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Complex Numbers Section 0.7

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What if it isnt Real?? We have found the square root of a positive number like = 4, Previously when asked to find the square root of a negative number like we said there is not a real solution. To find the square root of a negative number we need to learn about complex numbers

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Imaginary unit The imaginary unit is represented by What would i² be??

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Simplify the following

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This can not be simplified any further. Your solution is a complex number that contains a real part (the 7) and an imaginary part (the 6i).

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Defining a Complex Number Complex numbers in standard form are written a + bi a is the real part of the complex number and bi as the imaginary part of the solution. If a = 0 then our complex number will only have the imaginary part (bi) and is called a pure imaginary number. Imaginary Number example: Complex Number example:

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Adding and Subtracting Complex Numbers To add and subtract, simply treat the i like a typical variable.

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Adding and subtracting complex numbers.

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Multiplying complex numbers Always write in the form a + bi (real part first, imaginary second)

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Multiply (2 + 3i)(2 – i) 4 + 4i – 3(-1) 4 + 4i + 3 7 + 4i

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Complex Conjugate The product of complex conjugates is a real number (imaginary part will be gone) (a + bi) and (a – bi) are conjugates. (a + bi)(a – bi) = a² - abi + abi - b²i² =a² - b²(-1) =a² + b²

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z = 2 + 4i Find z ( the conjugate of z) and then multiply z times z z = 2 – 4i zz = (2 + 4i)(2 – 4i) = 4 – 16 i² = 4 + 16 = 20

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Write the quotient in standard form Multiply numerator and denominator by conjugate Simplify remembering i² = -1 Write in standard form a + b = a + b c c c

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Write in Standard Form

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Powers for i 1 1

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Complex Numbers We haven’t been allowed to take the square root of a negative number, but there is a way: Define the imaginary number For example,

Complex Numbers We haven’t been allowed to take the square root of a negative number, but there is a way: Define the imaginary number For example,

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