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kramer733
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Does it have something to do with the quadratic form? What would i type on google to search for more information to get better search results?
To summarize what MisterX and Jamma said, we often write complex numbers like that because it is convenient (it fits our commonly used xy- coordinate system format) but there are many different ways in which we could write complex numbers: [itex]re^{i\theta}[/itex] is one. Engineers often use the format "[itex]r cis(\theta)[/itex]" which is short for "[itex]r(cos(\theta)+ i sin(\theta))= re^{i\theta}[/itex]" as Jamma said.kramer733 said:Does it have something to do with the quadratic form? What would i type on google to search for more information to get better search results?
kramer733 said:But what about the "+" sign in between the real part and the imaginary part? Since graphing complex numbers on the complex plane is a lot like graphing real numbers on the real plane, why didn't we use a comma inbetween the real parts and complex parts?
How do we know there are nice properties such as addition and multiplication for something in the form "A+Bi"? If we had instead used the convention "(A,Bi)" to graph complex numbers, then would it still have had addition and multiplication for complex numbers? It seems a bit odd to me. You can of course multiply a scalar with (a,bi) or add another complex number but it would've done differently.
with the current system, (a+bi)^2 = a^2 + 2(a)(bi) -b^2. would it still have resulted in the same if we would have multiplied (a,bi) with (a,bi)? How do we even do that?
Does it have something to do with the quadratic form? What would i type on google to search for more information to get better search results?
HallsofIvy said:To summarize what MisterX and Jamma said, we often write complex numbers like that because it is convenient (it fits our commonly used xy- coordinate system format) but there are many different ways in which we could write complex numbers: [itex]re^{i\theta}[/itex] is one. Engineers often use the format "[itex]r cis(\theta)[/itex]" which is short for "[itex]r(cos(\theta)+ i sin(\theta))= re^{i\theta}[/itex]" as Jamma said.
There is no "mathematical" reason- it is just a convention.
Complex numbers are necessary to solve certain mathematical problems that cannot be solved with real numbers alone. They are particularly useful in solving equations involving square roots of negative numbers, such as in electrical engineering and quantum mechanics.
The imaginary unit, denoted by the letter i, is the square root of -1. It is used to represent the second dimension in a complex number, giving it both a real and imaginary component. This allows for the representation of numbers in the form a+bi, where a is the real part and bi is the imaginary part.
Complex numbers can be represented on a Cartesian plane, with the real part plotted on the x-axis and the imaginary part plotted on the y-axis. The point where the two axes intersect is the origin, and each complex number can be represented as a vector from the origin to its position on the plane.
Complex numbers have various applications in mathematics, physics, and engineering. They are used in electrical engineering for calculating AC circuits, in signal processing for analyzing signals, and in quantum mechanics for describing wave functions. They are also used in computer graphics and 3D modeling to represent rotations and transformations.
Yes, complex numbers have real-world applications in various fields, such as in modeling and predicting the behavior of electrical circuits and in analyzing the behavior of waves in different mediums. They are also used in economics and finance to model and predict stock market trends. Additionally, complex numbers are used in computer programming and simulations for creating realistic graphics and simulations.