Sec 2 5 b More with Complex Numbers
Sec. 2. 5 b More with Complex Numbers
Definition: Complex Conjugate The complex conjugate of the complex number is What happens when we multiply a complex number by its conjugate? ? ? This is a positive real number!!!
Practice Problems Write the given complex numbers in standard form.
Complex Solutions of Quadratic Equations Remind me of the quadratic formula!!! What’s this part called? The discriminant!!! It can be used to tell whether the solutions to a particular quadratic equation are real numbers…
Discriminant of a Quadratic Equation For a quadratic equation are real numbers and , where a, b, and c , • If , there are two distinct real solutions. • If , there is one repeated real solution. • If of solutions. , there is a complex conjugate pair
Practice Problems Solve a = b = c = 1 Use the quadratic formula! A complex conjugate pair
Guided Practice Write the given complex number in standard form.
Guided Practice Write the given expression in standard form.
Guided Practice Write the given expression in standard form.
The Complex Plane Imaginary Axis Real Axis
The Complex Plane Plot u = 1 + 3 i, v = 2 – i, and u + v in the complex plane. Imaginary Axis Notice that the two complex numbers, their sum, and the origin form a quadrilateral (what type? ) Real Axis A Parallelogram!!!
Definition: Absolute Value of a Complex Number The absolute value, or modulus, of the complex number , where a and b are real numbers, is Imaginary Axis Real Axis
A Few More New Formulas The distance between the points u and v in the complex plane: The midpoint of the line segment connecting u and v in the complex plane:
A Few More New Formulas Find the distance between u = – 4 + i and v = 2 + 5 i in the complex plane, and find the midpoint of the segment connecting u and v. Distance: Midpoint: Can we verify these answers graphically?
Whiteboard Problems… Write the given complex number in standard form.
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