SN PLUS 3
SN PLUS 3

Suggestions

Use up and down arrows to review and enter to select.
  • Shakespeare dark gray
    • No Fear Shakespeare Translations
    • Shakespeare Study Guides
    • Shakespeare Life & Times
    • Glossary of Shakespeare Terms
  • Literature dark gray
    • No Fear Literature Translations
    • Literature Study Guides
    • Glossary of Literary Terms
    • How to Write Literary Analysis
  • Other Subjects dark gray
    • Biography
    • Biology
    • Chemistry
    • Computer Science
    • Drama
    • Economics
    • Film
    • Health
    • History
    • Math
    • Philosophy
    • Physics
    • Poetry
    • Psychology
    • Short Stories
    • Sociology
    • US Government and Politics
  • Test Prep PLUS dark gray
    • Test Prep Lessons
    • AP® English Literature
    • AP® English Language
  • Teacher dark gray
    • SparkTeach
    • Teacher's Handbook
  • Blog
My PLUS Dashboard
  • My PLUS Activity dark gray
    • Notes
    • Bookmarks
    • Test Prep PLUS
    • No Fear Translations & Audio
    • Mastery Quizzes
    • Flashcards
    • Infographics
    • No Fear Graphic Novels
  • Account Details
  • Subscription & Billing

Please wait while we process your payment

Reset Password

  • Please wait while we process your payment

    Log in Sign up

    Sparknotes

  • By signing up you agree to our terms and privacy policy.

    Don’t have an account? Subscribe now

    Create Your Account

    Sign up for your FREE 7-day trial

    • Ad-free experience
    • Study notes
    • Flashcards
      & Quizzes
    • AP® English Test Prep
    • Plus much more

  • Already have an account? Log in

    Your Email

    Choose Your Plan

    BEST VALUE

    Save over 50% with a SparkNotes PLUS Annual Plan!

    Continuing to Payment will take you to astripe-imagepayment page

    Purchasing SparkNotes PLUS for a group?

    Get Annual Plans at a discount when you buy 2 or more!

  • Price

    $24.99 $18.74 /subscription + tax

    Subtotal $37.48 + tax

    Save 25% on 2-49 accounts

    Save 30% on 50-99 accounts

    Want 100 or more? Contact us for a customized plan.

    Your Plan

    Payment Details

  • We're sorry, SparkNotes Plus isn't available in your country.

  • Payment Details

    Payment Summary

    SparkNotes Plus

    You'll be billed after your free trial ends.

    7-Day Free Trial

    Not Applicable

    Renews April 4, 2023 March 28, 2023

    Discounts (applied to next billing)

    DUE NOW

    US $0.00

    SNPLUSROCKS20  |  20% Discount

    This is not a valid promo code.

    Discount Code (one code per order)

    SparkNotes Plus subscription is $4.99/month or $24.99/year as selected above. The free trial period is the first 7 days of your subscription. TO CANCEL YOUR SUBSCRIPTION AND AVOID BEING CHARGED, YOU MUST CANCEL BEFORE THE END OF THE FREE TRIAL PERIOD. You may cancel your subscription on your Subscription and Billing page or contact Customer Support at custserv@bn.com. Your subscription will continue automatically once the free trial period is over. Free trial is available to new customers only.

    Choose Your Plan

    Suggestions

    Use up and down arrows to review and enter to select.
    • My Account Icon My Account white
      • My PLUS Activity
        • Notes
        • Bookmarks
        • Test Prep PLUS
        • No Fear Translations
        • Mastery Quizzes
        • Flashcards
        • Infographics
        • No Fear Graphic Novels
      • Account Details
      • Subscription & Billing
      My PLUS Dashboard
    • Shakespeare white
      • No Fear Shakespeare Translations
      • Shakespeare Study Guides
      • Shakespeare Life & Times
      • Glossary of Shakespeare Terms
    • Literature white
      • No Fear Literature Translations
      • Literature Study Guides
      • Glossary of Literary Terms
      • How to Write Literary Analysis
    • Other Subjects white
      • Biography
      • Biology
      • Chemistry
      • Computer Science
      • Drama
      • Economics
      • Film
      • Health
      • History
      • Math
      • Philosophy
      • Physics
      • Poetry
      • Psychology
      • Short Stories
      • Sociology
      • US Government and Politics
    • Test Prep PLUS white
      • Test Prep Lessons
      • AP® English Literature
      • AP® English Language
    • Teacher white
      • SparkTeach
      • Teacher's Handbook
    • Blog
    • Help

    Please wait while we process your payment

    expired-logo

    Your PLUS subscription has expired

    • We’d love to have you back! Renew your subscription to regain access to all of our exclusive, ad-free study tools.
    Renew your subscription

    Please wait while we process your payment

    expired-logo

    snpromo-logo
    • Looking for exclusive, AD-FREE study tools? Look no further!

    Complex Numbers

    Math

    Study Guide
    • Study Guide
    • Topics
      • Complex Numbers
      • Introduction to Complex Numbers
      • Problems
      • Operations With Complex Numbers
      • Problems
      • Complex Roots
      • Problems
      • Polar Form of Complex Numbers
      • Problems
    • Terms and Formulae

    Please wait while we process your payment

    snplus-logo

    Unlock your FREE SparkNotes PLUS trial!

    Unlock your FREE Trial!

    Sign up and get instant access to bookmarks.
    • Ad-Free experience
    • Easy-to-access study notes
    • Flashcards & Quizzes
    • AP® English test prep
    • Plus much more
    Already have an account? Log in

    Terms and Formulae

    Terms and Formulae

      Terms

    • Argument

      The angle created by the positive real axis and the segment connecting the origin to the plot of a complex number in the complex plane.

    • Complex Conjugate

      The complex conjugate of a given complex number a + bı is a - bı.

    • Complex Number

      The set of all numbers of the form a + bı, where a and b are real numbers. The real numbers are all complex numbers.

    • Complex Plane

      A plane with two perpendicular axes, the real axis and the imaginary axis, on which a complex number a + bı is plotted at the coordinate (a, b). It is customary for the real axis to coincide with the x-axis of the rectangular coordinate system, and for the imaginary axis to coincide with the y-axis of the rectangular coordinate system.

    • Imaginary Axis

      The axis in the complex plane that customarily coincides with the y-axis of the rectangular coordinate system, and on which the imaginary part bı of the complex number a + bı is plotted.

    • Imaginary Number

      A number that can be expressed in the form bı, where b is a real number.

    • Imaginary Part

      The bı term in every complex number a + bı.

    • Imaginary Unit

      The imaginary unit is ı. ı = .

    • Modulus

      for a complex number a + bı. In the complex plane, it is the distance between the plot of a complex number and the origin.

    • Polar Form of a Complex Number

      The polar form of a complex number z = a + bı is this: z = r(cos(θ) + ısin(θ)), where r = | z| and θ is the argument of z.

    • Real Axis

      The axis in the complex plane that typically coincides with the x-axis of the rectangular coordinate system, and on which the real part a of a complex numbers a + bı is plotted.

    • Real Part

      In a complex number a + bı, a.

    • Standard Form

      For a complex number, a + bı.

    • Formulae

       
      De Moivre's Theorem Let z = r(cos(θ) + ısin(θ).Thenzn = [r(cos(θ) + ısin(θ)]n = rn(cos(nθ) + ısin(nθ), where n is any positive integer.
       
      Roots of a Complex Number A complex number z = r(cos(θ) + ısin(θ) has exactly nnth roots given by the equation [cos() + ısin()], where n is a positive integer, and k = 0, 1, 2,..., n - 2, n - 1.
    Did you know you can highlight text to take a note? x

    Please wait while we process your payment

    snplus-logo

    Unlock your FREE SparkNotes PLUS trial!

    Unlock your FREE Trial!

    Sign up and get instant access to creating and saving your own notes as you read.
    • Ad-Free experience
    • Easy-to-access study notes
    • Flashcards & Quizzes
    • AP® English test prep
    • Plus much more
    Already have an account? Log in

    Take a Study Break

    • QUIZ: Is This a Taylor Swift Lyric or a Quote by Edgar Allan Poe?

    • The 7 Most Embarrassing Proposals in Literature

    • The 6 Best and Worst TV Show Adaptations of Books

    • QUIZ: Which Greek God Are You?

    Sign up for our latest news and updates!
    By entering your email address you agree to receive emails from SparkNotes and verify that you are over the age of 13. You can view our Privacy Policy here. Unsubscribe from our emails at any time.

    SparkNotes—the stress-free way to a better GPA

    • Quick Links
    • No Fear Shakespeare
    • Literature Guides
    • Other Subjects
    • Blog
    • Teacher’s Handbook
    • Premium Study Tools
    • SparkNotes PLUS
    • Sign Up
    • Log In
    • PLUS Help
    • More
    • Help
    • How to Cite SparkNotes
    • How to Write Literary Analysis
    • About
    • Contact Us

    Copyright © SparkNotes LLC

    • Terms of Use
    • |
    • Privacy
    • |
    • Cookie Policy
    • |
    • Do Not Sell My Personal Information