Question on first year engineering mathematics complex numbers

Steps to Solve

1. Identify Complex Numbers

Complex numbers are usually represented in the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part. For example, in the complex number $3 + 4i$, $3$ is the real part and $4$ is the imaginary part.

2. Perform Operations

Depending on what the problem is asking (addition, subtraction, multiplication, etc.), perform the required operation. For example, to add two complex numbers $z_1 = a + bi$ and $z_2 = c + di$, you would do:

$$ z_1 + z_2 = (a + c) + (b + d)i $$

3. Conjugate of a Complex Number

The conjugate of a complex number $z = a + bi$ is $z^* = a - bi$. This is useful for dividing complex numbers.

4. Magnitude of a Complex Number

The magnitude (or modulus) of a complex number $z = a + bi$ is calculated using the formula:

$$ |z| = \sqrt{a^2 + b^2} $$

5. Express in Polar Form

To express a complex number in polar form, use the magnitude $r = |z|$ and the angle $\theta = \tan^{-1}(\frac{b}{a})$. The polar form is:

$$ z = r(\cos(\theta) + i\sin(\theta)) $$

6. Consider Applications

If the problem involves applications, such as electrical engineering, consider how complex numbers represent phasors in AC circuits.