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Complex Numbers

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The square root of 1 is called the imaginary unit and is denoted by i. ... The real numbers are a subset of the complex numbers. ... – PowerPoint PPT presentation

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Title: Complex Numbers


1
Complex Numbers
2
A First Observation
  • There is no real number solution to the equation
    x2 10.
  • The square root of 1 is called the imaginary
    unit and is denoted by i.
  • This provides a solution to x210, but it is not
    a real number.

3
Definition of Complex Number
  • A complex number z is a number of the form a
    bi, where i ?-1.
  • a is called the real part of z, and b is called
    the imaginary part of z.
  • The standard form of a complex number is a bi,
    where a and b are real numbers.

4
More on Complex Numbers
  • If b ? 0, z a bi is also called an imaginary
    number.
  • If b 0, z is a real number.
  • If a 0 and b ? 0, z is called a pure imaginary
    number.

5
Equality of Complex Numbers
  • Two complex numbers abi, and cdi are equal if
    and only if a c and b d

6
Addition and Subtraction of Complex Numbers
  • (abi) (cdi) (ac)(bd)i
  • (abi) (cdi) (a-c)(b-d)i

7
Multiplication of Complex Numbers
  • (abi)(cdi)
    (ac-bd)(adbc)i
  • Note that this is simply the
    first-outer-inner-last rule, and we replace i2 by
    1.

8
Complex Conjugates
  • If z abi, the conjugate of z is a bi.
  • Note that z and its conjugate are equal if and
    only if the imaginary part of z 0, i.e. z is a
    real number.
  • (abi)(a-bi) a2 b2

9
Division of Complex Numbers
10
Real and Complex Numbers
  • The real numbers are a subset of the complex
    numbers. I.e. every real number is a complex
    number.
  • The properties of addition, subtraction,
    multiplication, and division that hold for real
    numbers also hold for complex numbers.
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