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Complex Numbers: Study Sheet

College-Cram.com:: Trigonometry:: Complex Numbers:: Study Sheet
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Description: Use this printable study sheet to review or learn the techniques for working with complex numbers and the imaginary number i, including addition, subtraction, multiplication, and division.
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Contents

  1. Definitions
  2. Adding Complex Numbers
  3. Dividing Complex Numbers
  4. Multiplying Complex Numbers by Complex Numbers
  5. Subtracting Complex Numbers
  6. Related Links

Definitions

  • The Coefficient in a complex number is the real number that is multiplied by i. For example, the cofficient in the expression 2 + 4i is 4.
  • A Complex Conjugate of a complex number is found by changing the sign on the imaginary part. For example, the complex conjugate of 2 + 4i is 2 - 4i.
  • A Complex Number is any number that can be expressed in the form a + bi where a and b are real numbers. Complex numbers are not algebraic expressions, they are numbers that have a real part (a) and an imaginary part (bi).
  • The Constant in a complex number is the real number that is not multiplied by i. For example, the constant in the expression 2 + 4i is 2.
  • The Imaginary Number i is defined as the square root of -1. (It is an imaginary number, since the square root of any negative number is not real.) An important property to remember is that i2 = -1.
  • Like-Terms in a complex number are those terms that are similarly formatted. For example, 7i and -3i are like-terms since they are both a coefficient multiplied by i, whereas 3 and 3i are not like-terms.

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Adding Complex Numbers

  1. Put both numbers into the standard form a + bi. (Remember, 3i is the same as 0 + 3i.)
  2. Line up both numbers one under the other, grouping the like terms in the same column.
  3. Add the constants in the first column.
  4. Add the coefficients in the second column and carry the number i to get your answer.

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Dividing Complex Numbers

  1. Put both numbers into the standard form a + bi. (Remember, 3i is the same as 0 + 3i.)
  2. Identify the complex conjugate of the denominator (bottom complex number) by changing the sign in front of the imaginary part.
  3. Multiply the numerator and denominator by that complex conjugate.
  4. Set the new denominator to the first term squared plus the coefficient squared.
  5. Use the FOIL method to multiply the complex numbers in the numerator (First, Outer, Inner, Last).
  6. Since i2 = -1, multiply the last term by -1 and remove the i2.
  7. For both the numerator and denominator, combine the real terms (without the i).
  8. Combine the imaginary terms (with the i) in the numerator to get your answer.

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Multiplying Complex Numbers by Complex Numbers

  1. Put both numbers into the standard form a + bi. (Remember, 3i is the same as 0 + 3i.)
  2. Use the FOIL method to multiply the complex numbers (First, Outer, Inner, Last).
  3. Since i2 = -1, multiply the last term by -1 and remove the i2.
  4. Combine the real terms (without the i).
  5. Combine the imaginary terms (with the i) to get your answer.

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Subtracting Complex Numbers

  1. Put both numbers into the standard form a + bi. (Remember, 3i is the same as 0 + 3i.)
  2. Line up both numbers one under the other, grouping the like terms in the same column.
  3. Subtract the constants in the first column.
  4. Subtract the coefficients in the second column and carry the number i to get your answer.

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Related Links

Learn more about complex numbers and how to use them.

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Click one of these Keywords for more resources on the topic: adding complex numbers, addition, coefficient, complex conjugate, complex numbers, dividing complex numbers, division, FOIL, FOIL method, i, imaginary number, imaginary number i, imaginary numbers, like terms, like-terms, multiplication, multiplying complex numbers, multiplying complex numbers and real numbers, number i, real numbers, subtracting complex numbers, subtraction, trigonometry

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