The purpose of this guide is to solve the given set of **complex numbers** in** rectangular form** and find their **magnitude, angle, and polar form**.

The basic concept behind this article is the **Complex Numbers**, their **Addition or Subtraction**, and their **Rectangular** and **Polar forms**.

A **Complex number** can be thought of as a combination of a **Real Number** and an **Imaginary Number,** which is usually represented in **rectangular form** as follows:

\[z=a+ib\]

Where:

$a\ ,\ b\ =\ Real\ Numbers$

$z\ =\ Complex\ Number$

$i\ =\ Iota\ =\ Imaginary\ Number$

Part $a$ of the above equation is called the **Real Part,** whereas value $ib$ is called the **Imaginary Part**.

## Expert Answer

Given that:

**First Complex Number** $= 1+2i$

**Second Complex Number** $= 1+3i$

The **sum of two complex numbers** $(a+ib)$ and $(c+id)$ in **rectangular form** is calculated as follows by operating on **real** and **imaginary parts** separately:

\[(a+ib)+(c+id)\ =\ (a+c)+i(b+d)\]

By substituting the given **complex numbers** in above equation, we get:

\[\left(1+2i\right)+\left(1+3i\right)\ =\ \left(1+1\right)+i\left(2+3\right)\]

\[\left(1+2i\right)+\left(1+3i\right)\ =\ 2+5i\]

So:

\[Sum\ of\ Complex\ Numbers\ =\ 2+5i\]

This is the **binomial form** of the **sum of complex numbers** represented in $x$ and $y$ **coordinates** as $x=2$ and $y=5$.

In order to find the **magnitude** $A$ of the given **sum of complex numbers**, we will use **Pythagoras’s Theorem of Triangles** to find the **hypotenuse** of the **Triangular Form** of the **complex numbers**.

\[A^2\ =\ x^2+y^2\]

\[A\ =\ \sqrt{x^2+y^2}\]

By substituting the values of both $x$ and $y$, we get:

\[A\ =\ \sqrt{2^2+5^2}\]

\[A\ =\ \sqrt{4+25}\]

\[A\ =\ \sqrt{29}\]

Hence, the **magnitude** $A$ of the given** sum of complex numbers** is $\sqrt{29}$.

The **angle of the complex numbers** is defined as follows if their real numbers are positive:

\[\tan{\theta\ =\ \frac{y}{x}}\]

By substituting the values of both $x$ and $y$, we get:

\[\tan{\theta\ =\ \frac{5}{2}}\]

\[\theta\ =\ \tan^{-1}{\left(\frac{5}{2}\right)}\]

\[\theta\ =\ 68.2°\]

**Euler’s identity** can be used to convert **Complex Numbers** from a **rectangular form** into a** polar form** represented as follows:

\[A\angle\theta\ =\ x+iy\]

Where:

\[x\ =\ A\cos\theta \]

\[y\ =\ A\sin\theta \]

Hence:

\[A\angle\theta\ =\ A\cos\theta\ +\ iA\sin\theta \]

\[A\angle\theta\ =\ A(\cos\theta\ +\ i\sin\theta) \]

Substituting the value of $A$ and $\theta$, we get:

\[\sqrt{29}\angle68.2° = 29 [\cos(68.2°) + i \sin(68.2°)]\]

## Numerical Result

For the given **set of complex numbers** in **rectangular form** $(1+2i)+(1+3i)$

The **Magnitude** $A$ of the **Sum of Complex Numbers** is:

\[A\ =\ \sqrt{29}\]

The **Angle** $\theta$ of **Complex Number** is:

\[\theta\ =\ 68.2°\]

The **Polar Form** $A\angle\theta$ of **Complex Number** is:

\[\sqrt{29}\angle68.2° = 29 [\cos(68.2°) + i \sin(68.2°)]\]

## Example

Find the **magnitude** of the **Complex numbers** in the **rectangular form** represented by $(4+1i)\times(2+3i)$.

**Solution**

Given that:

**First Complex Number** $= 4+1i$

**Second Complex Number** $= 2+3i$

The **Multiplication** **of two complex numbers** $(a+ib)$ and $(c+id)$ in **rectangular form** is calculated as follows:

\[(a+ib)\times(c+id)\ =\ ac+iad+ibc+i^2bd\]

As:

\[i^2={(\sqrt{-1})}^2=-1\]

Hence:

\[(a+ib)\times(c+id)\ =\ ac+i(ad+bc)-bd\]

Now, by substituting the given complex number in above expression for multiplication:

\[(4+1i)\times(2+3i)\ =\ 8+12i+2i+3i^2\]

\[(4+1i)\times(2+3i)\ =\ 8+14i-3\ =\ 5+14i\]

By using **Pythagoras’ Theorem**:

\[A\ =\ \sqrt{x^2+y^2}\]

\[A\ =\ \sqrt{5^2+{14}^2}\]

\[A\ =\ \sqrt{221}=14.866\]