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Welcome Back!

MATH 1200 March 21, 2016

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House Keeping

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Definitions

• Definition. A complex number is an expression of the form a + bi, where

i2 = −1 and a and b are real numbers; a is called the real part of z and is denoted by Re(z), b is called the imaginary part of z and is denoted by Im(z).

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Question

Let z1 = 2 − 3i and z2 = −5 + i be complex numbers. Then Addition: z1 + z2 = Multiplication: z1 × z2 = Subtraction: z1 − z2 = Division: z1 z2 =

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Answer

Let z1 = 2 − 3i and z2 = −5 + i be complex numbers. Then Addition: z1 + z2 = (2 − 3i) + (−5 + i) = (2 + (−5)) + ((−3) + 1)i = −3 − 2i. Multiplication: z1 × z2 = (2 − 3i) × (−5 + i) = 2(−5) + 2i + (−3)(−5)i + (−3)1i2 = −10 + 2i + 15i + 3 = −7 + 17i. Subtraction: z1 − z2 = (2 − 3i) − (−5 + i) = (2 − (−5)) + ((−3) − 1)i = 3 − 4i.

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Division: z1 z2 = 2 − 3i −5 + i = (2 − 3i)(−5 − i) (−5 + i)(−5 − i) = 2(−5) + 3i2 (−5)2 − i2 + (−2)i + 15i (−5)2 − i2 = −13 26 + 13 26i.

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Definitions

• Definition. A complex number is an expression of the form a + bi, where

i2 = −1 and a and b are real numbers; a is called the real part of z and is denoted by Re(z), b is called the imaginary part of z and is denoted by Im(z). For a complex number z = a + ib, denote z = a − bi. Also, denote |z| = √ a2 + b2 (z is called the conjugate and |z| is called the absolute value of z).

• Definition. Let z1 = a + bi and z2 = c + di be two complex numbers.

Then z1 = z2 if and only if a = c and b = d.

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Summary

1 We have created a new type of numbers, called complex numbers.

We denoted the set of these numbers by C.

2 Complex numbers are expressions of the form a + bi, where a and b

are real numbers.

3 A complex number can be a real because we can think of a real

number a as a + 0i. Hence R (the set of real numbers) is a subset of C.

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In an Argand diagram, a complex number z = a + bi can be represented graphically as the vector from the origin to the point (a, b). Every complex number z = a + ib has the polar form z = r(cosθ + isinθ), where r = √ a2 + b2 and θ is the angle between real arrow and the vector from

• rigin to the point (a, b) in Argand diagram.

Imaginary Real b a z θ

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Question

• 1. Find the polar form of z = −1 + i. Graph z = −1 + i.
• 2. Compute (−1 + i)2. Find the polar form of (−1 + i)2. Graph (−1 + i)2.
• 3. Compute (−1 + i)3. Find the polar form of (−1 + i)3. Graph (−1 + i)3.
• 4. Is there a pattern? What do you notice? Write down a formula for the

pattern you noticed. Remember for questions 5 and 6 to explain your thinking :)

• 5. Compute (−1 + i)50.
• 6. Find all complex numbers z such that z3 = 8i.

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Thank you ...

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