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India
Class XI

Graphing Complex Numbers - Rectangular Form

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+bi.

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=125i $\left(a,b\right)=\left(12,-5\right)$(a,b)=(12,5)
$z_3=-4-4i$z3=44i $\left(a,b\right)=\left(-4,-4\right)$(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+bi.

 

   

Activity

Try this for yourself before checking out the solution.

If $z=1-i$z=1i, find $z,z^2,z^3,z^4,z^5,z^6$z,z2,z3,z4,z5,z6

Plot these points on an Argand Diagram.

Is there a geometric pattern?

Can you generalise your result?

(see here for the solution)

More Worked Examples

QUESTION 1

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

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QUESTION 2

Consider the following.

  1. Graph the number $-1+2i$1+2i.

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  2. Evaluate $\left(-1+2i\right)+\left(9+5i\right)$(1+2i)+(9+5i).

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

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QUESTION 3

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

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QUESTION 4

What is the complex number represented on the graph?

State the number in rectangular form.

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Outcomes

11.A.CNQE.1

Need for complex numbers, especially √-1, to be motivated by inability to solve every quadratic equation. Brief description of algebraic properties of complex numbers. Argand plane and polar representation of complex numbers. Statement of Fundamental Theorem of Algebra, solution of quadratic equations in the complex number system.

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