Non-algebraic numbers {transcendental number}| are roots of equations with trigonometric, inverse trigonometric, exponential, and logarithmic functions. Transcendental numbers can relate to circles, triangles, exponents, and logarithms, such as pi = 3.1415926535... and e = 2.7182818284... = Euler's number.
pi
pi is the limit of ratio between many-sided regular-polygon circumference and center-to-vertex line length.
e
e is base of expression e^-x, such that derivative of e^-x equals e^-x. Lower values make lower derivatives and higher make higher, so e is in middle. It also makes x^(1/x) maximum. It is the limit of (1 + 1/n)^n, when compound interest has many periods and interest is 1/n per period. Definite integral of (1/x) * dx from 1 to e equals 1.
e = 1/0! + 1/1! + 1/2! + 1/3! + ... = 1/1! + 2/2! + 3/3! + 4/4! + ...
e^a = 1 + a + a^2/2! + a^3/3! + ...
-1/(e^pi) = 1 + i - 1/2 - i/6 + 1/24 + i/120 + ...
-1/(e^pi) - 1/2 + 1/24 - 1/720 + ... = i*(1 - 1/6 + 1/120 + ...).
i = (-1/(e^pi) - 1/2 + 1/24 - 1/720 + ...)/(1 - 1/6 + 1/120 + ...).
integral from -infinity to +infinity of e^-x^2 * dx = pi^0.5.
trigonometry
sin(a) = a - a^3/3! + a^5/5! - a^7/7! + ...
cos(a) = 1 - a^2/2! + a^4/4! - a^6/6! + ...
i
i = e^(i*pi/2).
ln(i) = i*pi/2
e^(i*a) = cos(a) + i*sin(a), where a is in radians and is real.
sin(a) = (e^i*a - e^-i*a)/2i
cos(a) = (e^i*a + e^-i*a)/2.
e^i = 1 + i - 1/2 - i/6 + 1/24 + i/120 + ...
cos(i) = 1 + 1/2 + 1/24 + 1/720 + ...
i = arccos(1 + 1/2 + 1/24 + 1/720 + ...).
-e^-pi = e^i
i = ln(-e^-pi).
e^-pi = -e^i
pi = -ln(-e^i) = - ln(-1) - ln(i).
ln(i) = i*pi/2 = i/2 * (- ln(-1) - ln(i)).
ln(i) = -i*ln(-1)/2 - i*ln(i)/2
i*ln(i) = ln(-1)/2 + ln(i)/2
2*i*ln(i) = ln(-1) + ln(i).
2*i*ln(i) - ln(i) = ln(-1).
ln(i) * (2*i - 1) = ln(-1).
2^0.5 is minimum of rectangle-diagonal-length to average-side-length ratio.
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Date Modified: 2022.0224