Real or Imaginary ?

Was looking up the meaning of "imaginary numbers" and was quite impressed by wikipedia's explanation of imaginary numbers. Note, how in the explanation of numbers, you can see how science and philosophy converge .



In mathematics, an imaginary number (or purely imaginary number) is a complex number whose square is a negative real number. Imaginary numbers were defined in 1572 by Rafael Bombelli. At the time, such numbers were thought not to exist, much as zero and the negative numbers were regarded by some as fictitious or useless. Many other mathematicians were slow to believe in imaginary numbers at first, including Descartes who wrote about them in his La Géométrie, where the term was meant to be derogatory.[1]


Any complex number, z, can be written as
where and are real numbers and is the imaginary unit, which has the defined property that:


i(square) = -1

Applications of imaginary numbers

Despite their name, imaginary numbers are as "real" as real numbers.[2] (See the definition of complex numbers on how they can be constructed using set theory.) One way to understand this is by realizing that numbers themselves are abstractions, and the abstractions can be valid even when they are not recognized in a given context. For example, fractions such as ⅔ and ⅛ are meaningless to a person counting stones, but essential to a person comparing the sizes of different collections of stones. Similarly, negative numbers such as − 3 and − 5 are meaningless when keeping score in a football game, but essential when keeping track of monetary debits and credits.[1]

For most human tasks, real numbers (or even rational numbers) offer an adequate description of data, and imaginary numbers have no meaning; however, in many areas of science and mathematics, imaginary numbers (and complex numbers in general) are essential for describing reality. Imaginary numbers have essential concrete applications in a variety of sciences and related areas such as signal processing, control theory, electromagnetism, quantum mechanics, cartography, and many others.
For example, in electrical engineering, when analyzing AC circuitry, the values for the electrical voltage (and current) are expressed as imaginary or complex numbers known as phasors. These are real voltages that can cause damage/harm to either humans or equipment even if their values sometimes contain no "real part". The study of AC (alternating current) entails introduction to electricity governed by trigonometric (i.e. oscillating) functions. From calculus, one knows that differentiating or integrating either "+/- sin(t)" or "+/- cos(t)" four times (with respect to "t," of course) results in the original function "+/- sin(t)" or "+/- cos(t)." From complex algebra, one knows that multiplying the imaginary unit quantity "i" by itself four times will result in the number 1 (identity). Thus, calculus can be represented by the algebraic properties of the imaginary unit quantity (this was exploited by Charles Proteus Steinmetz).

Specifically, Euler's formula is used extensively to express signals (e.g., electromagnetic) that vary periodically over time as a combination of sine and cosine functions. Euler's formula accomplishes this more conveniently via an expression of exponential functions with imaginary exponents. Euler's formula states that, for any real number x,

Some programming languages also have built-in support for imaginary numbers. For example, in the Python interpreter, one may use them by appending a lowercase or uppercase J to the number:[3]