# 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?

Notes:

• $$i = \sqrt {-1}$$ 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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