Careers360 Logo
Engineering and Architecture
Search Colleges, Exams, Schools & more
JEE Main Registration 2025 Session 1 (Open) - Link, Last Date, Fees, How to Apply
  1. Home
  2. JEE Main
  3. Study Material
  4. Euler form of complex number - Practice Questions & MCQ

Euler form of complex number - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Euler form of complex number, Properties of argument of a complex number is considered one of the most asked concept.

  • 21 Questions around this concept.

Solve by difficulty

QuestionEASY

Let z,\omega  be complex numbers such that  \overline{z}+i\overline{\omega}=0  and  argz\omega =\pi Then arg z equals:

View Solution

Concepts Covered - 2

Euler form of complex number

The polar form of complex number z = r (cos ? + i sin ?)

In Euler form (cos ? + i sin ?) part of the polar form of complex numbers is represented by eiΘ. So, z = r (cos ? + i sin ?) is written as r.eiΘ in Euler's Form

We know the expansion of ex is

\\\text{The expansion of } e^x \,\,\text{is}\\\\\mathrm{e^x = 1 + \frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+.....} \\\\\mathrm{Replacing \;x\; by \; ix } \\\\\mathrm{e^{ix} = 1+\frac{ix}{1!}+\frac{(ix)^2}{2!} + \frac{(ix)^3}{3!}+ \frac{(ix)^4}{4!}+.... } \\\\\mathrm{e^{ix} = 1 + \frac{ix}{1!}-\frac{x^2}{2!}-\frac{ix^3}{3!}+\frac{x^4}{4!}+...+} \\\\\mathrm{rearranging\; the \; terms, we\; have} \\\\\mathrm{e^{ix} = (1-\frac{x^2}{2!}+\frac{x^4}{4!}) + i(x-\frac{x^3}{3!}+\frac{x^5}{5!})} \\\\\mathrm{We \; notice\; that\; first \; bracket\; is \; the \; expansion \; of \; \sin x \; } \\\mathrm{and \; 2nd \; bracket\; is \; the \; expansion\; of \;\cos x, so\; we \; have} \\\mathrm{e^{ix}=\sin x +i\cos x}

 

So, ei? = cos? + isin? and 

e-i? = cos? - isin?

 

Euler forms make algebra very simple for complex numbers in cases where multiplication, division or powers of complex numbers are involved. Any complex number can be expressed as

 

\\\mathrm{z=x+iy\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\(Cartesian\;form)}\\\mathrm{z=|z(\cos\theta+i\sin\theta)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(Polar\;form)}\\\mathrm{z=|z|e^{i\theta}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\(Eular's\;form)}

Application of Euler form:

1. Multiplication of two complex numbers:

\\\mathrm{Let \;z=\left | z \right |e^{i\theta_1}} \\\mathrm{And \;w=\left | w \right |e^{i\theta_2}} \\\mathrm{Multiplying\; these \;two \;number } \\\mathrm{z\cdot w= \left | z \right |e^{i\theta_1}\cdot \left | w \right |e^{i\theta_2}} \\\mathrm{= \left | z \right |\cdot \left | w \right |e^{i(\theta_1+\theta_2)}}

 

2. Division also can be done in the same way,

\\\mathrm{z=|z|e^{i\theta_1}\;\;and\;\;w=|w|e^{i\theta_2}\;be\;two\;complex\;number}\\\mathrm{\therefore\;\frac{z}{w}=\frac{|z|}{|w|}e^{i(\theta_1-\theta_2)}}

 

3. The logarithm of Complex Number

\\\mathrm{z=|z|e^{i\theta}}\\\mathrm{\log_e(z)=\log_e(|z|e^{i\theta})}\\\mathrm{\log_e(z)=\log_e(|z|)+\log_e(e^{i\theta})}\\\mathrm{\log_e(z)=\log_e(|z|)+{i\arg(z)}}

Properties of argument of a complex number

\\\mathrm{ i) \,\,arg(z_1z_2)=arg(z_1) + arg(z_2) + 2k\pi, \;\; k\; is \;an\;integer}   

Value of k should be chosen in such a way that arg lies between (-?,?]

This formula for argument can be generalised for n number of complex in similar way

\\\mathrm{ ii) arg\left ( \frac{z_1}{z_2} \right ) = arg(z_1) - arg(z_2) +2k\pi}   where k is an integer

\\\mathrm{iii)\,\, arg( \overline{z}) = - arg(z) }

\\\mathrm{iv) arg\left ( z^n \right ) = n\cdot arg(z) +2k\pi}  k belongs to an integer

\\\mathrm{(v)\;|z_1+z_2|=|z_1|+|z_2|\Rightarrow arg(z_1)=arg(z_2)}\\\mathrm{(vi)\;|z_1+z_2|=||z_1|-|z_2||\Rightarrow arg(z_1)-arg(z_2)=\pi}

Study it with Videos

Euler form of complex number
Properties of argument of a complex number

Study other Related Concepts

Euler form of complex number Current Topic

Powers of Iota
Complex number
Algebraic operation on Complex Numbers
Multiplication of Complex Numbers
Conjugates of Complex Numbers
Modulus of complex number and its Properties
Argument of complex number
Polar form of complex numbers
Euler form of complex number
Square root of complex numbers

"Stay in the loop. Receive exam news, study resources, and expert advice!"

Get Answer to all your questions

JEE Main Articles

NITs Cutoff 2024 for B. Tech CSE (Computer Science and Engineering)
Nov 08, 2024
JEE Main Information Bulletin 2025 Out - Direct Link To Download PDF
Nov 08, 2024
Is 75% Criteria Removed from JEE Main 2025? - Check Latest Updates
Nov 08, 2024
JEE Main New Rules announced by nta
Nov 08, 2024
Top 10 Engineering Entrance Exams in India 2025 - BTech Exams after 12th
Nov 08, 2024
JEE Main 2025 Syllabus PDF (Out) - Download Subject-Wise Detailed Syllabus Here
Nov 08, 2024
JEE Main 2025 - Registration (Open), Exam Dates Out, Syllabus, Pattern, Previous Year Papers
Nov 08, 2024
NITs Cutoff 2024 for B.Tech Mining Engineering - Check Opening and Closing Ranks
Nov 08, 2024

B.Tech/B.Arch Admissions OPEN

UPES B.Tech Admissions 2025
UPES B.Tech Admissions 2025
Apply

Ranked #42 among Engineering colleges in India by NIRF | Highest CTC 50 LPA , 100% Placements

Integral University B.Tech Admissions 2024
Integral University B.Tech Admissions 2024
Apply

NAAC A+ Accredited | Highest CTC 12 LPA | Scholarships Available

Parul University B.Tech Admissions 2025
Parul University B.Tech Admissions 2025
Apply

India's youngest NAAC A++ accredited University | NIRF rank band 151-200 | 2200 Recruiters | 45.98 Lakhs Highest Package

Dayananda Sagar University B.Tech Admissions 2025
Dayananda Sagar University B.Tech Admissions 2025
Apply

60+ Years of Education Legacy | UGC & AICTE Approved | Prestigious Scholarship Worth 6 Crores | H-CTC 35 LPA

Lovely Professional University B.Tech Admissions 2025
Lovely Professional University B.Tech Admissions 2025
Apply

India's Largest University | NAAC A++ Accredited | 100% Placement Record | Highest Package Offered : 3 Cr PA

Atlas SkillTech University | B.Tech Admissions
Atlas SkillTech University | B.Tech Admissions
Apply

Hands on Mentoring and Code Coaching | Cutting Edge Curriculum with Real World Application

View All Application Forms
News and Notifications
2 days ago :
October 18, 2024 - 09:55 AM IST :
BTech Admissions 2025 LIVE: JEE Main, MET registrations open; AEEE, VITEEE dates out
November 08, 2024 - 12:31 PM IST
BTech Admissions 2025: JEE Main registration last date, direct link
November 07, 2024 - 10:09 PM IST
JEE Main 2025 Registration: NTA issues advisory for Aadhaar authentication, name mismatch issues
November 06, 2024 - 05:53 PM IST
BTech Admissions 2025: JEE Main, AEEE registration underway; latest updates
November 05, 2024 - 10:30 PM IST
JEE aspirant found dead in Kota PG; police deny suicide
November 04, 2024 - 09:34 PM IST
JEE Main Syllabus 2025: Topic-wise weightage, marks for physics, chemistry, mathematics
November 04, 2024 - 06:11 PM IST
BTech Admissions 2025 LIVE: AEEE, JEE Mains registration underway; latest updates on MET, SRMJEE, VITEEE
November 04, 2024 - 10:49 AM IST
Download Careers360 App
All this at the convenience of your phone

  • Regular Exam Updates

  • Best College Recommendations

  • College & Rank predictors

  • Detailed Books and Sample Papers

  • Question and Answers

Scan and download the app

OR

Joint Entrance Examination (Main)      
JEE Main 2025 Sample Papers
Back to top