Math Problem Statement

Describe and correct the error a student made when dividing complex numbers.

Solution

Based on the image, here’s an analysis of the error in the problem related to dividing complex numbers:

The student's mistake is that they multiplied the numerator and denominator each by their own complex conjugate rather than multiplying both by the complex conjugate of the denominator. Here's how to correct it:

  1. Correct Process:

    • When dividing complex numbers, you should multiply both the numerator and denominator by the complex conjugate of the denominator.
    • The complex conjugate of 3i3 - i is 3+i3 + i.
  2. Proper Calculation: 1+i3i×3+i3+i\frac{1 + i}{3 - i} \times \frac{3 + i}{3 + i}

    The numerator becomes: (1+i)(3+i)=3+i+3i+i2=3+4i1=2+4i(1 + i)(3 + i) = 3 + i + 3i + i^2 = 3 + 4i - 1 = 2 + 4i

    The denominator becomes: (3i)(3+i)=32i2=9(1)=9+1=10(3 - i)(3 + i) = 3^2 - i^2 = 9 - (-1) = 9 + 1 = 10

    So the final result is: 2+4i10=210+4i10=15+2i5\frac{2 + 4i}{10} = \frac{2}{10} + \frac{4i}{10} = \frac{1}{5} + \frac{2i}{5}

  3. Error in the Work Shown:

    • The student incorrectly multiplied by separate conjugates, leading to the wrong calculations for both numerator and denominator.
  4. Correct Option: B is correct—the student multiplied the numerator and denominator each by their own complex conjugates rather than multiplying both by the complex conjugate of the denominator.


Let me know if you want a deeper breakdown of the multiplication or a step-by-step visualization.

Follow-Up Questions:

  1. Why do we use the complex conjugate of the denominator instead of separate conjugates?
  2. What would happen if the denominator were left in the form 3i3 - i?
  3. Can you generalize the steps to divide any complex numbers?
  4. How does simplifying i2i^2 as 1-1 impact calculations?
  5. Could this problem be solved using a different method, such as polar form?

Tip:

Always check your denominator to ensure it's simplified into a real number when dividing complex numbers.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Complex Numbers
Division of Complex Numbers
Algebraic Manipulations

Formulas

Multiply numerator and denominator by the complex conjugate of the denominator.
(a + bi)(c + di) = ac + adi + bci + bdi^2
i^2 = -1

Theorems

Properties of Complex Conjugates
Simplification of Complex Denominators

Suitable Grade Level

Grades 10-12