Zero

 

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Is zero transcendental?

In mathematics, a transcendental number is a real or complex number that is not algebraic – that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best-known transcendental numbers are π and e.

https://en.wikipedia.org/wiki/Transcendental_number

While π and e might be the best known transcendental numbers, it is noted that hyperbolic trigonometric functions, which are expressed as functions of powers of e,  which is itself a transcendental number, are transcendental functions.  These hyperbolic trigonometric functions are also periodic in i*π,  i.e. repeat on the imaginary axis with a frequency of multiples of π[1], another transcendental number.   

Euler’s formula, e^ix=cos(x)* sin(x)*i, is also the coordinate transformation of a complex number from polar coordinates to rectangular, Cartesian, coordinates with a real and an imaginary axis, where the polar radius is 1.   It includes the rotation of the imaginary axis by an angle of x.  Sin(π)=0 means that the coefficient of the imaginary axis is zero.  Does that mean that there is NO imaginary axis?  That depends on whether that zero is absolute or relative. 

Absolute zero is the absence of the absolute, i.e. a temperature of absolute zero means that there is an absence of temperature.  By contrast, zero on the Centigrade scale does not mean that there is no temperature, just that the temperature is relative to a zero point on the scale, which in the case of the Centigrade scale is the freezing point of water.  IOW, negative numbers are allowed on a relative scale.

Transcendental functions are not expressible as a finite combination of the algebraic operations of addition, subtraction, multiplication, division, raising to a power, and extracting a root. Examples include the functions sin(x), cos(x), cosh(x), sinh(x), e^x, ln(x), etc., and any functions containing them. Special transcendental functions are the reflection of sin(x), which is sin(‑x) and is equal to sin(x) and the reflection of cosh(x), which is cosh(-x) and is also identical to cosh(x). This is not true of cos(x), tan(x), sinh(x), tanh(x), e^x, ln(x) and other transcendental functions.  They are not self-reflective, reflections of themselves.

Because cos(x) is transcendental but not self-reflective this means that Euler’s formula is also transcendental but not self-reflective. In that equation x can take on any value between -∞ and ∞.  Because Euler’s Formula is a combination of a self-reflective sin(x) and a non-self-reflective cos(x), but trigonometric functions are repeating, there are many solutions with a zero coefficient of the imaginary axis, i.e. a rotation of the imaginary axis of zero: even multiples of π, including zero, which have values of cos =1 as the coefficient of the real axis; and odd multiples of π which have values of cos=  -1 as the coefficient of the real axis. Since in this case zero is not merely the absence of an absolute, and is in fact multiple numbers, it must be a relative zero, not an absolute zero. An absolute zero is the absence of a transcendental .  A relative zero is not.

So is zero transcendental?  That depends on if that zero is absolute or relative.

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[1] Cosh, sinh, and related functions, repeat with a period of 2*π*i.  Tanh and related functions repeat with a period of π*i.