Complex number in rectangular form what is (1+2j) + (1+3j)? Your answer should contain three significant figures.

This problem aims to find the real and the imaginary part of a complex number. The concept required to solve this problem includes complex numbers, conjugates, rectangular forms, polar forms, and magnitude of a complex number. Now, complex numbers are the numerical values that are represented in the form of:

$z = x + y\iota$

Where, $x$, $y$ are real numerals, and $\iota$ is an imaginary numeral and its value is$(\sqrt{-1})$. This form is called the rectangular coordinate form of a complex number.

The magnitude of a complex number can be obtained by taking the square root of the sum of squares of coefficients of the complex number, let’s say $z = x + \iota y$, the magnitude $|z|$, can be taken as:

$|z| = \sqrt{x^2 + y^2}$

One other way to think of magnitude is the distance of $(z)$ from the source of the complex number plane.

To find the polar form of the given complex number, we will first calculate their sum to construct a binomial form. Two complex numbers can be summed using the formula:

$= (a_1 + b_1\iota) + (a_2 + b_2\iota)$

$= (a_1 + a_2) + (b_1 + b_2)\iota$

$= (a + b\iota)$

The given complex numbers are $(1 + 2\iota) + (1 + 3\iota)$, substituting it gives us:

$= (1 + 2\iota) + (1 + 3 \iota)$

$= (1+ 1) + (2+ 3)\iota$

$= 2 + 5\iota$

The next step is to find the polar form, which is another way to express the rectangular coordinate form of a complex number. It is given as:

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

Where $(r)$ is the length of the vector, yielded as $r^2 = a^2+b^2$,

and $\theta$ is the angle created with the real axis.

Let’s calculate the value of $r$ by plugging in $a=2$ and $b=5$:

$r = \sqrt{a^2 + b^2}$

$r = \sqrt{2^2 + 5^2}$

$r = \sqrt{29}$

$r \approx 5.39$

Now finding the $\theta$:

$\theta = \tan^{-1}(\dfrac{b}{a})$

$\theta = \tan^{-1}(\dfrac{5}{2})$

$\theta = 68.2^{\circ}$

Plugging in these values in the above formula gives us:

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

$z = \sqrt{29}(\cos(68.2) +\iota \sin(68.2))$

Numerical Result

The polar form of the rectangular coordinate complex number is $z = \sqrt{29}(\cos(68.2) + \iota\sin(68.2))$.

Example

Express the rectangular form of $5 + 2\iota$ in polar form.

It is given as:

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

Calculating the value of $r$:

$r = \sqrt{a^2+b^2}$

$r = \sqrt{5^2+2^2}$

$r = \sqrt{29}$

Now finding the $\theta$:

$\tan\theta = (\dfrac{b}{a})$

$\theta = \tan^{-1}(\dfrac{b}{a})$

$\theta = \tan^{-1}(\dfrac{2}{5})$

$\theta = 0.38^{\circ}$

Plugging in these values in the above formula gives us:

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

$z = \sqrt{29}(\cos(0.38) +\iota\sin(0.38))$

$z = 5.39(\cos(0.38) + \iota\sin(0.38))$