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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.

Expert Answer

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:

\[ z = r(\cos\theta + \iota\sin\theta) \]

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

\[ z = 5.39(\cos(0.38) + \iota\sin(0.38)) \]

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