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1 and Euler's identity

With Euler's identity you can write the number 1 as

 


Explanation

In calculations with complex numbers you frequently use the equation

where k is an integer. Substitution of φ = π in Euler's formula gives

eiπ = cos π + i sin π = −1 + i·0 = −1

From this follows

so

 


Example 1

You can see that sin (2π) = 0, as according the definition of the sine applies

 


Example 2

The values of 2i can be calculated with

 


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