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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.

 


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