Angle and absolute value of complex numbers

Angle and absolute value of complex numbers

There are times when we are interested in obtaining a better understanding of the properties of a complex number, such as its argument and modulus. In this section, we will learn how to calculate the argument, also known as the angle, and the modulus, also known as the magnitude or the absolute value, of a complex number.
Basic Concepts: Distance formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}, Solving expressions using 45-45-90 special right triangles
Related Concepts: Imaginary zeros of polynomials, Magnitude of a vector, Direction angle of a vector

Lessons

Notes:

Magnitude = modulus = absolute value
∣z∣=a2+b2 |z|= \sqrt{a^2+b^2}

Argument = angle
arg(z)=θ arg(z)=\theta
  • 1.
    Given the complex number z=2+3iz=2+3i
    a)
    Find its absolute value

    b)
    Find the angle it makes in the complex plane in radians

  • 2.
    Given the complex number w=5i−3 w=5i-3
    a)
    Find its modulus

    b)
    Find its argument in radians

  • 3.
    Given that a complex number ww makes an angle θ=3π4\theta=\frac{3\pi}{4} in the complex plane and has an absolute value ∣w∣=5|w|=5, write the complex number w in rectangular form.
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15.
Complex Numbers and Complex Plane
15.1
Introduction to imaginary numbers
15.2
Complex numbers and complex planes
15.3
Adding and subtracting complex numbers
15.4
Complex conjugates
15.5
Multiplying and dividing complex numbers
15.6
Distance and midpoint of complex numbers
15.7
Angle and absolute value of complex numbers
15.8
Polar form of complex numbers
15.9
Operations on complex numbers in polar form
Precalculus
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