Unit circle
In the complex plane you can show the unit complex numbers (circle group).
Euler's formula
All points on this circle conform to Euler's formula
and therefor they give the points shown in the Argand diagram
Substitution of φ in the formula gives the values
→ → → →
Number circle
You can include the complex unit circle in the number circle. This shows the consistency of natural numbers and complex numbers. The inner border forms a square that starts with the number 1, and continues till 24.
21 22 23 24 1 2 3 20 4 19 5 18 i2 0 −i2 6 17 7 16 8 15 14 13 12 11 10 9
Because i2 = −1 and −i2 = +1 the number 1 is twice on the number circle.
Imaginary unit
You can take the imaginary unit as the base of an exponential function. For the powers follows clockwise that
i 0 = 1, i 1 = i, i 2 = –1, i 3 = –i
i 4 = 1, i 5 = i, i 6 = –1, i 7 = –i
or counter-clockwise
i 0 = 1, i –1 = –i, i –2 = –1, i –3 = i
i –4 = 1, i –5 = –i, i –6 = –1, i –7 = i