Imagining the imaginary

Algebra Level 4

A student has a lesson on Complex Numbers and makes the following conclusions:

  1. \(\mathbb{R} \subset \mathbb{I}\)
  2. \(\mathbb{I} \subset \mathbb{C}\)
  3. \(\sqrt[-1]{-i} = i\)
  4. \(\sqrt[]{\bar{z}z} = |z|\)
  5. \(z = r\cos(\theta)+ir\sin(\theta)\)

Which of the above is/are false?


  • \(i = \sqrt {-1}\) is the imaginary unit
  • \(\mathbb{R}\) is the set of all real numbers
  • \( \mathbb{I}\) is the set of all imaginary numbers
  • \( \mathbb{C}\) is the set of all complex numbers
  • \(z=a+ ib; a,b \in \mathbb R\) is the general form of a complex number
  • \(\bar{z} = a-ib\) is the conjugate of \(z\)
  • \(|z| = \sqrt{a^2+b^2}\) is the modulus of \(z\)

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