M.1 U.1 Complex Numbers.

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Presentation transcript:

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. “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)

Imaginary Numbers What is the square root of 9?

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

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

Imaginary Numbers Simplify these radicals: =6xi =2y√5yi

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

Multiples of i Powers of i:

Powers of i - Practice i28 i75 i113 i86 i1089 1 -i i -1

Solutions Involving i Solve:

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

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

Practice (4 + 7i) - (2 - 3i) (3 - i) + (7i) (-3 + 2i) - (-3 + i) Add these Complex Numbers: (4 + 7i) - (2 - 3i) (3 - i) + (7i) (-3 + 2i) - (-3 + i) = 2 +10i = 3 + 6i = i

Multiplying FOIL and replace i2 with -1:

Practice Multiply: 5i(3 - 4i) (7 - 4i)(7 + 4i) = 20 +15i = 65

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:

Rationalizing Monomial: multiply the top & bottom by i.

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

Practice Simplify:

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

Graphing Complex Numbers

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

Check Your Answers

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