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M.1 U.1 Complex Numbers

M.1 U.1 Complex Numbers. What are imaginary numbers?. Viewed the same way negative numbers once were How can you have less than zero? Numbers which square to give negative real numbers.

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M.1 U.1 Complex Numbers

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  1. M.1 U.1Complex Numbers

  2. What are imaginary numbers? • Viewed the same way negative numbers once were • How can you have less than zero? • Numbers which square to give negative real numbers. • “I dislike the term “imaginary number” — it was considered an insult, a slur, designed to hurt i‘s feelings. The number i is just as normal as other numbers, but the name “imaginary” stuck so we’ll use it.” • Imaginary numbers deal with rotations, complex numbers deal with scaling and rotations simultaneously (we’ll discuss this further later in the week)

  3. Imaginary Numbers • What is the square root of 9? • What is the square root of -9?

  4. Imaginary Numbers • The constant, i, is defined as the square root of negative 1:

  5. Imaginary Numbers • The square root of -9 is an imaginary number...

  6. Imaginary Numbers • Simplify these radicals:

  7. Multiples of i • Consider multiplying two imaginary numbers: • So...

  8. Multiples of i • Powers of i:

  9. i28 i75 i113 i86 i1089 Powers of i - Practice

  10. Solutions Involving i • Solve: • Solve:

  11. Complex Numbers • Have a real and imaginary part . • Write complex numbers as a + bi • Examples: 3 - 7i, -2 + 8i, -4i, 5 + 2i Real = a Imaginary = bi

  12. Add & Subtract • Like Terms • Example: (3 + 4i) + (-5- 2i) = -2 + 2i

  13. Practice Add these Complex Numbers: • (4 + 7i) - (2 - 3i) • (3 - i) + (7i) • (-3 + 2i) - (-3 + i)

  14. Multiplying • FOIL and replace i2 with -1:

  15. Practice Multiply: • 5i(3 - 4i) • (7 - 4i)(7 + 4i)

  16. Division/Standard Form • A complex number is in standard form when there is no i in the denominator. • Rationalize any fraction with i in the denominator. Monomial Denominator: Binomial Denominator:

  17. Rationalizing • Monomial: multiply the top & bottom by i.

  18. Complex #: Rationalize • Binomial: multiply the numerator and denominator by the conjugateof the denominator ... conjugate is formed by negating the imaginary term of a binomial

  19. Practice • Simplify:

  20. Absolute Value of Complex Numbers • Absolute Value is a numbers distance from zero on the coordinate plane. • a = x-axis • b = y axis • Distance from the origin (0,0) = • |z| = √x2+y2 Modulus

  21. Graphing Complex Numbers

  22. Exit Ticket • Simplify (-2+4i) –(3+9i) • Write the following in standard form 8+7i 3+4i • Find the absolute value 4-5i

  23. Check Your Answers

  24. Homework • Complex Numbers worksheet • For #7, remember the quadratic formula!

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