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| The sometimes 'Beauty' of mathematics??? |
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| rstofer:
--- Quote from: RJHayward on August 10, 2021, 01:12:02 am ---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. --- End quote --- Our 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. |
| rstofer:
--- Quote from: AntiProtonBoy on August 10, 2021, 02:38:59 am ---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. --- End quote --- I 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 = value M = [ 1 1 1 1 1 1 100000; ... % total number of attendees 25 40 60 70 32 0 4897000; ... % total revenue 0 1 -1 0 0 0 11000; ... % 11000 more alumni than faculty 0 1 0 1 -10 0 0; ... % public + alumni = 10 * veterans -1 1 1 0 0 0 0; ... % alumni + faculty = students 1 0 1 0 -4 -4 0]; % faculty + students = 4 (guests + veterans) A = M(:,1:6); % extract 6x6 matrix B = M(:,7); % extract value column vector P = M(2,1:6); % ticket price by attendee type R = inv(A)*B; % compute number of attendees by type T = sum(R); % check total attendees = 100,000 CHECK U = P*R; % total revenue = 4,897,000 CHECK V = 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 = 1:6 fprintf('%-8s%7d @ %2d = %7d\n',string(label(i)),... round(R(i)),... M(2,i),... round(V(i))) end fprintf('\nTotals %6d %8d\n',round(T), round(U)) --- End code --- [/font] Results: --- Code: ---Students 25000 @ 25 = 625000 Alumni 18000 @ 40 = 720000 Faculty 7000 @ 60 = 420000 Public 42000 @ 70 = 2940000 Veterans 6000 @ 32 = 192000 Guests 2000 @ 0 = 0 Totals 100000 4897000 --- End code --- [/font] BTW, the code is written for MATLAB and works in Octave as well. Octave is free! |
| Ground_Loop:
--- Quote from: tggzzz on August 04, 2021, 02:11:00 pm ---Euler's identity shows how five fundamental numbers are related in a single equation: \( e^{i \pi} + 1 = 0 \) --- End quote --- This one! Note though that this is an expression and not an equation. There are no variables. |
| TimFox:
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. |
| Ground_Loop:
--- Quote from: TimFox on August 10, 2021, 09:16:36 pm ---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. --- End quote --- Yep, my mistake. I get hung up on people calling it a formula, which it's not. |
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