General > General Technical Chat
The sometimes 'Beauty' of mathematics???
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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