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### 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

e = cos φ + i sin φ

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 i 5 18 i2 0 −i2 6 17 −i 7 16 8 15 14 13 12 11 10 9

The imaginary numbers i and i are on the unit circle. 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   The Argand diagram shows the values 1, i, –1 and –i on the indicated points.