What is a mystery behind the imaginary unit?

You know, electromagnetic waves bring information from the early Universe as well as provide the information exchange between our smartphones. In physics, to mathematically describe waves, we use so called complex exponential function. Its exponent includes an imaginary unit. That simplifies manipulations with both sine and cosine, because these trigonometric functions can be readily written using a complex exponential function, which was first published by Leonard Euler in 1748. This also allows us to write complex numbers in the so called polar form, alike a vector in polar coordinates. Thereby, apart from being a square root from minus one, an imaginary unit connects geometry to algebra. Thus it is just a language rather than something mysterious. It comes in handy when we write formulas for periodic phenomena such as waves, since we can simply perform algebraic operations with exponential functions, implying periodic trigonometric functions. The same language fits for probability waves in quantum mechanics.

Euler's formula connected geometry with algebra

Euler first published his formula when he worked at the Russian Academy of Sciences at the invitation of Peter the Great.


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