Complex II

the roots of unity

\[ z = r e^{i \theta} \]

\[ \tau = : 2 \pi \]

\[ e^{i \tau} = 1 \]

\[ e^{k i \tau} = 1 \]

\[ z = r e^{i \theta + k i \tau} \]

\[ z^n = 1 \]

\[ (r e^{i \theta} e^{k i \tau})^n = 1 \]

\[ r^n {e^{i \theta}}^n {e^{k i \tau}}^n = 1 \]

\[ r^n e^{i \theta n} e^{k i \tau n} = 1 \]

\[ r^n = 1 \]

\[ r = 1 \]

\[ e^{i \theta n} e^{k i \tau n} = 1 \]

\[ e^{i \theta n} = e^{m i \tau} \]

\[ (z^n)^m = 1^m = 1 \]

\[ i \theta n = m i \tau \]

\[ \theta n = \tau m \]

\[ \theta = \tau \frac{m}{n} \]

\[ rou_{m, n} = e^{i \tau \frac{m}{n}} \]

The "rou" means root of unity, because unity means \(1\), and root because \(z\) is the \(n\)'th root of \(1\). but all of the roots of unity can be constructed with what is called the principal value \(rou_{1, n}\) (actually no, the principal value is \(rou_{0, n}\), so just \(1\)), but that is:

\[ e^{i \frac{\tau}{n}} \]

This is because the more common \(e^{i \tau \frac{m}{n}}\) is just \((e^{i \frac{\tau}{n}})^m\), so the following statement is:

\[ rou_{m, n} = (rou_{1, n})^m \]

\[ rou_n = : rou_{1, n} \]

\[ rou_{m, n} = rou_n^m \]

this also looks kind of like notation, but it is just a power.