Worksheet: Introduction to Complex Numbers

In this worksheet, we will practice dealing with imaginary numbers, knowing that complex numbers are made of a real part and an imaginary part.

Q1:

Is the following statement true or false: Any complex number is also a real number.

  • Afalse
  • Btrue

Q2:

If 𝑧=5+2𝑖, what is 𝑅𝑒(𝑧)?

Q3:

If 𝑧=46𝑖, what is Im(𝑧)?

Q4:

Which of the following statements will be true of two equal complex numbers?

  • ATheir real parts will not be equal and their imaginary parts will not be equal.
  • BTheir real parts will be equal and their imaginary parts will not be equal.
  • CTheir real parts will be equal and their imaginary parts will be equal.
  • DTheir real parts will not be equal and their imaginary parts will be equal.

Q5:

Is the following statement true or false: Any imaginary number is also a complex number.

  • Afalse
  • Btrue

Q6:

What type of number is 3+5𝑖?

  • Areal
  • Bcomplex
  • Cpurely imaginary

Q7:

What is the real part of the complex number 3+4𝑖?

Q8:

What is the imaginary part of the complex number 22𝑖?

Q9:

If 4+𝑚𝑖 is a real number, what is the value of 𝑚?

Q10:

If 𝑝+2𝑖 is an imaginary number, what is the value of 𝑝?

Q11:

If 𝑧=43𝑖, which of the following statements is correct?

  • A R e ( 𝑧 ) = 3 , I m ( 𝑧 ) = 4
  • B R e ( 𝑧 ) = 4 , I m ( 𝑧 ) = 3
  • C R e ( 𝑧 ) = 3 , I m ( 𝑧 ) = 4
  • D R e ( 𝑧 ) = 3 , I m ( 𝑧 ) = 4
  • E R e ( 𝑧 ) = 4 , I m ( 𝑧 ) = 3

Q12:

Add 4 to 𝑖.

  • A 4 𝑖
  • B 4 + 𝑖
  • C 4 𝑖
  • D 4 + 𝑖
  • E 4 𝑖

Q13:

True or False: Any real number is also a complex number.

  • ATrue
  • BFalse

Q14:

Factor 𝑥+𝑦 over the complex numbers.

  • A ( 𝑥 + 𝑦 ) ( 𝑥 𝑦 )
  • B ( 𝑥 𝑦 𝑖 ) ( 𝑥 𝑦 𝑖 )
  • C ( 𝑥 + 𝑦 𝑖 ) ( 𝑥 𝑦 𝑖 )
  • D ( 𝑥 + 𝑦 ) ( 𝑥 + 𝑦 )
  • E ( 𝑥 + 𝑦 𝑖 ) ( 𝑥 + 𝑦 𝑖 )

Q15:

Express 54 in terms of 𝑖.

  • A 2 6 𝑖
  • B 3 6 𝑖
  • C 3 6
  • D 3 6 𝑖
  • E 3 2 𝑖

Q16:

The complex numbers are a field in part because .

  • A is a real vector space
  • Bfor every non-zero 𝑧 there is a 𝑤 such that 𝑧𝑤=1
  • C | 𝑧 | 0 for all 𝑧
  • D

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