8 5.4 – Complex NumbersImaginary number – created so that square roots of negative numbers can be foundImaginary unit – ii = √ i2 = – i3 = – I i4 = 1Pure imaginary number – square roots of negative real numbersEx. 3i, -5i, and i√2For any positive real number b,√-b2 = √b2 √-1 or bi
14 5.4 – Complex NumbersComplex number – any number that can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit. a is called the real part, and b is called the imaginary partEx i and 2 – 6i = 2 + (-6)IIf b = 0, the complex number is a real numberIf b ≠ 0, the complex number is imaginaryIf a = 0, the complex number is a pure imaginary number
15 5.4 – Complex NumbersTwo complex numbers are equal if and only if (IFF) their real parts are equal AND their imaginary parts are equal.a + bi = c + di IFF a = c and b = d
16 5.4 – Complex Numbers Example 6 Find the values of x and y that make that equation 2x + yi = -14 – 3i true.
17 5.4 – Complex NumbersTo add or subtract complex numbers, combine like terms.Combine the real partsCombine the imaginary parts
19 5.4 – Complex NumbersYou can also multiply 2 complex numbers using theFOIL method
20 5.4 – Complex Numbers Example 8 In an AC circuit, the voltage E, current I, and impedance Z are related by the formula E = I Z. Find the voltage in a circuit with current 1 + 3j amps and impedance 7 – 5j ohms.
22 5.4 – Complex NumbersComplex conjugates – two complex numbers of the form a + bi and a – bi.The product of complex conjugates is always a real number.You can use this to simplify the quotient of two complex numbers.