Complex Numbers
Section 1.3 Complex Numbers 
How can we solve the equation x ^{2}= −1? x = ± It turns out that this is not a real number ! So we have to "make up " some new numbers! We define the new number i as follows i = 
Complex Numbers Let i^{2}=1 then i= i is the square root of 1 
Examples

Complex Numbers A complex number has the form a + bi a is called the real part of the complex number b is called the imaginary part of the complex number 
The Complex Plane 
Adding Complex Numbers (a + bi ) + (c + di ) = (a + c) + (b + d ) i To add two complex numbers, add the real parts and add the imaginary parts, the result is a complex number. 
Examples (2 + 3i ) + (−5 + 4i ) = (5 + 2i ) − (2 − 7i ) = 
Multiplying Complex Numbers (a+bi ) (c+di ) = ac + adi + bci + bdi^{2} = (ac −bd ) + (ad + bc) i This follows from the distributive property and the fact that t^{2} = −1 
Examples (2 + 3i ) (−5 + 4i ) = (5 + 2i ) (2 − 7i ) = (a + bi ) (a − bi ) = ( a + bi) and (a − bi ) are called complex conjugates 
If z=a + bi is a complex number, then = a bi is the conjugate of z. The product of a complex number and its conjugate is a real number. 
Multiply the complex conjugates (2 + 3i ) (2 − 3i ) = (5 − 2i ) (5 + 2i ) = (c + di ) (c − di ) = 
Dividing Complex Numbers 
Magnitude of Complex Numbers If c = a + bi is a complex number, The absolute value (magnitude) of is the distance away from 0 0 in the complex plane. This follows from the distance formula! 
The Complex Plane 
Examples Find c = 2 + 3i c = 5 − 4i c = −4 + 3i c = −4 − 3i 
The Quadratic Formula If ax^{2} + bx + c = 0 then 
Use the quadratic formula to solve x^{2} + 4 = 0 3y^{2} + 2y +1= 0 2t^{2}− t = −2 
Analytical Solutions Given a quadratic equation ax ^{2} + bx + c = 0 Use the quadratic formula If b ^{2}− 4ac > 0 there are two real solutions If b^{2} − 4ac = 0 there is one real solution If b^{2} − 4ac< 0there are no real solutions, but two complex conjugate solutions. 
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