Complex Numbers

Lesson

The Argand Diagram is what we call the plane that will allow us to plot complex numbers. It is named after the Swiss mathematician Jean Argand (1768 - 1822). Using the $x$`x`-axis as the real axis, and the $y$`y`-axis as the imaginary axis, the ordered pairs $\left(a,b\right)$(`a`,`b`) reflect complex numbers of the form $a+bi$`a`+`b``i`.

Plotting points on the plane is as simple as identifying the real and imaginary components from a complex number.

$z_1=2+3i$z1=2+3i |
$\left(a,b\right)=\left(2,3\right)$(a,b)=(2,3) |

$z_2=12-5i$z2=12−5i |
$\left(a,b\right)=\left(12,-5\right)$(a,b)=(12,−5) |

$z_3=-4-4i$z3=−4−4i |
$\left(a,b\right)=\left(-4,-4\right)$(a,b)=(−4,−4) |

$z_4=7$z4=7 |
$\left(a,b\right)=\left(7,0\right)$(a,b)=(7,0) |

$z_5=-6i$z5=−6i |
$\left(a,b\right)=\left(0,-6\right)$(a,b)=(0,−6) |

Plotting these points on the Argand diagram would result in the following graph.

There is another way we can display complex number on the plane, and that is as a vector. The above complex numbers and all be represented as vectors on the plane with initial position $\left(0,0\right)$(0,0) and terminal position at the point $\left(a,b\right)$(`a`,`b`) as designated by the values of $a$`a` and $b$`b` in the number $a+bi$`a`+`b``i`.

Activity

^{Try this for yourself before checking out the solution.}

If $z=1-i$`z`=1−`i`, find $z,z^2,z^3,z^4,z^5,z^6$`z`,`z`2,`z`3,`z`4,`z`5,`z`6.

Plot these points on an Argand Diagram.

Is there a geometric pattern?

Can you generalise your result?

(see here for the solution)

Plot $6+2i$6+2`i` on the Argand diagram (complex plane).

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Consider the following.

Graph the number $-1+2i$−1+2

`i`.Loading Graph...Evaluate $\left(-1+2i\right)+\left(9+5i\right)$(−1+2

`i`)+(9+5`i`).Graph the result of $\left(-1+2i\right)+\left(9+5i\right)$(−1+2

`i`)+(9+5`i`).Loading Graph...

Graph the complex number $-6$−6 as a vector.

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What is the complex number represented on the graph?

State the number in rectangular form.

Loading Graph...

Manipulate complex numbers and present them graphically

Apply the algebra of complex numbers in solving problems