Post by Beanybag on Apr 11, 2008 20:01:38 GMT -5
Imaginary Numbers
Most of you should be familiar with the fact that the square root of -1 is i.
e and pi come into play here. Most people know what pi is, but e is a little different. It's a constant somewhat like pi in that it's a limit. It's the limit of (1 + 1/x)^x as x->infinity, and roughly equals 2.718
I've been looking into why imaginary numbers even exist if they simply stand for the square root of a negative number..
What I've found is making me poop bricks.
If you raise i^x power, you get a graph representing sin(x) using a complex axis for the y-axis.
It's the wavy graph like this:
Then there's e raised to the power of i.
So e raised to the power of 45i is equal to the inverse of the square root of two plus the i times the inverse of the square root of two...
Then it gets trippy. Since e has been tied into trig, pi comes into play.
Basically, e raised to the power of pi times i is equal to -1.
Expand the trig out and substitute and you get
More substitution and you find that
i^i = BLABLALBLA I JUST CRAPPED BRICKS.
That's what happens if you raise a real number to an imaginary power. If you raise it to itself 3 times, you get -i, and 5 times you get i again.
And then get this.
The cosine of i is REAL, but...
The sine of i is imaginary!
Number Theory and Tetration - Mathematical Art
I'll start with Tetration. Tetration is
It is basically, raising a number to itself by the amount specified.
It is an extremely fast growing function, faster than even double exponentiation (10^(10^x)). You play with Tetrarion enough, and you find you can you can tetrate by i. Yes, you can raise a number to itself i times.
The result is quite beautiful. This picture is a fractal of tetration in the complex plane. (click to enlarge pictures)
Hell Fractals in general are gorgeous.
upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Julia_set_%28indigo%29.png/800px-Julia_set_%28indigo%29.png
Now onto Number Theory...and the enigmatic prime numbers, my favorite subject of all.
This is a picture of a spiral of the natural numbers with the prime numbers emphasized.
Also, from the conjecture that every even number is the sum of two prime numbers, we find that when we graph the pairs of these numbers...we notice an interesting pattern.
What's more is Prime numbers...are quite musical. Terence Tao described one approach to proving the prime number theorem in poetic terms: listening to the "music" of the primes.
We start with a "sound wave" that produces noise at the prime numbers and silence at other numbers; this is the von Mangoldt function. Then we analyze its notes or frequencies by subjecting it to a process akin to Fourier transform; this is the Mellin transform. Then we prove, and this is the hard part, that certain "notes" cannot occur in this music. This exclusion of certain notes leads to the statement of the prime number theorem. According to Tao, this proof yields much deeper insights into the distribution of the primes than the "elementary" proofs others have thought of.
No one has come up with an accurate theory for prime number patterns. They are seemingly random in their placement, however we have come up with interesting conjectures as to how they occur and have limited their randomness somewhat.
Oh there is so much more...if only I had time to go into it all. @_@
How big's that pile of bricks? <:
Most of you should be familiar with the fact that the square root of -1 is i.
e and pi come into play here. Most people know what pi is, but e is a little different. It's a constant somewhat like pi in that it's a limit. It's the limit of (1 + 1/x)^x as x->infinity, and roughly equals 2.718
I've been looking into why imaginary numbers even exist if they simply stand for the square root of a negative number..
What I've found is making me poop bricks.
If you raise i^x power, you get a graph representing sin(x) using a complex axis for the y-axis.
It's the wavy graph like this:
Then there's e raised to the power of i.
So e raised to the power of 45i is equal to the inverse of the square root of two plus the i times the inverse of the square root of two...
Then it gets trippy. Since e has been tied into trig, pi comes into play.
Basically, e raised to the power of pi times i is equal to -1.
Expand the trig out and substitute and you get
More substitution and you find that
i^i = BLABLALBLA I JUST CRAPPED BRICKS.
That's what happens if you raise a real number to an imaginary power. If you raise it to itself 3 times, you get -i, and 5 times you get i again.
And then get this.
The cosine of i is REAL, but...
The sine of i is imaginary!
Number Theory and Tetration - Mathematical Art
I'll start with Tetration. Tetration is
It is basically, raising a number to itself by the amount specified.
It is an extremely fast growing function, faster than even double exponentiation (10^(10^x)). You play with Tetrarion enough, and you find you can you can tetrate by i. Yes, you can raise a number to itself i times.
The result is quite beautiful. This picture is a fractal of tetration in the complex plane. (click to enlarge pictures)
Hell Fractals in general are gorgeous.
upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Julia_set_%28indigo%29.png/800px-Julia_set_%28indigo%29.png
Now onto Number Theory...and the enigmatic prime numbers, my favorite subject of all.
This is a picture of a spiral of the natural numbers with the prime numbers emphasized.
Also, from the conjecture that every even number is the sum of two prime numbers, we find that when we graph the pairs of these numbers...we notice an interesting pattern.
What's more is Prime numbers...are quite musical. Terence Tao described one approach to proving the prime number theorem in poetic terms: listening to the "music" of the primes.
We start with a "sound wave" that produces noise at the prime numbers and silence at other numbers; this is the von Mangoldt function. Then we analyze its notes or frequencies by subjecting it to a process akin to Fourier transform; this is the Mellin transform. Then we prove, and this is the hard part, that certain "notes" cannot occur in this music. This exclusion of certain notes leads to the statement of the prime number theorem. According to Tao, this proof yields much deeper insights into the distribution of the primes than the "elementary" proofs others have thought of.
No one has come up with an accurate theory for prime number patterns. They are seemingly random in their placement, however we have come up with interesting conjectures as to how they occur and have limited their randomness somewhat.
Oh there is so much more...if only I had time to go into it all. @_@
How big's that pile of bricks? <:
