Math Problem Statement

What is the imaginary unit i defined as, and how can you use it? Complete the sentence for 5 + 2i, and identify the number that does not belong.

Solution

Let's work through the questions in the image:

  1. Vocabulary: The imaginary unit ii is defined as the square root of -1, that is, i=1i = \sqrt{-1}. It is used to express complex numbers, which consist of a real part and an imaginary part. For example, in a complex number like a+bia + bi, aa is the real part and bb is the imaginary part, with ii indicating the imaginary component.

  2. Complete the sentence: For the complex number 5+2i5 + 2i, the imaginary part is 2 and the real part is 5.

  3. Writing: To add complex numbers, you simply add the real parts together and add the imaginary parts together separately. For example, when adding (a+bi)(a + bi) and (c+di)(c + di), the result will be (a+c)+(b+d)i(a + c) + (b + d)i.

  4. Which one doesn't belong?: The number that does not belong with the others is 3+6i\sqrt{3} + 6i. This is because all the other numbers have either a purely real or purely imaginary part, whereas 3+6i\sqrt{3} + 6i has both a non-zero real part (3\sqrt{3}) and a non-zero imaginary part (6i6i).


Do you need further explanation or clarification on any of these concepts?

Here are 5 related questions to deepen your understanding:

  1. What is the result of squaring the imaginary unit ii?
  2. How would you multiply two complex numbers like (3+4i)(3 + 4i) and (12i)(1 - 2i)?
  3. Can a real number be considered a complex number? Why or why not?
  4. How do you find the conjugate of a complex number like 5+2i5 + 2i?
  5. What is the geometric interpretation of a complex number on the complex plane?

Tip: When working with complex numbers, always remember that i2=1i^2 = -1, which can simplify many calculations involving powers of ii.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Complex Numbers
Imaginary Unit
Real and Imaginary Parts

Formulas

i = sqrt(-1)
(a + bi) + (c + di) = (a + c) + (b + d)i

Theorems

Basic operations with complex numbers
Adding real and imaginary parts separately

Suitable Grade Level

Grades 9-10