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Complex Numbers

Complex Numbers. Modulo can be thought of as a round key ring. When we get the end, we simply start over. A number plus the modulo number gives us the original number back. Think of Monday plus 7 days – it’s Monday again!

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Complex Numbers

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  1. Complex Numbers • Modulo can be thought of as a round key ring. When we get the end, we simply start over. • A number plus the modulo number gives us the original number back. Think of Monday plus 7 days – it’s Monday again! • If we go past the starting point, we can find our position on the Modulo ring by subtracting the modulo number

  2. Modulo • Modulo can be thought of as a round key ring. When we get the end, we simply start over. • A number plus the modulo number gives us the original number back. Think of Monday plus 7 days – it’s Monday again! • If we go past the starting point, we can find our position on the Modulo ring by subtracting the modulo number

  3. Modulo • What is 4 + 5 modulo 7? Every time we get to 7, the next number is 1. Think days of the week. Every time we get the Sunday we go to Monday. • 4 + 5 = 9, 9 – 7 = 2, so the answer is 2 • (Thursday plus 5 days is Tuesday)

  4. Modulo • What is 10 + 3 modulo 12? Every time we get to 12, the next number is 1. Think months of the year. Every time we get to December, we go to January. • 10 + 3 = 13, 13 – 12 = 1, so the answer is 1 • (October plus 3 months is January)

  5. Imaginary Numbers • Definition: • Imaginary numbers are modulo 4

  6. Imaginary Numbers • Definition: • Do pages imaginary number task • Imaginary numbers are modulo 4

  7. Imaginary Numbers • Imaginary numbers are written in the form a + bi, where a is the real component, and b is the imaginary component. • Imaginary numbers can be graphed with the real part on the x-axis, and the imaginary part on the y-axis. • The magnitude of an imaginary number is found by Pythagorean Theorem

  8. Adding and Subtracting Imaginary Numbers • Imaginary numbers may be added by adding the real parts and then the imaginary parts • Imaginary numbers may be subtracted by subtracting the real and then the imaginary parts

  9. Multiplying Imaginary Numbers • Imaginary numbers may be multiplied by the distributive rule. • Remember that i2 = -1

  10. Dividing by Imaginary Numbers • Imaginary numbers may be divided into real or complex numbers by rationalizing the denominator.

  11. Rationalizing the Denominator • Rationalizing the denominator includes eliminating all radicals in the denominator, which includes i. • If the denominator has only the imaginary component, multiply the numerator and denominator by i. • If the denominator consists of both real and imaginary components, multiply the numerator and denominator by the conjugate of the denominator

  12. Rationalizing the Denominator • If the denominator has both the real and imaginary components of the complex number, multiply the numerator and denominator by the conjugate. • Example:

  13. Practice • Page 84, # 1 – 36 all (do some in class)

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