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## 11.2 Geometric Representation of Complex Numbers

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**11.2 Geometric Representation of Complex Numbers**Can be confusing, polar form has a degree with it, rectangular form does not, this all takes place in a plane that is not the x and y axis, but behaves similarly.**Complex numbers are numbers that involve i.**• They are of the form a+bi. • We cannot graph these numbers on the Cartesian Plane because both axis on the Cartesian plane are representative of REAL numbers (not imaginary) • We do however have a method for graphing these complex numbers.**Jean Robert Argand**• We can represent complex numbers geometrically thanks to Argand who made the argument that we can replace the y-axis with an imaginary axis . The following method will be how we plot complex numbers. Plot 6+5i. imaginary axis x axis This is called the complex plane or Argand Diagram.**This point that is representing a+bican be represented in**rectangular coordinates (a,b) or in polar coordinates (r,ϴ). Lets see how we can work through the translation. In general we refer to the point by the name “z” Now that z=(a,b)=a+bi But in polar coordinates we know that a=rcosϴ and b=rsinϴ. So now replace a and b you get … rcosϴ + (rsinϴ)i Then you get rcos ϴ + r i sin ϴ…factor out r you get … r(cos ϴ+ i sin ϴ) We define (cos ϴ+i sin ϴ) as “cis” so z=rcis ϴ z = (a,b) In addition we can still find r by pythagoras, r or ||z||= r b ϴ a**Rewrite the following complex numbers in polar form.**• 3-2i • -4+2i • -4i**Rewrite each complex number in rectangular form.**• 8 cis 110 • 12 cis 250**Product of 2 complex numbers in polar form**• When we want to multiply two complex numbers we multiply the radii r and s, and we add the angles. i x**Express each product in polar form**• (4 cis 25o)(6 cis 35o) = 24 cis (25o + 35o) =24 cis (60o) z1 z2 Point z2 a distance of 6 away from the origin and 35 degrees counterclockwise of the polar axis Point z1 a distance of 4 away from the origin and 25 degrees counterclockwise of the polar axis Point z1z2 a distance of 24 away from the origin and 60 degrees counterclockwise of the polar axis**Find z1z2 in rectangular form by multiplying z1 and z2.**• Find z1, z2, and z1z2 in polar form. Show that z1z2 in polar form agrees with z1z2 in rectangular form. • Show z1, z2, and z1z2 in an Argand diagram.