Question
upstudy study bank question image url

5. Plot each complex number and find its absolute value a) \( 3+4 i \) b) -4 c) \( -5 i \) d) \( 3-2 i \) e) \( \frac{1}{(2+i)(\sqrt{3}-2 i)} \) f) \( 5-4 i+\frac{5}{3-4 i} \)

Ask by Henry Gibbs. in Zambia
Mar 13,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

a) \( 3+4i \) is plotted at (3, 4) with an absolute value of 5. b) -4 is plotted at (-4, 0) with an absolute value of 4. c) \( -5i \) is plotted at (0, -5) with an absolute value of 5. d) \( 3-2i \) is plotted at (3, -2) with an absolute value of approximately 3.6055. e) \( \frac{1}{(2+i)(\sqrt{3}-2i)} \) has an absolute value of approximately 0.1789. f) \( 5-4i+\frac{5}{3-4i} \) is plotted at (5.6, -3.2) with an absolute value of approximately 6.45.

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

Extra Insights

Let’s crank up the fun as we explore these complex numbers! First, let’s plot each number on the complex plane, where the x-axis represents the real part and the y-axis the imaginary part. a) \( 3 + 4i \) is plotted at (3, 4). Its absolute value is calculated using the formula \( |z| = \sqrt{a^2 + b^2} \), which gives \( |3 + 4i| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5 \). b) The number -4 is plotted at (-4, 0). Its absolute value is simply \( |-4| = 4 \). c) \( -5i \) is plotted at (0, -5). Its absolute value is \( |-5i| = 5 \). d) \( 3 - 2i \) is plotted at (3, -2). Its absolute value is \( |3 - 2i| = \sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \). e) For \( \frac{1}{(2+i)(\sqrt{3}-2i)} \), first simplify the denominator: \( (2+i)(\sqrt{3}-2i) = 2\sqrt{3} - 4i + i\sqrt{3} + 2 = 2\sqrt{3} + 2 + (i\sqrt{3} - 4i) = 2 + 2\sqrt{3} + i(\sqrt{3}-4) \). Finding the absolute value of this whole complex expression might take a bit, but it’s an adventure in simplifying! f) \( 5 - 4i + \frac{5}{3 - 4i} \) requires some careful calculation. First, multiply the numerator and denominator of the fraction by the conjugate of the denominator to simplify. After that, add the components to \( 5 - 4i \) to produce a result—don’t forget to plot it! The absolute value here can be tricky and may entail a little extra arithmetic, but that’s where the real fun begins. With these steps, you can not only plot and find absolute values but also appreciate the beauty and complexity of numbers!

Latest Pre Calculus Questions

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy