# Circle group $S^1$

Can someone describe the circle group $$S^1$$ in a easily understandable way? Are elements in $$S^1$$ in the form of $$e^{2i\theta\pi}$$? What does this look like pictorially? Doesn't necessarily need a graph but maybe at least describe it? Appreciated!

• It's... a circle. The group operation is multiplication like you describe. I'm not quite sure what your question is asking. Do you know that complex numbers have that representation, and that $e$ to an imaginary power is norm 1? – Alfred Yerger Apr 16 at 2:53
• I guess I'm not sure about e to an imaginary number power. I will look into that, thank you – yyyy Apr 16 at 3:00
• Instead of considering elements of the group as $e$ to an imaginary power, simply consider them to be angles. Then the group operation is addition of angles. – Solomonoff's Secret Apr 16 at 3:04
• Oooooor consider them to be the real numbers modulo $1$, or modulo $2\pi$. Or rigid rotations of a circle. Pictorically it's just a circle. – Jackozee Hakkiuz Apr 16 at 3:05