11 2 GEOMETRIC REPRESENTATION OF COMPLEX NUMBERS Can

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+5 i. imaginary axis This is called the complex plane or Argand Diagram. x axis

This point that is representing a+bi can 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 = (a, b) r ϴ b a z=rcis ϴ

• Rewrite the following complex numbers in polar form. • 3 -2 i • -4+2 i • -4 i

• Rewrite each complex number in rectangular form. • 8 cis 110 • 12 cis 250

PG. 406 1 -12

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 25 o)(6 cis 35 o) z 1 z 2 = 24 cis (25 o + 35 o) =24 cis (60 o) Point z 1 a distance of 4 away from the origin and 25 degrees counterclockwise of the polar axis Point z 2 a distance of 6 away from the origin and 35 degrees counterclockwise of the polar axis Point z 1 z 2 a distance of 24 away from the origin and 60 degrees counterclockwise of the polar axis

• Find z 1 z 2 in rectangular form by multiplying z 1 and z 2. • Find z 1, z 2, and z 1 z 2 in polar form. Show that z 1 z 2 in polar form agrees with z 1 z 2 in rectangular form. • Show z 1, z 2, and z 1 z 2 in an Argand diagram.

HWK PG. 406 13 -22