Welcome Back! MATH 1200 March 21, 2016 MATH 1200 Welcome Back! - PowerPoint PPT Presentation

Welcome Back! MATH 1200 March 21, 2016 MATH 1200 Welcome Back! March 21, 2016 1 / 11

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Definitions Definition. A complex number is an expression of the form a + bi , where i 2 = − 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 ). MATH 1200 Welcome Back! March 21, 2016 3 / 11

Question Let z 1 = 2 − 3 i and z 2 = − 5 + i be complex numbers. Then Addition: z 1 + z 2 = Multiplication: z 1 × z 2 = Subtraction: z 1 − z 2 = Division: z 1 = z 2 MATH 1200 Welcome Back! March 21, 2016 4 / 11

Answer Let z 1 = 2 − 3 i and z 2 = − 5 + i be complex numbers. Then Addition: z 1 + z 2 = (2 − 3 i ) + ( − 5 + i ) = (2 + ( − 5)) + (( − 3) + 1) i = − 3 − 2 i . Multiplication: z 1 × z 2 = (2 − 3 i ) × ( − 5 + i ) = 2( − 5) + 2 i + ( − 3)( − 5) i + ( − 3)1 i 2 = − 10 + 2 i + 15 i + 3 = − 7 + 17 i . Subtraction: z 1 − z 2 = (2 − 3 i ) − ( − 5 + i ) = (2 − ( − 5)) + (( − 3) − 1) i = 3 − 4 i . MATH 1200 Welcome Back! March 21, 2016 5 / 11

Division: = 2 − 3 i z 1 − 5 + i = z 2 (2 − 3 i )( − 5 − i ) ( − 5 + i )( − 5 − i ) = 2( − 5) + 3 i 2 ( − 5) 2 − i 2 + ( − 2) i + 15 i ( − 5) 2 − i 2 = − 13 26 + 13 26 i . MATH 1200 Welcome Back! March 21, 2016 6 / 11

Definitions Definition. A complex number is an expression of the form a + bi , where i 2 = − 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 √ a 2 + b 2 ( z is called the conjugate and | z | is called the absolute | z | = value of z ). Definition. Let z 1 = a + bi and z 2 = c + di be two complex numbers. Then z 1 = z 2 if and only if a = c and b = d . MATH 1200 Welcome Back! March 21, 2016 7 / 11

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 + 0 i . Hence R (the set of real numbers) is a subset of C . MATH 1200 Welcome Back! March 21, 2016 8 / 11

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 ( cos θ + isin θ ), where √ a 2 + b 2 and θ is the angle between real arrow and the vector from r = origin to the point ( a , b ) in Argand diagram. Imaginary z b θ Real a MATH 1200 Welcome Back! March 21, 2016 9 / 11

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 z 3 = 8 i . MATH 1200 Welcome Back! March 21, 2016 10 / 11

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