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Why does [tex]|e^i| = 1 [/tex] ?
mr. vodka said:Thank you Dickfore. Could it be true though that your invalid formula is true in the special case of f being real and analytical?
The absolute value of e(i), or the absolute value of the complex number e to the power of the imaginary unit i, is equal to 1. This is because e to the power of any complex number will always have a magnitude of 1.
The absolute value of e(i) is calculated by taking the magnitude (or distance from the origin) of the complex number e to the power of i. This can be done using the Pythagorean theorem, where the absolute value is equal to the square root of the sum of the squares of the real and imaginary components.
The absolute value of e(i) is important because it is a fundamental constant in mathematics and has many applications in various fields such as physics, engineering, and finance. It is also closely related to the unit circle and plays a key role in understanding trigonometric functions.
No, the absolute value of any number is always positive, including the absolute value of e(i). This is because the purpose of absolute value is to measure the distance from zero, not to indicate the sign of a number.
The absolute value of e(i) is closely related to Euler's formula, which states that e to the power of i multiplied by theta (expressed in radians) is equal to cosine theta plus i times sine theta. This can be seen by substituting theta as 0, which results in e to the power of 0 (equal to 1) being equal to cosine 0 (equal to 1) plus i times sine 0 (equal to 0), giving the equation 1 = 1 + 0i.