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Mathematics LibreTexts

18.1: Complex numbers

  • Page ID
    23696
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    Informally, a complex number is a number that can be put in the form

    \[z=x+i\cdot y, \]

    where \(x\) and \(y\) are real numbers and \(i^2=-1\).

    The set of complex numbers will be further denoted by \(\mathbb{C}\). If \(x\), \(y\), and \(z\) are as in 18.1.1, then \(x\) is called the real part and \(y\) the imaginary part of the complex number \(z\). Briefly it is written as

    \[x=\text{Re} z \ \ \ \ \text{and} \ \ \ \  y=\text{Im} z.\]

    On the more formal level, a complex number is a pair of real numbers \((x,y)\) with the addition and multiplication described below; the expression \(x + i\cdot y\) is only a convenient way to write the pair \((x,y)\).

    \[\begin{aligned} (x_1+i\cdot y_1) + (x_2+i\cdot y_2) &:= (x_1+x_2) + i\cdot(y_1+y_2); \\ (x_1+i\cdot y_1)\cdot(x_2+i\cdot y_2) &:= (x_1\cdot x_2-y_1\cdot y_2) + i\cdot(x_1\cdot y_2+y_1\cdot x_2). \end{aligned}\]