COMPLEX NUMBERS Complex numbers in polar form are often writen as \(r \angle 0\). where \(r\) is the modulus and \(\theta\) is the argument. Express \(1+2 j\) in this way,

Understanding the modulus of a complex number is an essential part of grasping complex number theory. Think of the modulus as the distance from the origin to the point in the complex plane that represents the complex number. It's similar to how you would measure the straight-line distance between two points on a map. To calculate the modulus, you use the formula \( r = \sqrt{a^2 + b^2} \), where \( a \) and \( b \) represent the real and imaginary components of the complex number, respectively.

Take our example \( 1 + 2j \): the real part \( a = 1 \) and the imaginary part \( b = 2 \). Plugging these into the formula, we get the modulus \( r = \sqrt{1^2 + 2^2} = \sqrt{5} \). This modulus represents the 'length' of the vector from the origin to the point \( (1, 2) \) in the complex plane.

The next element of a complex number in polar form is its argument. If the modulus tells us how far the number is from the origin, the argument tells us the direction. More precisely, it's the angle the line segment between the origin and the number makes with the positive real axis, measured in a counter-clockwise direction.

To find this angle, we use the formula \( \theta = \arctan(\frac{b}{a}) \), with \(a\) as the real part and \(b\) as the imaginary part. For \(1 + 2j\), the argument is \( \theta = \arctan(\frac{2}{1}) = \arctan(2) \). This calculation tells us the exact angle we need to rotate a vector with the same length as the modulus from the positive real axis to point towards the complex number in question.

Polar form is a different way of expressing complex numbers that showcases their geometric properties by utilizing a vector's magnitude and direction. After calculating the modulus and argument, as shown in the previous sections, we can represent any complex number in the form \( r \angle \theta \), where \( r \) is the modulus and \( \theta \) is the argument. It's a concise and visually intuitive way to describe complex numbers.

Looking at our example \(1 + 2j\), after applying the modulus and argument calculations, we express it in polar form as \( \sqrt{5} \angle \arctan(2) \). This notation succinctly captures both the distance of the point from the origin and its angle relative to the positive real axis.

Converting complex numbers between their standard \(x + yj\) form and their polar form can be thought of as translating between two languages. The standard form gives us a clear numerical picture of the real and imaginary parts, while the polar form gives us a geometric interpretation.

In converting \(1 + 2j\) to its polar form, we used the formulas for modulus and argument to transform into \( \sqrt{5} \angle \arctan(2) \). Conversely, to return to the standard form from polar form, we would use the modulus and argument to reconstruct the original real and imaginary parts using the cos and sin functions:

• The real part is \( r \cdot \cos(\theta) \)
• The imaginary part is \( r \cdot \sin(\theta) \)

This back-and-forth is a crucial skill for anyone working with complex numbers in various applications and fields.