Cogitation
26-08-2005, 22:46
[from signature]
i^1 = i. i^2 = -1. i^3 = -i. i^4 = 1. i^5 = i.
So the fifth root of i is itself. Therefore the tenth root of -1 is i - as are the 18th, 26th, and so on roots. Do all negative numbers have these particular even roots? Not exactly, no.
All negative real numbers will have {2nd, 10th, 18th, 26th, and so on roots} with the same angle on the complex plane; more specifically, they'll all be i times a positive real number. However, they will not all have the same magnitude.
This is going to get tricky without actually drawing this, but let me try, anyway.
Take some paper and pencil and draw a horizontal and vertical axis. The horixontal axis is your real axis, with positive reals pointing to the right. The vertical axis is your imaginary axis, with positive multiples of i pointing up. This is the "complex plane" and any complex number may be expressed as "x + y * i" where x and y are real numbers. You can plot any complex number on the complex plane.
Now, what you have is a cartesian plane, but you can treat it with a polar coordinate system, instead, and this has interesting properties. For example, if you want to multiply two complex numbers, there are two ways you can do it.
First, you can do a straightforward binomial multipllication:
(a + b * i) * (c + d * i) = (e + f * i)
This will turn out to be:
e = a * c - b * d
f = a * d + b * c
Second, you can convert the cartesian coordinates into polar coordinates. The "angle" is the counterclockwise angle starting from the positive real axis. So, i has an angle of 90 degrees, -1 has an angle of 180 degrees, and -i has an angle of 270 degrees. The "magnitude" is just sqrt(a^2 + b^2).
Without proof, I assert that to multiply two complex numbers together, you just add the angles and multiply the magnitudes.
For example, calculate (1 + i) * (1 + i). Both numbers have a magnitude of sqrt(2) and an angle of 45 degrees. So, add the angles (you get 90 degrees) and multiply the magnitudes (you get 2). The answer is 2*i.
Double-check this against straight binomial multiplication:
(1 + i) * (1 + i) = 1 + i + i - 1 = 2 * i
You get the same answer.
So, any (positive real times i) raised to the 2nd, 10th, 18th, and so on powers will bring you onto the negative real axis. So will the 6th, 14th, 22nd, and so on powers. So, really, the progression is 2, 6, 10, 14, 18, 22, and so on.
One more fun fact: Whatever the power is, there are that many roots and they are scattered symetrically around the origin of the complex plane. For example, there are four different numbers that are the 4th root of -1. They are
+ sqrt(1/2) + sqrt(1/2) * i
+ sqrt(1/2) - sqrt(1/2) * i
- sqrt(1/2) + sqrt(1/2) * i
- sqrt(1/2) - sqrt(1/2) * i
This has been a crash course in complex arithmetic. :)
--The Democratic States of Cogitation
"Think about it for a moment."
Founder and Delegate of The Realm of Ambrosia
i^1 = i. i^2 = -1. i^3 = -i. i^4 = 1. i^5 = i.
So the fifth root of i is itself. Therefore the tenth root of -1 is i - as are the 18th, 26th, and so on roots. Do all negative numbers have these particular even roots? Not exactly, no.
All negative real numbers will have {2nd, 10th, 18th, 26th, and so on roots} with the same angle on the complex plane; more specifically, they'll all be i times a positive real number. However, they will not all have the same magnitude.
This is going to get tricky without actually drawing this, but let me try, anyway.
Take some paper and pencil and draw a horizontal and vertical axis. The horixontal axis is your real axis, with positive reals pointing to the right. The vertical axis is your imaginary axis, with positive multiples of i pointing up. This is the "complex plane" and any complex number may be expressed as "x + y * i" where x and y are real numbers. You can plot any complex number on the complex plane.
Now, what you have is a cartesian plane, but you can treat it with a polar coordinate system, instead, and this has interesting properties. For example, if you want to multiply two complex numbers, there are two ways you can do it.
First, you can do a straightforward binomial multipllication:
(a + b * i) * (c + d * i) = (e + f * i)
This will turn out to be:
e = a * c - b * d
f = a * d + b * c
Second, you can convert the cartesian coordinates into polar coordinates. The "angle" is the counterclockwise angle starting from the positive real axis. So, i has an angle of 90 degrees, -1 has an angle of 180 degrees, and -i has an angle of 270 degrees. The "magnitude" is just sqrt(a^2 + b^2).
Without proof, I assert that to multiply two complex numbers together, you just add the angles and multiply the magnitudes.
For example, calculate (1 + i) * (1 + i). Both numbers have a magnitude of sqrt(2) and an angle of 45 degrees. So, add the angles (you get 90 degrees) and multiply the magnitudes (you get 2). The answer is 2*i.
Double-check this against straight binomial multiplication:
(1 + i) * (1 + i) = 1 + i + i - 1 = 2 * i
You get the same answer.
So, any (positive real times i) raised to the 2nd, 10th, 18th, and so on powers will bring you onto the negative real axis. So will the 6th, 14th, 22nd, and so on powers. So, really, the progression is 2, 6, 10, 14, 18, 22, and so on.
One more fun fact: Whatever the power is, there are that many roots and they are scattered symetrically around the origin of the complex plane. For example, there are four different numbers that are the 4th root of -1. They are
+ sqrt(1/2) + sqrt(1/2) * i
+ sqrt(1/2) - sqrt(1/2) * i
- sqrt(1/2) + sqrt(1/2) * i
- sqrt(1/2) - sqrt(1/2) * i
This has been a crash course in complex arithmetic. :)
--The Democratic States of Cogitation
"Think about it for a moment."
Founder and Delegate of The Realm of Ambrosia