# Find the modulus and argument of the complex number $\dfrac{1+2i}{1-3i}$.

Answer

Verified

363.3k+ views

Hint: To solve this question, we should be aware about the basic properties of complex numbers. For a complex number, say a+bi, we have,

Modulus of this complex number = $\sqrt{{{a}^{2}}+{{b}^{2}}}$

Argument = ${{\tan }^{-1}}\dfrac{b}{a}$

Further, to solve this problem, we will make use of rationalization along with the above two properties to solve. Also, keep in mind that ${{i}^{2}}=-1$.

Complete step-by-step answer:

We have, the complex number,

=$\dfrac{1+2i}{1-3i}$

To bring it in the form, a+bi, we do rationalization, thus, we have,

=$\dfrac{1+2i}{1-3i}\times \dfrac{1+3i}{1+3i}$

Thus, multiplying, we have,

=$\dfrac{(1+2i)(1+3i)}{(1-3i)(1+3i)}$

=$\dfrac{1+3i+2i+6{{i}^{2}}}{({{1}^{2}}-{{(3i)}^{2}})}$

Now, we use, ${{i}^{2}}=-1$ to solve further,

=$\dfrac{1+5i-6}{1+9}$

=$\dfrac{5i-5}{10}$

=$\dfrac{5(i-1)}{10}$

=$\dfrac{i-1}{2}$

=-0.5+0.5i -- (A)

Now, we have got the expression in the form a+bi, thus, we can find modulus and argument using the following properties-

Modulus of this complex number = $\sqrt{{{a}^{2}}+{{b}^{2}}}$ -- (1)

Argument = ${{\tan }^{-1}}\dfrac{b}{a}$ -- (2)

Thus, we have,

Modulus of -0.5+0.5i =$\sqrt{{{(-0.5)}^{2}}+{{0.5}^{2}}}=\sqrt{0.25+0.25}=\sqrt{0.5}$=0.707 (approximately)

Argument of -0.5+0.5i = ${{\tan }^{-1}}\left( \dfrac{0.5}{-0.5} \right)={{\tan }^{-1}}(-1)=\dfrac{3\pi }{4}$

It is important to note that the argument would not be $\dfrac{-\pi }{4}$since the real part (concerning cos$\theta $) is negative and imaginary part (concerning sin$\theta $) is negative, thus the angle should belong to the second quadrant. Thus, for angles in the second quadrant, only $\tan \left( \dfrac{3\pi }{4} \right)=-1$.

Hence, the modulus and argument of the complex number $\dfrac{1+2i}{1-3i}$ are 0.707 and $\dfrac{3\pi }{4}$respectively.

Note: While solving problems related to complex numbers concerning modulus and argument of the complex number, it is important to keep in mind the basic properties of complex numbers like addition, subtraction, multiplication and division of the complex number. While, addition and subtraction are quite intuitive since they are similar to the properties of real numbers, one should be careful while performing multiplication and division. While multiplication, one must keep in mind the fact that ${{i}^{2}}=-1$ and for division, in most cases, one needs to perform rationalization. Further, it is also suggested to know about the basic properties of inverse trigonometric functions since they are used while finding the argument of the complex number.

Modulus of this complex number = $\sqrt{{{a}^{2}}+{{b}^{2}}}$

Argument = ${{\tan }^{-1}}\dfrac{b}{a}$

Further, to solve this problem, we will make use of rationalization along with the above two properties to solve. Also, keep in mind that ${{i}^{2}}=-1$.

Complete step-by-step answer:

We have, the complex number,

=$\dfrac{1+2i}{1-3i}$

To bring it in the form, a+bi, we do rationalization, thus, we have,

=$\dfrac{1+2i}{1-3i}\times \dfrac{1+3i}{1+3i}$

Thus, multiplying, we have,

=$\dfrac{(1+2i)(1+3i)}{(1-3i)(1+3i)}$

=$\dfrac{1+3i+2i+6{{i}^{2}}}{({{1}^{2}}-{{(3i)}^{2}})}$

Now, we use, ${{i}^{2}}=-1$ to solve further,

=$\dfrac{1+5i-6}{1+9}$

=$\dfrac{5i-5}{10}$

=$\dfrac{5(i-1)}{10}$

=$\dfrac{i-1}{2}$

=-0.5+0.5i -- (A)

Now, we have got the expression in the form a+bi, thus, we can find modulus and argument using the following properties-

Modulus of this complex number = $\sqrt{{{a}^{2}}+{{b}^{2}}}$ -- (1)

Argument = ${{\tan }^{-1}}\dfrac{b}{a}$ -- (2)

Thus, we have,

Modulus of -0.5+0.5i =$\sqrt{{{(-0.5)}^{2}}+{{0.5}^{2}}}=\sqrt{0.25+0.25}=\sqrt{0.5}$=0.707 (approximately)

Argument of -0.5+0.5i = ${{\tan }^{-1}}\left( \dfrac{0.5}{-0.5} \right)={{\tan }^{-1}}(-1)=\dfrac{3\pi }{4}$

It is important to note that the argument would not be $\dfrac{-\pi }{4}$since the real part (concerning cos$\theta $) is negative and imaginary part (concerning sin$\theta $) is negative, thus the angle should belong to the second quadrant. Thus, for angles in the second quadrant, only $\tan \left( \dfrac{3\pi }{4} \right)=-1$.

Hence, the modulus and argument of the complex number $\dfrac{1+2i}{1-3i}$ are 0.707 and $\dfrac{3\pi }{4}$respectively.

Note: While solving problems related to complex numbers concerning modulus and argument of the complex number, it is important to keep in mind the basic properties of complex numbers like addition, subtraction, multiplication and division of the complex number. While, addition and subtraction are quite intuitive since they are similar to the properties of real numbers, one should be careful while performing multiplication and division. While multiplication, one must keep in mind the fact that ${{i}^{2}}=-1$ and for division, in most cases, one needs to perform rationalization. Further, it is also suggested to know about the basic properties of inverse trigonometric functions since they are used while finding the argument of the complex number.

Last updated date: 28th Sep 2023

â€¢

Total views: 363.3k

â€¢

Views today: 4.63k

Recently Updated Pages

What do you mean by public facilities

Difference between hardware and software

Disadvantages of Advertising

10 Advantages and Disadvantages of Plastic

What do you mean by Endemic Species

What is the Botanical Name of Dog , Cat , Turmeric , Mushroom , Palm

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

One cusec is equal to how many liters class 8 maths CBSE

The equation xxx + 2 is satisfied when x is equal to class 10 maths CBSE

What is the color of ferrous sulphate crystals? How does this color change after heating? Name the products formed on strongly heating ferrous sulphate crystals. What type of chemical reaction occurs in this type of change.

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE