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# Complex Squares

Send us your thoughts on squaring complex numbers, any conjectures that arise, and any explanations for what you find.

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Age 16 to 18

Challenge Level

*You may find it useful to try A Brief Introduction to Complex Numbers and A Brief Introduction to the Argand Diagram before tackling this problem.*

Any complex number $z=a+ib$ can be represented as a point $(a, b)$ on the Argand diagram.

**What is special about complex numbers which square to give a real number?**

Given any real number $x$, is there always a complex number that squares to give $x$?

Represent your findings on the Argand diagram.

**Find some complex numbers which square to an imaginary number.**

What can you say about such complex numbers?

Given any imaginary number, is there always a complex number that squares to give that number?

Represent your findings on the Argand diagram.

**Explore the effects of squaring on other complex numbers** as they are represented on the Argand diagram. You may wish to use GeoGebra.

**Can you find any complex numbers which square to give the complex numbers you found in the first two parts of the problem?**

Send us your thoughts on squaring complex numbers, any conjectures that arise, and any explanations for what you find.