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danne89
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Hi! I'd a look at complex numbers and can't understand how they can be applied to "the real world". Can anyone give me some concrete examples, please. Or a site that does.
Danne
Danne
NeutronStar said:To begin with forget about the terms "real" and "imaginary". Think instead in terms of "quantities" and ask whether complex numbers can be thought of as representing quantitative properties. I think that you'll find that complex numbers are just that,.. complex. They have what is called a "real" component and an "imaginary" component. Then what you need to ask is what do these different components represent quantitatively.
The best concrete example I can think of is an electric circuit that contains an inductor. A dynamic current flowing in such a circuit can be described by a complex number. The real part of that description refers to the electron current flow (or hole current flow if you're a semiconductor nut). The imaginary part of that description refers to the magnetic field associated with the inductor. What you lose in electron current you gain in magnetic field and vice versa.
So both the real part and the imaginary part of the complex number represent "real" quantities in the "real" world.
Think of it this way,… electron current in a circuit can only flow in two directions. They are described by the sign of the number that represents the quantity of current flow (the real part of the complex number). Current flow can never be less than zero (no current flow at all), but it can have a magnitude in either of two directions (+ or -) . So the real part of the complex number always represent a quantity of electron current greater than or equal to zero and the sign represents the direction that the current is moving.
However, when some of the electron current gets converted into magnetic field energy by the inductor we can't represent that by either positive or negative. We need a new "direction". The new direction is called i for "imaginary". This new direction can also be positive or negative. In other words, the magnetic field can either be growing (the positive imaginary direction) or collapsing (the negative imaginary direction). So the imaginary part of the complex number represents the quantity of magnetic field present as well as its dynamical state (positive or negative).
In this case the complex number isn't any more mysterious than any other number. It's really just a shorthand way of combining various coordinate systems and quantitative ideas.
I've found that no matter how abstractly we take this idea of complex numbers, if we stop to think about what we are doing, we can break down very abstract concepts into their intuitive counterparts. Assuming, of course, that we have any intuitive understanding to begin with of the concepts that we are working with.
It does appear that some mathematicians have absolutely no intuitive clue concerning the objects that they are working with. This can usually be revealed by simply asking them to offer an intuitive explanation of their project. When they start talking in axiomatic circles and can't reduce their idea to a simple intuitive explanation then you can rest assured that they, themselves, have absolutely no intuitive clue about what the heck they are doing.
Many mathematicians do indeed work entirely from this axiomatic approach. I personally like to keep the intuitive insight alive along the way. But then, I'm a scientist.
Complex numbers are numbers that consist of a real part and an imaginary part, usually written in the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1). Real numbers only have a single real value, while complex numbers have both a real and imaginary value. This allows complex numbers to represent both real and imaginary quantities, making them useful in a variety of applications.
Complex numbers have various applications in mathematics, including solving polynomial equations, studying fractals, and representing periodic phenomena such as waves and vibrations. They are also used in fields such as calculus, differential equations, and signal processing.
In physics and engineering, complex numbers are used to represent physical quantities that have both magnitude and direction, such as in electrical circuits and mechanical vibrations. They are also used in fields such as quantum mechanics, electromagnetism, and fluid dynamics to model and analyze complex systems.
Yes, complex numbers can be graphed on a coordinate plane known as the complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. This allows for visual representation and manipulation of complex numbers in a similar way to real numbers.
While complex numbers may not be directly used in everyday life, they are essential in many modern technologies. For example, they are used in the development of computer graphics, GPS systems, and electrical engineering. Complex numbers also have applications in finance, such as in modeling stock market trends. Additionally, they are used in music and art to create complex and unique patterns and structures.