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Which of the following complex numbers lies on the unit circle?

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Question: Which of the following complex numbers lies on the unit circle?

Options:

  1. 1 + 0i
  2. 0 + 1i
  3. 1 + 1i
  4. 0 + 0i

Correct Answer: 1 + 0i

Solution: 

A complex number z lies on the unit circle if |z| = 1. For z = 1 + 0i, |z| = √(1² + 0²) = 1. For z = 0 + 1i, |z| = √(0² + 1²) = 1. Both 1 + 0i and 0 + 1i lie on the unit circle.

Which of the following complex numbers lies on the unit circle?

Practice Questions

Q1
Which of the following complex numbers lies on the unit circle?
  1. 1 + 0i
  2. 0 + 1i
  3. 1 + 1i
  4. 0 + 0i

Questions & Step-by-Step Solutions

Which of the following complex numbers lies on the unit circle?
  • Step 1: Understand that a complex number is in the form z = a + bi, where a is the real part and b is the imaginary part.
  • Step 2: Know that the unit circle is defined as the set of complex numbers where the distance from the origin (0,0) is 1.
  • Step 3: The distance (or modulus) of a complex number z = a + bi is calculated using the formula |z| = √(a² + b²).
  • Step 4: To check if a complex number lies on the unit circle, we need to see if |z| equals 1.
  • Step 5: For the complex number z = 1 + 0i, calculate |z|: |z| = √(1² + 0²) = √(1) = 1.
  • Step 6: For the complex number z = 0 + 1i, calculate |z|: |z| = √(0² + 1²) = √(1) = 1.
  • Step 7: Since both calculations give us 1, we conclude that both 1 + 0i and 0 + 1i lie on the unit circle.
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