Question

find the modulus and the argument of the following complex number 3 - 3 i


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Answer 1

Given us the complex number, 3 - 3i

We have to find,

• Modulus of the complex number.
• Argument of the complex number.

We know, for any complex number in the form,

• x + iy

Where,

x and y are real and imaginary part of the complex number respectively.

Then, the modulus is given by

⇒ | x + iy | = √( x² + y² )

In the given complex number,

• Real part, x = 3
• Imaginary part, y = -3

So,

⇒ | 3 - 3i | = √{ (3)² + (-3)² }

⇒ | 3 - 3i | = √( 2 × 9 )

⇒ | 3 - 3i | = 3√2

So, The modulus of the given complex number is 3√2.

Let us find the argument of the given complex number, z = 3 - 3i where x = 3 and y = -3

So,

⇒ arg(z) = tan⁻¹ (3/-3)

⇒ arg(z) = tan⁻¹ (-1)

⇒ arg(z) = tan⁻¹ (tan 135°)

⇒ arg(z) = tan⁻¹ (tan 27π/36)

⇒ arg(z) = tan⁻¹ (tan 9π/12)

⇒ arg(z) = tan⁻¹ (tan 3π/4)

⇒ arg(z) = 3π / 4

Hence,

• Modulus = 3√2
• Argument = 3π / 4

Answer 2

Step-by-step explanation:

ANSWER ✍️

Given us the complex number, 3 - 3i

We have to find,

Modulus of the complex number.

Argument of the complex number.

We know, for any complex number in the form,

x + iy

Where,

x and y are real and imaginary part of the complex number respectively.

Then, the modulus is given by

⇒ | x + iy | = √( x² + y² )

In the given complex number,

Real part, x = 3

Imaginary part, y = -3

So,

⇒ | 3 - 3i | = √{ (3)² + (-3)² }

⇒ | 3 - 3i | = √( 2 × 9 )

⇒ | 3 - 3i | = 3√2

So, The modulus of the given complex number is 3√2.

Let us find the argument of the given complex number, z = 3 - 3i where x = 3 and y = -3

So,

⇒ arg(z) = tan⁻¹ (3/-3)

⇒ arg(z) = tan⁻¹ (-1)

⇒ arg(z) = tan⁻¹ (tan 135°)

⇒ arg(z) = tan⁻¹ (tan 27π/36)

⇒ arg(z) = tan⁻¹ (tan 9π/12)

⇒ arg(z) = tan⁻¹ (tan 3π/4)

⇒ arg(z) = 3π / 4

Hence,

Modulus = 3√2

Argument = 3π / 4

hope this helps you!!

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