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<ul><li>Complex Numbers </li></ul>
If  a  and  b  are real numbers and  i  is the imaginary  unit, then  a  +  bi   is called a  complex number .  ▪  a   is ...
Definition:   The number  i , called the  imaginary unit , is the number such that i  = ____ √-1 __  and  i 2  = __ -1 ___...
Powers of i
Let  a  +  bi   and  c  +  di   be complex numbers. 1.  Add/Subtract the Real parts. 2.  Add/Subtract the Imaginary parts ...
Let  a  +  bi   and  c  +  di   be complex numbers. 1.  Multiply the binomials. 2.  Convert i 2  to -1 and add the like te...
<ul><li>A complex number  z  is a number of the form  z = x + yi .  Its  conjugate  is a number of the form    =  x - yi ....
Division Of Complex Numbers Let  a  +  bi   and  c  +  di   be complex numbers.   Multiply the numerator and denominator o...
Real Axis Imaginary Axis y x  The angle formed from the real axis and a line from the origin to ( x ,  y ) is called the ...
The Principal Argument is between -   and     Real Axis Imaginary Axis y x z  = r   1  The unique value of  θ  such th...
The magnitude or modulus of  z  is the same as  r. We can take complex numbers given as and convert them to polar form :  ...
 
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complex numbers

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complex numbers

  1. 1. <ul><li>Complex Numbers </li></ul>
  2. 2. If a and b are real numbers and i is the imaginary unit, then a + bi is called a complex number . ▪ a is the real part ▪ bi is the imaginary part . Definition of Complex Numbers
  3. 3. Definition: The number i , called the imaginary unit , is the number such that i = ____ √-1 __ and i 2 = __ -1 ______
  4. 4. Powers of i
  5. 5. Let a + bi and c + di be complex numbers. 1. Add/Subtract the Real parts. 2. Add/Subtract the Imaginary parts . <ul><li>(3 + 4i) + (2 - i) = (3 + 2) + (4i - i) = (5 + 3i) </li></ul><ul><li>(7 + i) - (3 - i) = (7 - 3) + i(1 - (-1)) = 4 + 2i </li></ul>
  6. 6. Let a + bi and c + di be complex numbers. 1. Multiply the binomials. 2. Convert i 2 to -1 and add the like terms. (3 + 2i)(4 + 5i) = (3 × 4) + (3 × (5i)) + ((2i) × 4) + ((2i) × (5i))                              = 12 + 15i + 8i + 10i²                              = 12 + 23i -10 (Remenber that 10i² = 10(-1) = -10)                              = 2 + 23i  Therefore, (3 + 2i)(4 + 5i) = 2+23i
  7. 7. <ul><li>A complex number z is a number of the form z = x + yi . Its conjugate is a number of the form = x - yi . The complex number and its conjugate have the same real part. Re( z ) = Re( ). The sign of the imaginary part of the conjugate complex number is reversed. Im( z ) = - Im( ). </li></ul><ul><li>The conjugate numbers have the same modulus and opposite arguments. | z | = | |, arg( z ) = - arg( ). Any complex number multiplied by its complex conjugate is a real number, equal to the square of the modulus of the complex numbers z . z = ( x + yi )( x - yi ) = x 2+ y 2 = | z |2 </li></ul>
  8. 8. Division Of Complex Numbers Let a + bi and c + di be complex numbers. Multiply the numerator and denominator of the fraction by the Complex Conjugate of the Denominator . Then to perform the operation 2+6i x 4-i = (2+6i) (4-i) = 14+22i = 14 + 22 i 4+i 4-i (4+i) (4-i) 17 17 17
  9. 9. Real Axis Imaginary Axis y x  The angle formed from the real axis and a line from the origin to ( x , y ) is called the argument of z , with requirement that 0   < 2  . modified for quadrant and so that it is between 0 and 2  Let a complex number be Z such that : z Modulus and Argument of Complex Numbers The magnitude or modulus of z denoted by z is the distance from the origin to the point ( x , y ).
  10. 10. The Principal Argument is between -  and  Real Axis Imaginary Axis y x z = r  1  The unique value of θ such that – π < θ < π is called principle value of the argument. but in Quad II
  11. 11. The magnitude or modulus of z is the same as r. We can take complex numbers given as and convert them to polar form : Real Axis Imaginary Axis y x z = r  Plot the complex number: Find the polar form of this number. 1  factor r out but in Quad II

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