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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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