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Complex Numbers

Complex Numbers. Section 0.7. What if it isn’t 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.

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Complex Numbers

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  1. Complex Numbers Section 0.7

  2. What if it isn’t 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

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

  4. Simplify the following

  5. Simplify the following 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).

  6. Defining a Complex Number • Complex numbers in standard form are written a + bi a is the real part of the complex number andbi 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:

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

  8. Adding and subtracting complex numbers.

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

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

  11. 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

  12. z = 2 + 4iFind 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

  13. 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

  14. Write in Standard Form

  15. Powers for i 1 -1 1

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