2 April 2018

i

\(i=\sqrt {-1}\) is the imaginary unit of any complex number, discovered by the Italian mathematician Girolamo Cardano.

A purely imaginary is a number in the form \(bi\), where \(b\) is a real number and \(i\) is the square root of \(-1\) for \(b\neq0\).

The complex numbers in general, and the purely imaginary numbers, in particular, are essential to describe physical phenomena and has concrete applications in electromagnetism, signal processing, control theory, quantum mechanics, cryptography, cartography, etc.

\(i\) is the result of the following equations:

  • \(x^2+1=0\);
  • \(x^3+x=0\), for \(x\neq0\).

Square roots of negative numbers can be written in the form: \(\sqrt {-n}=\sqrt {-1} \times \sqrt{n}=i \sqrt{n}\).

Another forms of representing the number \(i\):

\(e^{i\frac{\pi}{2}}=\cos \left ( \frac{\pi}{2} \right )+i\sin \left ( \frac{\pi}{2} \right )=i\);

\(e^{i\pi}+1=0\) (Euler's identity).

Powers of $i$ repeat a particular pattern \(\left ( i,-1,-i,1,... \right )\):

\(i\)
\(i^{2}=-1\)
\(i^{3}=i^{2} \times i=\left (-1  \right ) \times i=-i\)
\(i^{4}=i^{2} \times i^{2}=\left (-1  \right ) \times \left (-1  \right )=1\)
\(i^{5}=i^{4} \times i=1 \times i=i\)
\( \cdots \)

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