Hearted Youtube comments on Mathologer (@Mathologer) channel.
-
98
-
98
-
98
-
97
-
97
-
97
-
97
-
96
-
96
-
95
-
95
-
94
-
94
-
94
-
93
-
92
-
91
-
91
-
91
-
91
-
89
-
88
-
88
-
87
-
86
-
86
-
86
-
85
-
85
-
85
-
84
-
84
-
84
-
84
-
84
-
84
-
83
-
83
-
82
-
82
-
82
-
81
-
For the demonstration of A(X) + A(Y) = A(XY)
Definitions :
- we define A’(x,y) as the area under the curve between x and y (we notice that A’(1,y) = A(y) and that A’(a,b) + A’(b,c) = A’(a,c))
- we define f_p(z) the transformation of an area z by a factor p
Let’s find an expression for f_p:
Given z=A’(a,b), the top left hand corner of the area is mapped to:
1/p * 1/a = 1/(ap) (squeezing)
which corresponds to the inverse function for a value ap (shifting). Same goes for the top right hand corner.
We can conclude that:
f_p(A’(a,b)) = A’(pa,pb) = A’(a,b)
Let now prove the theorem :
A(x) + A(y) = A’(1,x) + A’(1,y)
= A’(1,x) + f_x(A’(1,y))
= A’(1,x) + A’(x,xy)
= A’(1,xy)
= A(xy)
81
-
81
-
81
-
81
-
81
-
80
-
80
-
80