Euler's Number

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Euler's Number

Post by Jason » Thu Oct 04, 2018 10:57 pm

What is it good for? :ask:
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Re: Euler's Number

Post by pErvinalia » Thu Oct 04, 2018 11:48 pm

Stuff.
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Re: Euler's Number

Post by Jason » Fri Oct 05, 2018 12:01 am

Can you use it to calculate trig ratios or something? :think:
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Re: Euler's Number

Post by pErvinalia » Fri Oct 05, 2018 12:03 am

Can't remember. I remember using it for natural logarithms at school, but not sure what natural logarithm is good for. :dunno:
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Re: Euler's Number

Post by JimC » Fri Oct 05, 2018 12:14 am

Wiki article: https://en.wikipedia.org/wiki/E_(mathematical_constant)

From the article:
The number e is a mathematical constant that is the base of the natural logarithm: the unique number whose natural logarithm is equal to one. It is approximately equal to 2.71828,[1] and is the limit of (1 + 1/n)n as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series
The number e is of eminent importance in mathematics,[6] alongside 0, 1, π and i. All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler's identity.
It also has important applications in the areas of probability theory, calculus, and the standard normal distribution, a key principle in statistics.
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Re: Euler's Number

Post by pErvinalia » Fri Oct 05, 2018 12:21 am

It's a tool of the patriarchy!
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Re: Euler's Number

Post by Jason » Fri Oct 05, 2018 1:21 am

Hmm ok. So how the hell do you calculate trig ratios then?

I mean I've seen tables of them, but how are they derived?
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Re: Euler's Number

Post by pErvinalia » Fri Oct 05, 2018 1:55 am

SOHCAHTOA?
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Re: Euler's Number

Post by Jason » Fri Oct 05, 2018 2:19 am

I mean the tables that give the ratios. How are they calculateD?
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Re: Euler's Number

Post by JimC » Fri Oct 05, 2018 3:56 am

From Wiki:
Modern computers and calculators use a variety of techniques to provide trigonometric function values on demand for arbitrary angles (Kantabutra, 1996). One common method, especially on higher-end processors with floating-point units, is to combine a polynomial or rational approximation (such as Chebyshev approximation, best uniform approximation, and Padé approximation, and typically for higher or variable precisions, Taylor and Laurent series) with range reduction and a table lookup — they first look up the closest angle in a small table, and then use the polynomial to compute the correction. Maintaining precision while performing such interpolation is nontrivial, however; and methods like Gal's accurate tables, Cody and Waite reduction, and Payne and Hanek reduction algorithms can be used for this purpose. On simpler devices that lack a hardware multiplier, there is an algorithm called CORDIC (as well as related techniques) that is more efficient, since it uses only shifts and additions. All of these methods are commonly implemented in hardware for performance reasons.

The particular polynomial used to approximate a trig function is generated ahead of time using some approximation of a minimax approximation algorithm.

For very high precision calculations, when series-expansion convergence becomes too slow, trigonometric functions can be approximated by the arithmetic-geometric mean, which itself approximates the trigonometric function by the (complex) elliptic integral (Brent, 1976).

Trigonometric functions of angles that are rational multiples of 2π are algebraic numbers. The values for a/b·2π can be found by applying de Moivre's identity for n = a to a bth root of unity, which is also a root of the polynomial xb - 1 in the complex plane. For example, the cosine and sine of 2π ⋅ 5/37 are the real and imaginary parts, respectively, of the 5th power of the 37th root of unity cos(2π/37) + sin(2π/37)i, which is a root of the degree-37 polynomial x37 − 1. For this case, a root-finding algorithm such as Newton's method is much simpler than the arithmetic-geometric mean algorithms above while converging at a similar asymptotic rate. The latter algorithms are required for transcendental trigonometric constants, however.
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Re: Euler's Number

Post by Jason » Fri Oct 05, 2018 4:47 am

I'm afraid I don't understand you Jim. It seems that we're dealing with approximations instead of a simple formula. Could you explain?
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Re: Euler's Number

Post by JimC » Fri Oct 05, 2018 6:11 am

Śiva wrote:
Fri Oct 05, 2018 4:47 am
I'm afraid I don't understand you Jim. It seems that we're dealing with approximations instead of a simple formula. Could you explain?
If you want trigonometric ratios as decimals, then other than a handful such as sin 30 = 0.5, the answers are irrational numbers, so are always approximations, to whatever level of accuracy you want. Essentially, they are derived from summing infinite converging series, to whatever number of decimal places may be required in any given application (such summing is easily done by computers/calculators). Many, but not all trig ratios have exact values expressed in surd form, but the trig values of some degree values are even stranger, and have no value which can be expressed as algebraic numbers containing surds. Trigonometry at grade school level, with right angled triangles is quite straightforward, but if you enter it fully, the whole strangeness of pure mathematics begins to unfold... :levi:
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Re: Euler's Number

Post by Svartalf » Fri Oct 05, 2018 8:41 am

JimC wrote:
Fri Oct 05, 2018 12:14 am
Wiki article: https://en.wikipedia.org/wiki/E_(mathematical_constant)

From the article:
The number e is a mathematical constant that is the base of the natural logarithm: the unique number whose natural logarithm is equal to one. It is approximately equal to 2.71828,[1] and is the limit of (1 + 1/n)n as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series
The number e is of eminent importance in mathematics,[6] alongside 0, 1, π and i. All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler's identity.
It also has important applications in the areas of probability theory, calculus, and the standard normal distribution, a key principle in statistics.
Just your simple explanation was enough to give me a headache... I'm really not good for advanced maths at all.
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Re: Euler's Number

Post by Jason » Fri Oct 05, 2018 11:57 am

JimC wrote:
Fri Oct 05, 2018 6:11 am
Śiva wrote:
Fri Oct 05, 2018 4:47 am
I'm afraid I don't understand you Jim. It seems that we're dealing with approximations instead of a simple formula. Could you explain?
If you want trigonometric ratios as decimals, then other than a handful such as sin 30 = 0.5, the answers are irrational numbers, so are always approximations, to whatever level of accuracy you want. Essentially, they are derived from summing infinite converging series, to whatever number of decimal places may be required in any given application (such summing is easily done by computers/calculators). Many, but not all trig ratios have exact values expressed in surd form, but the trig values of some degree values are even stranger, and have no value which can be expressed as algebraic numbers containing surds. Trigonometry at grade school level, with right angled triangles is quite straightforward, but if you enter it fully, the whole strangeness of pure mathematics begins to unfold... :levi:
Thanks Jim! :D
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Re: Euler's Number

Post by cronus » Fri Oct 05, 2018 12:35 pm

Like anything mathematical it's helpful so long as you don't put it where it don't belong.
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