In Mathematics, What Is a Complex Conjugate?

August 3, 2021

In mathematics, a complex conjugate is a pair of two-component numbers called complex numbers. Each of these complex numbers possesses a real number component added to an imaginary component. Though their value is equal, the sign of one of the imaginary components in the pair of complex conjugate numbers is opposite to the sign of the other. Despite having imaginary components, complex conjugates are used to describe physical realities. The use of complex conjugates works despite the presence of imaginary components, because when the two components are multiplied together, the result is a real number.


Imaginary numbers are defined as any numbers that when squared result in a real negative number. This may be restated in other terms for simplification. An imaginary number is any real number multiplied by the square root of negative one (-1) — by itself unintelligible. In this form, a complex conjugate is a pair of numbers that can be written, y=a+bi and y=a–bi, where “i” is the square root of -1. Formalistically, to distinguish the two y-values, one is generally written with a bar over the letter, ӯ, although occasionally an asterisk is used.

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Demonstrating that multiplication of two complex conjugate numbers produces a real result, consider an example, y=7+2i and ӯ=7–2i. Multiplying these two gives yӯ=49+14i–14i–4i2=49+4=53. Such a real result from complex conjugate multiplication is important, particularly in considering systems at the atomic and sub-atomic levels. Frequently, mathematical expressions for tiny physical systems include an imaginary component. The discipline in which this is especially important is quantum mechanics, the non-classical physics of the very small.


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In quantum mechanics, the characteristics of a physical system consisting of a particle are described by a wave equation. All that is to be learned about the particle in its system can be revealed by these equations. Frequently, wave equations feature an imaginary component. Multiplying the equation by its complex conjugate results in a physically interpretable “probability density.” The characteristics of the particle may be determined by mathematically manipulating this probability density.

By way of example, use of probability density is important in the discrete spectral emission of radiation from atoms. Such application of probability density is called “Born probability,” after German physicist Max Born. The important closely related statistical interpretation that the measurement of a quantum system will give certain specific results is called the Born rule. Max Born was a recipient of the 1954 Nobel Prize in Physics for his work in this area. Unfortunately, attempts to derive the Born rule from other mathematical derivations has met with mixed results.


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