1.3: Terminology and Basic Arithmetic
- Page ID
- 6468
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Complex numbers are defined as the set of all numbers
\[z = x + yi \nonumber \]
where \(x\) and \(y\) are real numbers.
- We denote the set of all complex numbers by \(\mathbb{C}\).
- We call \(x\) the real part of \(z\). This is denoted by \(x = \text{Re} (z)\).
- We call \(y\) the imaginary part of \(z\). This is denoted by \(y = \text{Im} (z)\).
Important: The imaginary part of \(z\) is a real number. It does not include the \(i\).
The basic arithmetic operations follow the standard rules. All you have to remember is that \(i^2 = -1\). We will go through these quickly using some simple examples. It almost goes without saying that it is essential that you become fluent with these manipulations.
- Addition: \((3 + 4i) + (7 + 11i) = 10 + 15i\)
- Subtraction: \((3 + 4i) - (7 + 11i) = -4 - 7i\)
- Multiplication:
\((3 + 4i)(7 + 11i) = 21 + 28i + 33i + 44i^2 = -23 + 61i.\)
Here we have used the fact that \(44i^2 = -44\).
Before talking about division and absolute value we introduce a new operation called conjugation. It will prove useful to have a name and symbol for this, since we will use it frequently.
Complex conjugation is denoted with a bar and defined by
\[\overline{x + iy} = x - iy \nonumber \]
If \(z = x + iy\) then its conjugate is \(\bar{z} = x - iy\) and we read this as "z-bar = \(x - iy\)".
\(\overline{3 + 5i} = 3 - 5i\).
The following is a very useful property of conjugation: If \(z = x + iy\) then
\(z\bar{z} = (x + iy)(x - iy) = x^2 + y^2\)
Note that \(z\bar{z}\) is real. We will use this property in the next example to help with division.
Write \(\dfrac{3 + 4i}{1 + 2i}\) in the standard form \(x + iy\).
Solution
We use the useful property of conjugation to clear the denominator:
\(\dfrac{3 + 4i}{1 + 2i} = \dfrac{3 + 4i}{1 + 2i} \cdot \dfrac{1 - 2i}{1 - 2i} = \dfrac{11 - 2i}{5} = \dfrac{11}{5} - \dfrac{2}{5} i\).
In the next section we will discuss the geometry of complex numbers, which gives some insight into the meaning of the magnitude of a complex number. For now we just give the definition.
The magnitude of the complex number \(x + iy\) is defined as
\[|z| = \sqrt{x^2 + y^2} \nonumber \]
The magnitude is also called the absolute value, norm or modulus.
The norm of \(3 + 5i = \sqrt{9 + 25} = \sqrt{34}\).
Important. The norm is the sum of \(x^2\) and \(y^2\). It does not include the \(i\) and is therefore always positive.