Angle and absolute value of complex numbers - 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: $d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
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:

Notes:

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

Argument = angle
$arg(z)=\theta$

1.
Given the complex number $z=2+3i$
2.
Given the complex number $w=5i-3$

Angle and absolute value of complex numbers

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AlgebraDistance formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}d=(x2​−x1​)2+(y2​−y1​)2​

TrigonometrySolving expressions using 45-45-90 special right triangles

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Distance formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}d=(x2​−x1​)2+(y2​−y1​)2​

Solving expressions using 45-45-90 special right triangles

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Notes:

Notes: Magnitude = modulus = absolute value ∣z∣=a2+b2 |z|= \sqrt{a^2+b^2}∣z∣=a2+b2​ Argument = angle arg(z)=θ arg(z)=\thetaarg(z)=θ

1.

Given the complex number z=2+3iz=2+3i z=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 w=5i−3

a)

Find its modulus

b)

Find its argument in radians

3.

Given that a complex number www makes an angle θ=3π4\theta=\frac{3\pi}{4} θ=43π​ in the complex plane and has an absolute value ∣w∣=5|w|=5∣w∣=5, write the complex number w in rectangular form.

Angle and absolute value of complex numbers

Don't just watch, practice makes perfect.

We have over 1850 practice questions in Algebra for you to master.

Algebra
Distance formula: $d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Trigonometry
Solving expressions using 45-45-90 special right triangles

Distance formula: $d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

or

Notes:

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

Argument = angle
$arg(z)=\theta$

Given the complex number $z=2+3i$

Given the complex number $w=5i-3$

Given that a complex number $w$ makes an angle $\theta=\frac{3\pi}{4}$ in the complex plane and has an absolute value $|w|=5$ , write the complex number w in rectangular form.