Friday, May 13, 2011

Imaginary numbers are silly

So, the imaginary constant is a pretty silly constant. While there is nothing wrong with them, they're defined in a very awkward fashion. A much more elegant way of doing it is by using 2x2 matrices.

Define the imaginary constant to be
\[
i=\left(\begin{matrix}0 & -1\\
1 & 0\end{matrix}\right)
\]
which is the rotation matrix for a 90 degree clockwise rotation of a two dimensional vector.

Let's further implicitly put an identity matrix next to all numbers. For example, consider a complex number
\[
z=a+bi=\left(\begin{matrix}a & 0\\
0 & a\end{matrix}\right)+\left(\begin{matrix}0 & -b\\
b & 0\end{matrix}\right)=\left(\begin{matrix}a & -b\\
b & a\end{matrix}\right)
\]
It's trivial to show that this obeys all that we'd expect from complex algebra (and complex conjugate is the same as transposing the matrix). Wasn't that nice? No obscure voodoo mathematics. Just a simple 2x2 matrix. We don't need to justify the existence of matrices. They are not just some ugly trick that's used to simplify calculations.

5 comments:

  1. Awesome!!! I never thought imaginary numbers are so silly.Interesting to learn about imaginary numbers.Thanks for sharing the information.


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  2. This makes sense, I already thought of i as a pi/2 rotation but I hadn't thought of using matrices. I don't get what you mean by "voodoo mathematics" as math with complex numbers is quite simple.

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  3. The voodoo part is i. As demonstrated we don't need this obscure black box constant. It is a 2x2 matrix.

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  4. A number whose square is less than or equal to zero is termed as an imaginary number. Let's take an example, √-5 is an imaginary number and its square is -5. An imaginary number can be written as a real number but multiplied by the imaginary unit.in a+bi complex number i is called the imaginary unit,in given expression "a" is the real part and b is the imaginary part of the complex number. The complex number can be identified with the point (a, b).

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