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Superposed_Cat
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Why do we represent waves as complex numbers? Why won't real suffice? Thanks for any help.
There is not really any difference between a pair of real numbers (with a given relationship) and a single complex number. In the frequency domain you need amplitude and phase, so that is two numbers and so complex numbers is a reasonable mathematical representation.Superposed_Cat said:Why do we represent waves as complex numbers? Why won't real suffice? Thanks for any help.
Philip Wood said:However, in wave mechanics, complex numbers are an essential part of the Physics.
But there are cases where the physical quantity inherently requires two real numbers to represent it and those two real numbers are related in such a way that representing them as a single complex number is reasonably, both mathematically and physically.WannabeNewton said:All we have to do is carefully take the real part at the end of a calculation in order to get physical quantities.
DaleSpam said:But there are cases where the physical quantity inherently requires two real numbers to represent it and those two real numbers are related in such a way that representing them as a single complex number is reasonably, both mathematically and physically.
For example, in MRI you detect the amount of magnetization in the plane transverse to the main magnetic field. There is a strength of the magnetization and a direction, requiring two real numbers to describe. In such cases, the physical quantity of interest is actually a complex number.
Waves can be represented as complex numbers because they exhibit both amplitude (size) and phase (position) properties. Complex numbers allow us to easily represent and manipulate these two properties simultaneously, making it a useful tool in studying and analyzing wave behavior.
Complex numbers provide a convenient way to represent the behavior of waves, which can be incredibly complex. They allow us to easily perform mathematical operations such as addition, subtraction, multiplication, and division, which helps us to better understand how waves interact and behave.
While waves can be represented using real numbers, complex numbers offer a more complete and efficient way to represent and analyze wave behavior. Real numbers only represent amplitude, while complex numbers also incorporate phase, making them more useful in understanding the behavior of waves.
Yes, complex numbers are used in various fields of science, including physics, engineering, and mathematics. They are especially useful in areas that involve waves, such as optics, acoustics, and signal processing. They are also used in quantum mechanics to describe the behavior of particles.
No, not all types of waves need to be represented using complex numbers. Simple waves, such as sound waves, can be adequately described using real numbers. However, for more complex and intricate wave behaviors, such as interference and diffraction, complex numbers are necessary for a more accurate representation.