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MathematicalPhysicist
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do those constants have any relation to each other?
does something like pi-e or pi/e has any significance?
does something like pi-e or pi/e has any significance?
r=[oo]
[pi] = 4 * [sum] ((-1)^r) = 4 - 4 + 4 - 4 + 4
r=1 (------) - - - - ... etc.
( 2r-1 ) 3 5 7 9
And
r=[oo]
e = [sum] ( 1 ) = 1 + 1 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1
r=1 (------) -- -- -- -- -- ... etc. - - -- ... etc.
((r-1)!) 0! 1! 2! 3! 4! 2 6 24
A few years back I took Complex Analysis from Dr. King, then Chairman of the Lehigh U Math Department. He spent a fair amount of time with this relationship. He preferred to write itOriginally posted by mathman
e(pi)i=-1
i forgot about this equation.Originally posted by mathman
e(pi)i=-1
the condition for the summations in both cases is the same, ie r=infinity r=1.Originally posted by lavalamp
If it's any help these are the power series for [pi] and e:
Code:r=[oo] [pi] = 4 * [sum] ((-1)^r) = 4 - 4 + 4 - 4 + 4 r=1 (------) - - - - ... etc. ( 2r-1 ) 3 5 7 9 And r=[oo] e = [sum] ( 1 ) = 1 + 1 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 r=1 (------) -- -- -- -- -- ... etc. - - -- ... etc. ((r-1)!) 0! 1! 2! 3! 4! 2 6 24
[pi] can also be obtained like this:
x * Sin (180/x) where x is a very large number and 180/x is in degrees.
I've attached a script to calculate pi and e using the above power series', however I have not been able to calculate pi using the Sin method as JavaScript assumes that the angle is measured in radians and it does not have a built in Math.pi method to allow me to convert the angle from radians into degrees.
Be careful if you are calculating pi to 1,000,000 iterations, I have an Athlon 1800+ and it caused my PC to hang for a couple of seconds, although I was listening to music at the time.
If you want to view the source, generally in Windows browsers, you can go View > Source.
And I put that, what do you think this is:Originally posted by loop quantum gravity
the condition for the summations in both cases is the same, ie r=infinity r=1.
r=[oo]
e = [sum]
r=1
+---+--+
| | |
| | |
| | |
+---+--+
is there any reason why this condition applies in both of them?Originally posted by lavalamp
And I put that, what do you think this is:
It's just that if I were to make a script that would run forever you'd never get an answer so what would the point of it be?Code:r=[oo] e = [sum] r=1
Anyway I've re-posted the script if anyone's interested, it includes the (1 + (1/k))^k way to calculate e.
By the way, does anyone know the formula for finding the decimal places of [pi]? I have heard of a formula that when you put in a number (say n, for the nth decimal place), you get an answer. I assume there is one for e as well, so does anyone have that?
another way to write this (which i hope no one has yet written it) is:Originally posted by mathman
e(pi)i=-1
Originally posted by loop quantum gravity
another way to write this (which i hope no one has yet written it) is:
e^(i*pi)=-1
e^[(i*pi)/2]=-1^0.5
e^[(i*pi)/2]=i
Originally posted by loop quantum gravity
i forgot about this equation.
any significance to it?
Originally posted by lavalamp
If you would like I can post how it is possible to arrive at that solution (by that solution, I mean this - e^(i[pi])+1=0).
It uses the power series of e^x, but replaces x with i[pi], and you wind up with the power series for cos and sin, then when substituting in [pi], you get the equation mentioned above.
Originally posted by synergy
phi+phi^2 which is phi^3
Originally posted by jcsd
sin x = (eix - e-ix)/2i
cos x = (eix + e-ix)/2
The constants E, pi, and phi are all mathematical constants that are related to circles and curves. E, also known as Euler's number, is the base of the natural logarithm and is approximately equal to 2.718. Pi, denoted by the symbol π, is the ratio of a circle's circumference to its diameter and is approximately equal to 3.14159. Phi, also known as the golden ratio, is a mathematical constant that appears in nature and is approximately equal to 1.618.
E, pi, and phi are used in a variety of mathematical and scientific applications. E is commonly used in calculus and exponential functions. Pi is used in geometry, trigonometry, and physics to calculate the circumference, area, and volume of circles and spheres. Phi is used in art, architecture, and design to create aesthetically pleasing proportions and patterns.
Yes, there are many examples of the relationship between E, pi, and phi in the natural world. For instance, the spiral patterns found in seashells and galaxies follow the golden ratio. The human body also exhibits the golden ratio in the proportions of its limbs and facial features. Additionally, pi is used in the design of circular structures such as wheels and gears.
E, pi, and phi are significant in mathematics because they are fundamental constants that appear in many mathematical equations and formulas. They are also important in understanding the properties of circles, curves, and exponential growth. Furthermore, the relationship between these constants has been studied and appreciated by mathematicians and scientists for centuries.
The relationship between E, pi, and phi can be visualized in various ways. One common visualization is the Fibonacci spiral, which is created by connecting arcs with radii based on the Fibonacci sequence, a series of numbers related to the golden ratio. Another visualization is the unit circle, which shows the relationship between E and pi by representing the values of sine and cosine on the unit circle. There are also many other geometric constructions and diagrams that can illustrate the relationship between these constants.