Editing Euler's formula (section)

==== Interpretation of the formula ====
This formula can be interpreted as saying that the function {{math|''e''<sup>''iφ''</sup>}} is a [[unit complex number]], i.e., it traces out the [[unit circle]] in the [[complex plane]] as {{mvar|φ}} ranges through the real numbers. Here {{mvar|φ}} is the [[angle]] that a line connecting the origin with a point on the unit circle makes with the [[positive real axis]], measured counterclockwise and in [[radian]]s.

The original proof is based on the [[Taylor series]] expansions of the [[exponential function]] {{math|''e''<sup>''z''</sup>}} (where {{mvar|z}} is a complex number) and of {{math|sin ''x''}} and {{math|cos ''x''}} for real numbers {{mvar|x}} ([[Euler's formula#Using power series|see above]]). In fact, the same proof shows that Euler's formula is even valid for all ''complex'' numbers&nbsp;{{mvar|x}}.

A point in the [[complex plane]] can be represented by a complex number written in [[Coordinates (elementary mathematics)#Cartesian coordinates|cartesian coordinates]]. Euler's formula provides a means of conversion between cartesian coordinates and [[Polar coordinate system|polar coordinates]]. The polar form simplifies the mathematics when used in multiplication or powers of complex numbers. Any complex number {{math|1 = ''z'' = ''x'' + ''iy''}}, and its complex conjugate, {{math|1 = {{overline|''z''}} = ''x'' − ''iy''}}, can be written as
<math display="block">\begin{align}
z &= x + iy = |z| (\cos \varphi + i\sin \varphi) = r e^{i \varphi}, \\
\bar{z} &= x - iy = |z| (\cos \varphi - i\sin \varphi) = r e^{-i \varphi},
\end{align}</math>
where
*{{math|1=''x'' = Re ''z''}} is the real part,
*{{math|1=''y'' = Im ''z''}} is the imaginary part,
*{{math|1=''r'' = {{abs|''z''}} = {{sqrt|''x''<sup>2</sup> + ''y''<sup>2</sup>}}}} is the [[magnitude (mathematics)|magnitude]] of {{mvar|z}} and
*{{math|1=''φ'' = arg ''z'' = [[atan2]](''y'', ''x'')}}.

{{mvar|φ}} is the [[arg (mathematics)|argument]] of {{mvar|z}}, i.e., the angle between the ''x'' axis and the vector ''z'' measured counterclockwise in [[radian]]s, which is defined [[up to]] addition of {{math|2''π''}}. Many texts write {{math|1=''φ'' = tan<sup>−1</sup> ''{{sfrac|y|x}}''}} instead of {{math|1= ''φ'' = atan2(''y'', ''x'')}}, but the first equation needs adjustment when {{math|''x'' ≤ 0}}. This is because for any real {{mvar|x}} and {{mvar|y}}, not both zero, the angles of the vectors {{math|(''x'', ''y'')}} and {{math|(−''x'', −''y'')}} differ by {{pi}} radians, but have the identical value of {{math|1=tan ''φ'' = {{sfrac|''y''|''x''}}}}.