyatt
author_rank: Super Contributor
timestamp: August 08, 2021, 04:22:19 pm
content: Using a little elementary calculus.C = 2piRdC = 2pi*(dR)delta C = 2pi*(deltaR)So the change in circumference wrt the change in radius isindependentof the radius!!!Thus the comment, about using the Earth, Moon, Jupiter, Solar System and Universe on one extreme, and a beachball, basketball, baseball, pingpong ball and atom on the other.Best,

author: basinstreetdesign
author_rank: Frequent Contributor
timestamp: August 09, 2021, 03:40:12 am
content: Quote from: The Electrician on August 05, 2021, 11:17:07 pmI'm surprised that no one has mentioned that a simple extension of your sequence is that the "diagonal" (space-diagonal) of a 1 x 1 x 1 x 1 hypercube or tesseract is root(4) = 2

author: RJSV
author_rank: Super Contributor
timestamp: August 09, 2021, 04:05:42 am
content: Magic and Mathsquid:Just so I can sleep tonight:. 1/ 2 * pintimetersSo that mouse has easy time, right ? That's around 6 inches of clearance, right ? Uhh 20 centimeters ?

author: magic
author_rank: Super Contributor
timestamp: August 09, 2021, 06:42:27 am
content: Yep. Even a cat would pass

author: TimFox
author_rank: Super Contributor
timestamp: August 09, 2021, 01:41:02 pm
content: You would be surprised how small a hole a standard household mouse can fit through.

author: mathsquid
author_rank: Frequent Contributor
timestamp: August 09, 2021, 09:04:04 pm
content: Yes. The rope is \$\frac{1}{2\pi}\$ meters above the ground because the additional length was 1 meter.

author: RJSV
author_rank: Super Contributor
timestamp: August 10, 2021, 01:12:02 am
content: Thanks, phew that ALGEBRA is hard, for me. Seriously, I've come to realize: College level Calculus got built on foundation of 'crappy' high school Algebra.Heck, I've heard Einstein lamented, "...couldn't factor a polynomial, to save my life!..."Oh, and those smaller four legged 'Rodents' can run straight up a slightly roughened exterior wall.

author: CatalinaWOW
author_rank: Super Contributor
timestamp: August 10, 2021, 02:38:26 am
content: Sometimes the beauty comes when an (almost) totally non-math explanation is available. The following is apparently related to a problem in topology, but has practical applications, and has delighted me ever since I came across it.It is the proof that you can place a square table on any uneven floor that meets a couple of simple criteria. The simple criteria: 1. No discontinuities in the floor. 2. Irregularities have to be a fair amount smaller than the length of the legs. 3. The legs are equal length.The proof. First, by inspection, you can get three legs to touch by putting two in contact with the floor and rotating the table about the line between the contact points until another leg touches.Now the pretty part. Since the table is symmetric for 90 degree rotations about its center there are three other rotational positions where three legs touch the same three points. But they aren't the same three legs. And if you keep the legs in contact with the surface as you rotate between the four positions there will be a point at which the current three legs transition to the next three. At that point four legs are touching the ground.

author: AntiProtonBoy
author_rank: Frequent Contributor
timestamp: August 10, 2021, 02:38:59 am
content: One particular construct I find super elegant in mathematics is transformation matrices used in visualising geometry and inverse kinematics. These matrices are almost like magic. You can combine a sequence of transform operations into a single matrix and get the correct result in a single operation.

author: Circlotron
author_rank: Super Contributor
timestamp: August 10, 2021, 03:22:16 am
content: Can someone please explain how these cool mathematical principles came about as a consequence of a mindless universe beginning with a whopping big explosion? Or did they exist beforehand?

author: AntiProtonBoy
author_rank: Frequent Contributor
timestamp: August 10, 2021, 03:42:09 am
content: Most mathematical principles is a form of "abstract logic" (for a lack of a better term), which may not need be based on the properties of the universe. Of course the universe needs to exist for us to have the mind to think 1+1=2, but the math alone does not have to define anything about the universe.

author: GlennSprigg
author_rank: Super Contributor
timestamp: August 10, 2021, 01:18:26 pm
content: Quote from: basinstreetdesign on August 09, 2021, 03:40:12 amAh!! Yes... I've thought of that... and although 'logical', it is some-what impossible to 'imagine' such a 'diagonal'. Here is a Mental 'representation' ofaHyperCube/Tesseract, as it can be perceived in 3-Dimesions in our mind, before going on here...A 'square' has 4 right-angles. a 'Cube' has all right-angles too!, but we 'draw' one withdistortedangles to represent it in3-D. A 'HyperCube' consistsof technically aCube within a Cube, with lines joining the cubes too, and even THESE are at right-angles to everything else within! Of course, ourbrains can't fathom/picture that, so like with the 'cube' drawing, we can only imagine it as a3-D 'shadow, and seeing distorted angles, as above!We can't even IMAGINE where toplacesuch a 'diagonal', within such an elusive concept!!!P.S. I haven't finished with orig/prior discussions yet, though !!!

author: tggzzz
author_rank: Super Contributor
timestamp: August 10, 2021, 02:15:45 pm
content: At this point, Flatland: A Romance of Many Dimensions by George Abbott is relevant.

author: rstofer
author_rank: Super Contributor
timestamp: August 10, 2021, 04:18:42 pm
content: Quote from: RJHayward on August 10, 2021, 01:12:02 amOur local community college has a test of Pre-Calculus math skills before they will enroll a student in Calc I. If the student isn't sufficiently prepared, there is a year long Pre-Calculus program that will get them up to speed. But it adds a year to the graduation date.There's a truism about Calculus: "Calculus is easy, it's the Pre-Calc that will bury you" Hint: Don't take either semester of Pre-Calculus during the abbreviated summer session. Same work in much less time.In High School, they should teach Algebra, Geometry, Algebra II and Trigonometry as a standard curriculum. There should also be classes in Pre-Calculus and Calculus for those so inclined.There are many good resources for learning math and one of the best is Khan Academy.

author: rstofer
author_rank: Super Contributor
timestamp: August 10, 2021, 04:34:38 pm
content: Quote from: AntiProtonBoy on August 10, 2021, 02:38:59 amI have always like the matrix equation A*x=b or x = Ainv*b. For some reason, I prefer this approach to solving simultaneous linear equations.Here's a simple 6x6 matrix problem that I would not care to solve by hand.Code:% A football stadium has a capacity of 100,000 attendees% Ticket prices:% Student = $25% Alumni = $40% Faculty = $60% Public = $70% Veterans = $32% Guests = $ 0% A full capacity game netted $4,987,000% Let S = number of students, A = number of alumni, F = number of faculty% P = number of public, V = number of veterans, G = number of guests% S A F P V G = valueM = [ 1 1 1 1 1 1 100000; ... % total number of attendees25 40 60 70 32 0 4897000; ... % total revenue0 1 -1 0 0 0 11000; ... % 11000 more alumni than faculty0 1 0 1 -10 0 0; ... % public + alumni = 10 * veterans-1 1 1 0 0 0 0; ... % alumni + faculty = students1 0 1 0 -4 -4 0]; % faculty + students = 4 (guests + veterans)A = M(:,1:6); % extract 6x6 matrixB = M(:,7); % extract value column vectorP = M(2,1:6); % ticket price by attendee typeR = inv(A)*B; % compute number of attendees by typeT = sum(R); % check total attendees = 100,000 CHECKU = P*R; % total revenue = 4,897,000 CHECKV = R'.*P; % revenue by attendee type - transpose R from 6x1 to 1x6% then do element by element multiplication%% fancy output, all previous default output semicoloned%label = {'Students' 'Alumni' 'Faculty' 'Public' 'Veterans' 'Guests'};for i = 1:6fprintf('%-8s%7d @ %2d = %7d\n',string(label(i)),...round(R(i)),...M(2,i),...round(V(i)))endfprintf('\nTotals %6d %8d\n',round(T), round(U))Results:Code:Students 25000 @ 25 = 625000Alumni 18000 @ 40 = 720000Faculty 7000 @ 60 = 420000Public 42000 @ 70 = 2940000Veterans 6000 @ 32 = 192000Guests 2000 @ 0 = 0Totals 100000 4897000BTW, the code is written for MATLAB and works in Octave as well. Octave is free!

author: Ground_Loop
author_rank: Frequent Contributor
timestamp: August 10, 2021, 08:48:28 pm
content: Quote from: tggzzz on August 04, 2021, 02:11:00 pmThis one! Note though that this is an expression and not an equation. There are no variables.

author: TimFox
author_rank: Super Contributor
timestamp: August 10, 2021, 09:16:36 pm
content: No, an "equation" is an expression where the two sides are related by an "equals sign". 2 + 2 = 4 is another example of an equation.

author: Ground_Loop
author_rank: Frequent Contributor
timestamp: August 10, 2021, 09:39:52 pm
content: Quote from: TimFox on August 10, 2021, 09:16:36 pmYep, my mistake. I get hung up on people calling it a formula, which it's not.

author: TimFox
author_rank: Super Contributor
timestamp: August 10, 2021, 09:43:51 pm
content: Another type of expression is the "inequality", as in 2 < 3.If I remember correctly, the inventor of "=" said that "no two things could be more equal".