What are imaginary numbers actually used for?

An imaginary number is a number whose square is negative, usually written with the symbol i, where i squared equals minus one. Despite the name, these numbers are used in practical work. Their main uses are completing the solutions of polynomial equations and describing anything that behaves like a wave, including alternating current, radio and radar signals, and some physical motion. Engineers often write j instead of i, because i already stands for current. [1] [2]

Imaginary numbers are usually introduced through the question of whether the equation x squared plus 1 equals 0 can have a solution. Historically they first proved necessary in the 16th century, when the work of Cardano and Bombelli on cubic equations required square roots of negative numbers. Calling that solution i gives a new number, and minus i also works. Combining real and imaginary parts produces complex numbers of the form a plus b times i. The square roots of minus 25, for example, are plus and minus 5i, and the equation x squared plus 2x plus 5 has the roots minus 1 plus 2i and minus 1 minus 2i. [2]

Allowing complex numbers also makes the theory tidier. The Fundamental Theorem of Algebra says that every polynomial equation has a solution, and that a polynomial of degree n has exactly n roots when repeated roots are counted. One source adds that complex numbers unified trigonometry and exponents, so the rules of one become applications of the rules of the other. [2]

Household electricity is alternating current, in which voltage and current rise and fall as sine or cosine waves. Electrical engineers describe these waves with complex numbers. Euler's identity links an exponential with an imaginary exponent to cosine and sine, so a single complex number can hold both the amplitude and the phase of a voltage. Plus Maths says this makes the mathematics of power networks much easier. The same source notes that complex numbers were introduced in the 18th century and were originally thought to be abstract objects of no possible use. [3]

The University of California, Santa Barbara science line explains that AC switching creates a time delay between voltage and current, which imaginary numbers describe. It says this modelling helped engineers design devices such as washing machines, microwaves and toasters. [4]

Live Science says that wherever a measurement relies on a sine or cosine wave, imaginary numbers can be used. It lists signal processing for cellular and wireless technology, radar, and brain waves. In a BBC Teach clip, mathematician Marcus du Sautoy says air traffic radar uses them to separate moving planes from a stationary background, and that doing the calculation with ordinary numbers would be too slow to track planes in real time. [1] [5]

A Guardian Notes and Queries answer says complex numbers occur naturally in quantum physics and are useful for modelling periodic motion such as water or light waves. It says they are also needed to handle damping, such as air friction slowing a pendulum, and that complex analysis has helped solve fluid dynamics problems, including some about turbulence and how earthquakes shake buildings. [6]