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Thread: Maths!
- 03-18-2009 01:39 PM #1Look away
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Maths!
Complex numbers
(cos(x)+isin(x))^n=cos(nx)+isin(nx)
Now using maclaurine series
e^ix=1+x+x^2/2!+x^3/3!+....+x^r/r!+...
this can be used to that
e^ix=cos(x)+isin(x)
which means
e^inx=cos(nx)+isin(nx)
this is then used to show that when n=1 x=pi
e^(ipi)=-1
e^(ipi)+1=0
This result has been describe by many as the most beautiful equation in Maths.Xaqaria
The planet Earth exhibits all of these properties and therefore can be considered alive and its own single organism by the scientific definition.does the planet Earth reproduce, well no unless you count the moon.7. Reproduction: The ability to produce new organisms.
- 03-18-2009 01:55 PM #2Seeker of the Impossible Achievements:
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I am the Alpha and the Omega, the First and the Last, the Beginning and the End. I am the Resurrection and the Life.
I am the Light at the End of the Tunnel.
Rimor Somnium Universitas
- 03-18-2009 04:08 PM #3Fireworks and Hurricanes Achievements:
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It is pretty wonderful.
soft she stirs on starlit sand,
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she wonders at this strange new land.
- 03-18-2009 05:22 PM #4Fireworks and Hurricanes Achievements:
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Oooh... try drawing f(x) = (-1)^x.
It always disturbed me that I couldn't really draw that one properly. Now I see that the plane of my graph paper was simply intersecting a line in a conceptual three dimensional space. Makes sense.
soft she stirs on starlit sand,
and clasps wet shore within her hand.
she turns to face the silent seas,
and through her heart, a vital breeze.
she wonders at this strange new land.
- 03-19-2009 12:52 PM #5Member
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How do you draw it?
Originally Posted by Xei 
- 03-19-2009 01:20 PM #6Seeker of the Impossible Achievements:
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I think he means graphing it on a line.
I am the Alpha and the Omega, the First and the Last, the Beginning and the End. I am the Resurrection and the Life.
I am the Light at the End of the Tunnel.
Rimor Somnium Universitas
- 03-19-2009 01:39 PM #7Member
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Yeah, but I don't know how you draw it on a graph. What if you get f(0.5)=(-1)^0.5 How do you draw this?
I is sorry, I doesn't know math too well.
- 03-19-2009 03:41 PM #8Fireworks and Hurricanes Achievements:
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You need to know Euler's relationship (that the OP was talking about) to be able to do it.
What you just asked is a special case;
-1 ^ 0.5 = sqrt-1 = i.
Complex numbers (look 'em up if you don't know what you are, they're very simple really) can be written as a + bi. Alternatively you can write this as |z|*(cosx + isinx), where |z| is the length of the complex number and x is the angle it makes with the horizontal (this should be clear by looking at an Argand diagram). It turns out that (cosx + isinx) can be shown to be equal to e^ix.
If you set x = pi, you get e^ipi = cospi + isinpi = -1.
Therefore you can write f(x) = -1^x = (e^ipi)^x = e^ipix = cospix + isinpix.
Basically this traces out a circle radius 1 in the Argand plane as x increases, so if you plotted this plane against x you would get a helix.
Thought it was pretty.
soft she stirs on starlit sand,
and clasps wet shore within her hand.
she turns to face the silent seas,
and through her heart, a vital breeze.
she wonders at this strange new land.
- 03-19-2009 04:58 PM #9Member
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Oh so it's like a 3D graph? This is what I didn't get. It never occurred to me to put the complex plane in there too. If I'm understanding this now anyway. Thanks for your explanation. Originally Posted by Xei
- 03-19-2009 05:15 PM #10Fireworks and Hurricanes Achievements:
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Yeah; it's neat if you consider how this relates to trying to plot the function in 2D. The only answers you can get normally are integer powers of -1, where even powers are 1 and odd powers are -1. If you visualise the 3D answer, you can see that these few real answers are simply where the graph intersects the plane of the paper.
y= (-1/e)^x is a nice extension. Like a hurricane.
soft she stirs on starlit sand,
and clasps wet shore within her hand.
she turns to face the silent seas,
and through her heart, a vital breeze.
she wonders at this strange new land.
- 03-19-2009 05:55 PM #11Member
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Which program are you using? (Unless you're one of those savants who visualize it all.) It might be smart to get one
- 04-22-2009 12:24 PM #12Member
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e^(i*pi) = -1 is my favourite formula! Although it took me a while to get my head round it at first.
Xei: I've never thought about that kind of graph before. I can't really visualise what it would look like...
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