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Complex Numbers

A First Observation. There is no real number solution to the equation x2 1=0.The square root of 1 is called the imaginary unit and is denoted by i.This provides a solution to x2 1=0, but it is not a real number.. Definition of Complex Number. A complex number z is a number of the form a

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Complex Numbers

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    1. Complex Numbers

    2. A First Observation There is no real number solution to the equation x2 + 1=0. The square root of –1 is called the imaginary unit and is denoted by i. This provides a solution to x2+1=0, 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 a+bi, and c+di are equal if and only if a = c and b = d

    6. Addition and Subtraction of Complex Numbers (a+bi) + (c+di) = (a+c)+(b+d)i (a+bi) – (c+di) = (a-c)+(b-d)i

    7. Multiplication of Complex Numbers (a+bi)(c+di) = (ac-bd)+(ad+bc)i Note that this is simply the first-outer-inner-last rule, and we replace i2 by –1.

    8. Complex Conjugates If z = a+bi, 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. (a+bi)(a-bi) = a2 + b2

    9. Division of Complex Numbers

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