Complex Numbers




We will use complex numbers, denoted by $\QTR{Bbb}{C}$, in Section 5.5. Below we recall some basic information about complex numbers.

  1. A complex number $x\in $ $\QTR{Bbb}{C}$ is a number of the form $x=a+bi$ where MATH and $i^{2}=-1$.

  2. For $x=a+bi$, we call $a$ the real part of $x$ and denote it by $\func{Re}x$. We call $b$ the imaginary part of $x$ and denote it by $\func{Im}x$.

  3. The complex conjugate of $x=a+bi$ is the complex number $\overline{x}=a-bi$; that is, we change the sign of the imaginary part of $x$.

  4. If MATH is a vector of complex numbers, then $\func{Re}x$ is the vector of real parts of the components of $x$ and $\func{Im}x$ is the vector of imaginary parts of the components of $x$.

  5. If $x$ is a real number, then $x=a=a+0i$, so that MATH. Thus, if $x$ is a real number, then $\overline{x}=x$.

  6. Complex conjugation distributes over products: MATH for any appropriate multiplication (scalar, vector, matrix).

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