Find the modulus and amplitude of 1+√3i

To find the modulus and amplitude (or argument) of the complex number \( 1 + \sqrt{3}i \), we proceed as follows:

### 1. Modulus:
The modulus \( r \) of a complex number \( z = a + bi \) is given by:

\[
r = \sqrt{a^2 + b^2}
\]

For \( z = 1 + \sqrt{3}i \):
- \( a = 1 \)
- \( b = \sqrt{3} \)

So,

\[
r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2
\]

Thus, the modulus of \( 1 + \sqrt{3}i \) is \( 2 \).

### 2. Amplitude (Argument):
The amplitude (or argument) \( \theta \) of a complex number \( z = a + bi \) is given by:

\[
\theta = \tan^{-1}\left(\frac{b}{a}\right)
\]

For \( z = 1 + \sqrt{3}i \):
- \( a = 1 \)
- \( b = \sqrt{3} \)

So,

\[
\theta = \tan^{-1}\left(\frac{\sqrt{3}}{1}\right) = \tan^{-1}(\sqrt{3})
\]

We know that:

\[
\tan\left(\frac{\pi}{3}\right) = \sqrt{3}
\]

Thus,

\[
\theta = \frac{\pi}{3}
\]

Therefore, the amplitude (argument) of \( 1 + \sqrt{3}i \) is \( \frac{\pi}{3} \) radians.

### Final Answer:
- **Modulus:** \( 2 \)
- **Amplitude (Argument):** \( \frac{\pi}{3} \) radians

Do you want details or have any questions?

Here are some related questions:
1. What is the modulus and amplitude of \( 2 + 2i \)?
2. How do you convert a complex number from rectangular to polar form?
3. What is the general formula for finding the modulus of a complex number?
4. How do you find the argument of a complex number in different quadrants?
5. What is the geometric interpretation of modulus and amplitude in the complex plane?

**Tip:** The modulus represents the distance from the origin, while the amplitude is the angle with respect to the positive real axis.

Discover how to find the modulus and amplitude (argument) of the complex number 1 + √3i. This guide covers key concepts in complex numbers, including the formulas for modulus and argument calculation, with a step-by-step solution. Ideal for students in Grades 11-12.

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