jeffrey samuel
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you want to solve f(x)=x^2+4 equation.
You see that there is no actual solution for this equation.No actual solution means to that "If you draw this eq. in your coordinate system it will not cross over x axis" But to handle and make it usefull crossing the x axis is a must for the real world of an engineer.
"Why are you responding to a post almost 6 years old?" Is it important?
"Why do you say that the above equation has no actual solution?" & "Why is crossing the x-axis a "must"? " Explain me that why we find the roots of an equation?
What is the "real world"? In real world, only "real numbers" exist.
Actually i made an approximation. all that approximations belong to my imagination.
For a better expression you can visit **broken link removed**
You are right i have some missing points but your explanation explain nothing. Just like the classical idiom.
Numbers and math are used to MODEL real world events.
They are good at modeling one dimensional situations.
Complex numbers are really just pairs or numbers with a system of math attached to them that defines things like addition and multiplication in a manner that makes them useful to model two dimensional phenomenon.
We can and do use vector notation for two dimensions (and three and more) but the math is messier. At least, it takes up more paper.
It is a lot neater with complex numbers, especially when advanced things like Fourier analysis is involved.
So, they are used simply because they work. No other reason, really.
Chips & Dips,
There are many other uses for mathematics.Numbers and math are used to MODEL real world events.
Also multidimensionals. I went through solid geometry (3 dimensional geometry) using only real numbers.They are good at modeling one dimensional situations.
Wrong, and too general a statement. Complex numbers can have three parts, like i, j, and k in the physical world. Specifically, complex numbers are composed of parts that are orthogonal to each other. That stipulation makes them unique to other number pairs.Complex numbers are really just pairs or numbers with a system of math attached to them that defines things like addition and multiplication in a manner that makes them useful to model two dimensional phenomenon.
Vectors are not restricted to orthogonal relationships.We can and do use vector notation for two dimensions (and three and more) but the math is messier. At least, it takes up more paper.
Provided you work with all things orthogonal.
Isn't that a rather obvious statement? Why would anyone use a method that does not work?So, they are used simply because they work. No other reason, really.
First, it is Chips & Chips, not Chips & Dips. I do both electronics and machining and have shops for both. Machinists are said to make chips (metal chips).
Oh really, name one. Except for the theoretical musings of mathematicians, I can't think of a single use of math that is not a MODEL of a real world situation. We use math to describe what happens in the world and that makes it a modeling medium, just like a sculpture uses clay or marble.
I said they are a good way. I did not say they were the only way.
Do read the reference to Dr. Math brought up by Anomaly earlier in this thread. Here it is again for your convenience.
https://mathforum.org/library/drmath/view/53809.html
In it, this mathematician explains his way of teaching complex numbers. The first thing he does is take the i or j notation out of them. He describes them as simply pairs of numbers with some new rules for manipulating them in manners that are analogous to the common arithmetic operations like addition and multiplication. He shows, in a round about way, that the i or j notation is simply a way of writing these number pairs. It (i) does have a real world idea associated with it, unless you want to call the square root of a negative number a real world thing. For me, that is a stretch and I have never had any idea of any real world association for i. Read below for my real world association for the number pairs.
The point is, complex numbers can be used to represent (model) exactly the same thing that vectors are used for. A vector will only be orthogonal (parallel to the x or y axis) if one of the two components (x or y) is zero. Likewise, if you are using complex numbers to model the same real world thing that the vectors are modeling, then the complex numbers will only be orthogonal if one of two numbers of that number pair is zero. Otherwise, vector or complex number, it will be representing a non-orthogonal angle. In truth, any angle can be represented by either notation.
No, not at all. See the answer above.
Well, yes it is obvious. But it is also deeply profound. So deeply profound that none of the previous answers even suggested it.
They do work as an accurate model of real world phenomenon. Scientists struggle for entire careers trying to find mathematical models for real world events. But one thing to remember is that these models are rarely perfect. Newton described gravity with a simplistic equation that was revolutionary for his day. Einstein came along and basically said that even though Newton's equations (math model) did a pretty good job, they did fail under some circumstances. He, Einstein, came up with a better set of equations (math models) of real world events. But later scientists have come up with other theories (math models). Which of these is the real, real world reality? The true answer to that is "None!", because ultimately all the models that we come up with will be shown to be only an approximation and better models will come to light. We only use any particular model because it works for us in some particular set of circumstances. This is true for the representation of real world phenomenon by complex numbers. We only use it because it works, at least in some circumstances. Not because it has any absolute relationship to the actual real world; because, as the Newton/Einstein example shows, no model created by human beings is ultimately completely true. All of them are just approximations.
Complex number is used to represent imaginary number with real numbers,...
... a special imaginary number i is used to express √-1 which is not possible but sometimes it needs to express results.
If u take example of current. It has two parts. active part (a) and reactive part (b). To show total current we represent I=a+ib and magnitude as |I|.
You are limiting your thinking too much. And you are trying too hard to find a real world existence for these numbers.
I am using the word "model" in a very general sense. I have built scale models. I know about clothing models. And there are computer models. There are many meanings for that word. In science, a model is any useful analogy. The key word here is "useful". If a model helps to describe reality, then it is used. This is judged by the results. Scientists measure, theorize, and then compare the further predictions of that theory with further measurements.
That theory is a model.
The theory/model is not actual reality.
Newton came up with a theory/model of gravity. It works well, but it does break down under some circumstances.
So there is no ABSOLUTE, real world association here.
In modern science there is usually a lot of math associated with any theory. That math is part of, probably the essence of, that model. So, if the math works, we use it. And in some cases, the math associated with complex/imaginary numbers does provide a good framework for a theory. So we use it.
It does not matte what i or j mean.
It does not matter how that math was invented
The only thing that matters is THE FACT THAT IT DOES WORK. It works, at least in some circumstances, so we use it. That's it. Period. And when it stops working, we stop using it and look for another model with different math. Think: Einstein vs Newton.
So the answer to the original question is simply because it works.
If u take example of current. It has two parts. active part (a) and reactive part (b). To show total current we represent I=a+ib and magnitude as |I|.
Chips & Chips,
How am I limiting in my thinking? I am not looking, I found a real world existence for duplex numbers. Lots of things exist as a real part combined with an orthogonal part.
What does "modeling" have to do with complex numbers?
No it's not. A theory is a guess about how something exists. A model is based on assumptions of how something works. They are two different things.
No more than a description of how something exists or works.
No, he observed and documented the Law of Gravity. He never advanced a theory of how it exists.
Whoever said there was?
And the point is?
Of course it does. "i" or "j" is an operator. That is like saying it does not matter what the math operators like =,-,/,^ mean.
As long as the concept is correct.
When has math ever stopped working? The advancement from Newton to Einstein was a physics addendum, not a math advancement.
Isn't that stating the obvious?
Ratch
And I rechecked all your responses and did not see any "real world" example of a duplex or complex number. Did I miss something?
They are used as part of the model.
I don't see the difference. A theory and a model are essentially the same thing.
An assumption about how to explain experimental observations that must be tested. If it predicts further things well, it becomes stronger.
What do you mean by that.
He advanced a theory to explain the facts/observations that existed in his day and time. It tried to explain how part of the universe worked. That is a theory. It is also a model when you add the equations that predict how things will act. Then others can test it with more observations. And the cycle repeats: observation, theory, prediction, more observation. His thinking was just as advanced for his day as that of Einstein or Hawkings or whoever comes up with the latest theory. And it is no less a theory just because it operates at a coarser level. When a future scientist comes up with a more basic theory of how the universe works, then the works of Einstein and Hawkings and other greats of today will also fall by the wayside. But their work is no less important or no less of a theory.
What we call "Laws" of science are nothing more or less than well tested theories. They are tested so well and have passed so many tests that they are considered to be a lot better than most, run of the mill, theories. But they are still theories and they can still be replaced with better theories. I personally do not like the use of the word "Law" in this sense. But then I still consider Pluto to be a planet: I don't care what the astronomers say.
Calling "i" or "j" an operator is just one way of looking at them. I am not saying that is not valid. In fact, that is really how I am looking at them. Just as the multiplication operator is useful for finding the total distance traveled at a certain speed for a certain time, so these operators are good for other mathematical operations. One of these operations is representing orthogonal quantities. This is what I have been saying, we use it because it is useful. Because it works. Not because it has some magical existence in reality.
Math does exist by itself. I imagine there are mathematicians that never worry a single second about any practical application of their work. Scientists, like physicists, try to apply those mathematical constructs to real world situations in order to explain, to model, to predict them. When I say that the math "stops working" I only mean in relation to those theories or models.
Sometimes the obvious thing is also the most profound. So sometimes it does need to be stated.
Perhaps to expand on it just a bit, complex numbers are used in electronic analysis because they allow orthogonal quantities to be manipulated mathematically and this is needed for the analysis of some systems. They allow us to make accurate predictions about how these systems act in the real world.
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