Presentation on theme: "M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once were – –How can you have less than zero? n n Numbers."— Presentation transcript:
M.1 U.1 Complex Numbers
What are imaginary numbers? n n Viewed the same way negative numbers once were – –How can you have less than zero? n n 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.” n n Imaginary numbers deal with rotations, complex numbers deal with scaling and rotations simultaneously (we’ll discuss this further later in the week)
Imaginary Numbers n What is the square root of 9? n What is the square root of -9?
Imaginary Numbers n The constant, i, is defined as the square root of negative 1:
Imaginary Numbers n The square root of -9 is an imaginary number...
Imaginary Numbers n Simplify these radicals: =6xi=2y√5yi
Multiples of i n Consider multiplying two imaginary numbers: n So...
Multiples of i n Powers of i:
Powers of i - Practice n i 28 n i 75 n i 113 n i 86 n i i-iii
Solutions Involving i n Solve: Solve: Solve:
Complex Numbers n Have a real and imaginary part. n Write complex numbers as a + bi n Examples: 3 - 7i, i, -4i, 5 + 2i Real = a Imaginary = bi
Add & Subtract n Like Terms n Example: (3 + 4i) + (-5 - 2i) = i
Practice Add these Complex Numbers: n (4 + 7i) - (2 - 3i) n (3 - i) + (7i) n (-3 + 2i) - (-3 + i) = 2 +10i= 3 + 6i= i
Multiplying n FOIL and replace i 2 with -1:
Practice Multiply: n 5i(3 - 4i) n (7 - 4i)(7 + 4i) = i= 65
n A complex number is in standard form when there is no i in the denominator. n Rationalize any fraction with i in the denominator. Binomial Denominator: Monomial Denominator: Division/Standard Form
Rationalizing n Monomial: multiply the top & bottom by i.
Complex #: Rationalize n Binomial: multiply the numerator and denominator by the conjugate of the denominator... conjugate is formed by negating the imaginary term of a binomial
Practice n Simplify:
Absolute Value of Complex Numbers n 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| = √x 2 +y 2
Graphing Complex Numbers
Exit Ticket n Simplify (-2+4i) –(3+9i) n Write the following in standard form 8+7i 3+4i 3+4i n Find the absolute value 4-5i
Check Your Answers n
Homework n Complex Numbers worksheet –For #7, remember the quadratic formula!