Complex Number

Complex Number Form: 𝑥+𝑦𝑖

Complex Number Form: 𝑥+𝑦𝑖 x and y are real number i is the imaginary unit: square root of -1 (𝑖= −1 )

Complex Number Elementary operations Conjugate: negative the imaginary part Say 𝑧=𝑥+𝑦𝑖, then the conjugate of z will be: 𝑧 =𝑥−𝑦𝑖

Complex Number Elementary operations Conjugate: 𝑧 =𝑥−𝑦𝑖 Addition and Subtraction: complex number a and b 𝑎±𝑏= 𝑅𝑒 𝑎 ±𝑅𝑒 𝑏 + 𝐼𝑚 𝑎 ±𝐼𝑚 𝑏

Complex Number Elementary operations 𝑎±𝑏= 𝑅𝑒 𝑎 ±𝑅𝑒 𝑏 + 𝐼𝑚 𝑎 ±𝐼𝑚 𝑏 Conjugate: 𝑧 =𝑥−𝑦𝑖 𝑎±𝑏= 𝑅𝑒 𝑎 ±𝑅𝑒 𝑏 + 𝐼𝑚 𝑎 ±𝐼𝑚 𝑏 Multiplication and Division: follow the distributive property 𝑥+𝑦𝑖 × 𝑢+𝑣𝑖 =𝑥𝑢+𝑥𝑣𝑖+𝑦𝑢𝑖+𝑦𝑖𝑣𝑖 =𝑥𝑢+ 𝑥𝑣+𝑦𝑢 𝑖+𝑦𝑣 𝑖 2 = 𝑥𝑢−𝑦𝑣 + 𝑥𝑣+𝑦𝑢 𝑖

Complex Number Elementary operations 𝑎±𝑏= 𝑅𝑒 𝑎 ±𝑅𝑒 𝑏 + 𝐼𝑚 𝑎 ±𝐼𝑚 𝑏 Conjugate: 𝑧 =𝑥−𝑦𝑖 𝑎±𝑏= 𝑅𝑒 𝑎 ±𝑅𝑒 𝑏 + 𝐼𝑚 𝑎 ±𝐼𝑚 𝑏 Multiplication and Division: follow the distributive property 𝑥+𝑦𝑖 × 𝑢+𝑣𝑖 = 𝑥𝑢−𝑦𝑣 + 𝑥𝑣+𝑦𝑢 𝑖 𝑥+𝑦𝑖 𝑢+𝑣𝑖 = 𝑥𝑢+𝑦𝑣 𝑢 2 + 𝑣 2 + 𝑦𝑢−𝑥𝑣 𝑢 2 + 𝑣 2 𝑖

Complex Number Complex Number on Cartesian Coordinate System x is the real axis, y is the imaginary axis

Complex Number Complex Number on Cartesian Coordinate System x is the real axis, y is the imaginary axis If z = x + yi, then the angle of z is the 𝜃= tan −1 ( 𝑦 𝑥 )

Complex Number Complex Number on Cartesian Coordinate System x is the real axis, y is the imaginary axis If z = x + yi, then the angle of z is the 𝜃= tan −1 ( 𝑦 𝑥 ) Absolute value of z: 𝑧 = 𝑥 2 + 𝑦 2

Complex Number Complex Number on Cartesian Coordinate System x is the real axis, y is the imaginary axis If z = x + yi, then the angle of z is the 𝜃= tan −1 ( 𝑦 𝑥 ) Absolute value of z: 𝑧 = 𝑥+𝑦𝑖 = 𝑥 2 + 𝑦 2 Square root of z: 𝑧 = 𝑥+𝑦𝑖 = 𝑥+𝑦𝑖 +𝑥 2 ±𝑖 𝑥+𝑦𝑖 −𝑥 2