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5.9 complex numbers

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Jessica Garcia
Jessica Garcia

The document introduces complex numbers and their properties. It defines the imaginary unit i as the square root of -1. Complex numbers have both a real and imaginary part and can be added, subtracted, multiplied and divided. Powers of i rotate through the values of i, -1, -i, and 1, depending on whether the exponent is 1, 2, 3, or 4 modulo 4. Real and imaginary numbers are subsets of complex numbers.

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Complex Numbers
Consider the quadratic equation x 2 + 1 = 0. Solving for x , gives x 2 = – 1 Complex Numbers
Since there is not real number whose square is -1, the equation has no real solution. French mathematician Rene Descartes (1596 -1650) proposed that i be defined such that,
Complex Numbers Note that squaring both sides yields: therefore and so and And so on…
Finding Powers of i The successive powers of i rotate through the four values of i , -1, - i , and 1. i n = i if n = 1, 5, 9, … i n = -1 if n = 2, 6, 10, … i n = - i if n = 3, 7, 11, … i n = 1 if n = 4, 8, 12, …
*For larger exponents, divide the exponent by 4, then use the remainder as your exponent instead. Example:
Real numbers and imaginary numbers are subsets of the set of complex numbers. Real Numbers Imaginary Numbers Complex Numbers
Imaginary Unit Until now, you have always been told that you can’t take the square root of a negative number. If you use imaginary units, you can! The imaginary unit is ¡ . ¡ = It is used to write the square root of a negative number.
Property of the square root of negative numbers If r is a positive real number, then Examples :
Examples
Complex Numbers A complex number has a real part & an imaginary part. Standard form is: Real part Imaginary part Example: 5+4i
Adding and Subtracting To Add or Subtract Complex Numbers Change all imaginary numbers to bi form. Add (or subtract) the real parts of the complex numbers. Add (or subtract) the imaginary parts of the complex numbers. Write the answer in the form a + bi .
Adding and Subtracting (add or subtract the real parts, then add or subtract the imaginary parts) Ex: Ex: Ex:
Multiplying To Multiply Complex Numbers Change all imaginary numbers to bi form. Multiply the complex numbers as you would multiply polynomials. Substitute –1 for each i 2 . Combine the real parts and the imaginary parts. Write the answer in a + bi form.
Multiplying Treat the i’s like variables, then change any that are not to the first power Ex: Ex:
CAUTION!
Dividing To Divide Complex Numbers Change all imaginary numbers to bi form. Rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator. Substitute –1 for each i 2 .
Dividing Examples:
 

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5.9 complex numbers

  • 2. Consider the quadratic equation x 2 + 1 = 0. Solving for x , gives x 2 = – 1 Complex Numbers
  • 3. Since there is not real number whose square is -1, the equation has no real solution. French mathematician Rene Descartes (1596 -1650) proposed that i be defined such that,
  • 4. Complex Numbers Note that squaring both sides yields: therefore and so and And so on…
  • 5. Finding Powers of i The successive powers of i rotate through the four values of i , -1, - i , and 1. i n = i if n = 1, 5, 9, … i n = -1 if n = 2, 6, 10, … i n = - i if n = 3, 7, 11, … i n = 1 if n = 4, 8, 12, …
  • 6. *For larger exponents, divide the exponent by 4, then use the remainder as your exponent instead. Example:
  • 7. Real numbers and imaginary numbers are subsets of the set of complex numbers. Real Numbers Imaginary Numbers Complex Numbers
  • 8. Imaginary Unit Until now, you have always been told that you can’t take the square root of a negative number. If you use imaginary units, you can! The imaginary unit is ¡ . ¡ = It is used to write the square root of a negative number.
  • 9. Property of the square root of negative numbers If r is a positive real number, then Examples :
  • 11. Complex Numbers A complex number has a real part & an imaginary part. Standard form is: Real part Imaginary part Example: 5+4i
  • 12. Adding and Subtracting To Add or Subtract Complex Numbers Change all imaginary numbers to bi form. Add (or subtract) the real parts of the complex numbers. Add (or subtract) the imaginary parts of the complex numbers. Write the answer in the form a + bi .
  • 13. Adding and Subtracting (add or subtract the real parts, then add or subtract the imaginary parts) Ex: Ex: Ex:
  • 14. Multiplying To Multiply Complex Numbers Change all imaginary numbers to bi form. Multiply the complex numbers as you would multiply polynomials. Substitute –1 for each i 2 . Combine the real parts and the imaginary parts. Write the answer in a + bi form.
  • 15. Multiplying Treat the i’s like variables, then change any that are not to the first power Ex: Ex:
  • 17. Dividing To Divide Complex Numbers Change all imaginary numbers to bi form. Rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator. Substitute –1 for each i 2 .
  • 19.