Why do we use complex numbers

jeffrey samuel · Sep 9, 2012

the very fact i feel that complex reality is brought into existent is that to find the root of negative real nos

there is nother possible explanation till date with so much development to find a concise value for it other than

√-1=i

Ratch · Sep 10, 2012

savage67,

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?

Why do you say that the above equation has no actual solution?

What is the "real world"?

Why is crossing the x-axis a "must"?

A number like a + j b simply means it is a duplex number. Duplex means means it has a two components, a real number component and a orthogonal component. An orthogonal component means it is oriented at 90° CCW to the real component. A complex number does not mean that it is other worldly, not imaginable, or from another dimension. It simply means that the number has more than one component. A number like 3j does not mean j+j+j . It means instead to rotate 3 CCW for 90° . 3jj means rotate 3 twice, which equals -3 . So a complex number is every bit as "real" and definite as any number you can name.

Now lets take your equation, x^2+4 =0 ===> x²-4jj = (x+j2)(x-j2) . The solutions x = ±2j means that one solution 2 is rotated 90° CCW and the other solution rotated 90° CW. Both solutions are orthogonal numbers, are understandable, are definite, and have a real meaning. In polar form they would be expressed as 2/_±90°

Ratch

savage67 · Sep 10, 2012

"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 https://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/
You are right i have some missing points but your explanation explain nothing. Just like the classical idiom.

jeffrey samuel · Sep 10, 2012

many times stirring old stories brings more info out

Ratch · Sep 10, 2012

savage67,

"Why are you responding to a post almost 6 years old?" Is it important?

Not to me, but you have not answered the question.

"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?

Answer the question first.

What is the "real world"? In real world, only "real numbers" exist.

So why do you give complex roots in your first post, "So your real world roots are now 2i and -2i "?

Actually i made an approximation. all that approximations belong to my imagination.

What approximation is that?

For a better expression you can visit **broken link removed**

No need for me to do so, I already know the material.

You are right i have some missing points but your explanation explain nothing. Just like the classical idiom.

I answered all the questions you asked. What is a classical idiom?

Ratch

Chips & Chips · Sep 23, 2012

Since the old pot has been stirred, how about a new slant on it.

Why do we use complex numbers? Numbers and math are used to MODEL real world events. Real numbers are essentially one dimensional. 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.

Ratch · Sep 23, 2012

Chips & Dips,

Numbers and math are used to MODEL real world events.

There are many other uses for mathematics.

They are good at modeling one dimensional situations.

Also multidimensionals. I went through solid geometry (3 dimensional geometry) using only real numbers.

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.

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.

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.

Vectors are not restricted to orthogonal relationships.

It is a lot neater with complex numbers, especially when advanced things like Fourier analysis is involved.

Provided you work with all things orthogonal.

So, they are used simply because they work. No other reason, really.

Isn't that a rather obvious statement? Why would anyone use a method that does not work?

Ratch

Chips & Chips · Sep 23, 2012

Ratch,

Chips & Dips,

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).

Numbers and math are used to MODEL real world events.

Click to expand...

There are many other uses for mathematics.

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.

They are good at modeling one dimensional situations.

Click to expand...

Also multidimensionals. I went through solid geometry (3 dimensional geometry) using only real numbers.

I said they are a good way. I did not say they were the only way. Obviously, scientists and engineers who are better than both of us have made this choice and the rest of us are stuck with it. Personally, I think that they did make a good choice.

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.

Click to expand...

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.

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.

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.

Click to expand...

Vectors are not restricted to orthogonal relationships.

Well, vectors are not restricted to orthogonal relationships, but neither are the number pairs of complex numbers. Consider two vectors using orthogonal (x,y) notation:

Vector 1: x=2 and y=3
Vector 2: x=1 and y=4

In other words, using standard (x, y) notation,

Vector 1: (2, 3)
Vector 2: (1, 4)

Now we add the two vectors (as in modeling the sum of two forces):

VectorSum = (2+1, 3+4)
VectorSum = (3, 7)
or
VectorSum: x=3 and y=7

Now lets look at the same vectors as represented by complex numbers.
Complex number 1 = 2+3i
Complex number 2 = 1+4i

To add imaginary numbers we add the real components and we add the imaginary components:

ComplexNumberSum = (2+1)+(3+4)i
ComplexNumberSum = (3+7i)

Exactly the same result (numbers) that we got from vector addition. Just a different notation. And that ##**#@ confusing "i".

If we use the number pair notation suggested by Dr. Math in the earlier reference, then it looks almost exactly like the vector addition and I will not take the space to do it here.

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.

Provided you work with all things orthogonal.

No, not at all. See the answer above.

So, they are used simply because they work. No other reason, really.

Click to expand...

Isn't that a rather obvious statement? Why would anyone use a method that does not work?

Well, yes it is obvious. But it is also deeply profound. So deeply profound that none of the previous answers even suggested it. After all the various explanations of complex numbers in the other posts above, it really is the ultimate reason why complex numbers are used. 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.

Paul A.

Ratch · Sep 23, 2012

Chips & Chips,

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).

Sorry, I guess I was hungry when I answered your last post.

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.

Not everything is a model. The dictionary defines a model as "A standard or example for imitation or comparison." It would be a stretch to say adding up a grocery bill is a "model". The solution of a arbitary differential equation is not a model unless it is declared to be based on a physical example.

I said they are a good way. I did not say they were the only way.

Because you only mentioned one dimension, I assumed you meant only one dimension. Why did you not say reals could be applied to all dimensions?

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.

That has got to be the most unintutive way I have ever seen to describe duplex (complex) numbers. You really have to be an algebra jockey to get anywhere with that method. How does he do complex exponentials like j^j (j raised to the j power)? Look at this link where I go around and around with a person about what complex numbers are. https://www.electro-tech-online.com...t-imaginary-numbers-make-so-much-sense-2.html

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, they cannot. Have you ever heard of the dot product, box product, cross product, or triple product of a complex number? A spacial vector defines magnitude and direction. A complex number defines the magnitude of two or more quantities orthogonal to each other.

No, not at all. See the answer above.

See rebuttal above.

Well, yes it is obvious. But it is also deeply profound. So deeply profound that none of the previous answers even suggested it.

Why mention the obvious?

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.

All the above history and methodology is irrelevant with respect as why we use complex numbers. We use complex numbers because there are many things like voltage/current phase and wave mechanics that have orthogonal relationships. It is not because of any scientific methodology.

Ratch

rajput9571 · Sep 27, 2012

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. Complex number could write as (a+ib), where a a and b and i stands for √-1.
**broken link removed**

Ratch · Sep 27, 2012

rajput9571,

Complex number is used to represent imaginary number with real numbers,...

It appears you have not read or assimulated the above response I posted, or the link I submitted.

... a special imaginary number i is used to express √-1 which is not possible but sometimes it needs to express results.

Didn't you "get it", that "i" is an operator, not a number, and 1j is a definite number which can be expressed as 1/_90° in polar form?

What does that link defining rational numbers have to do with anything?

Ratch

Chips & Chips · Oct 8, 2012

Ratch,

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. Just something that works under some circumstances. 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 we use it simply because it works.

rabel126 · Oct 8, 2012

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|.

LvW · Oct 8, 2012

rabel126 said:
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|.

Hey - for my opinion an interesting discussion/definition regarding the term "model".

In this context, I am afraid that even the phenomenon we call "current" is nothing else than a model. Or am I wrong?

Ratch · Oct 8, 2012

Chips & Chips,

You are limiting your thinking too much. And you are trying too hard to find a real world existence for these numbers.

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.

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.

What does "modeling" have to do with complex numbers?

That theory is a model.

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.

The theory/model is not actual reality.

No more than a description of how something exists or works.

Newton came up with a theory/model of gravity. It works well, but it does break down under some circumstances.

No, he observed and documented the Law of Gravity. He never advanced a theory of how it exists.

So there is no ABSOLUTE, real world association here.

Whoever said there was?

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.

And the point is?

It does not matte what i or j mean.

Of course it does. "i" or "j" is an operator. That is like saying it does not matter what the math operators like =,-,/,^ mean.

It does not matter how that math was invented

As long as the concept is correct.

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.

When has math ever stopped working? The advancement from Newton to Einstein was a physics addendum, not a math advancement.

So the answer to the original question is simply because it works.

Isn't that stating the obvious?

Ratch

- - - Updated - - -

Rabel126,

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|.

Reactive current is also active, otherwise it would not exist. You must mean "real" and "orthogonal". Aren't the definitions of current and its magnitude you gave stating the obvious? What is your point?

Ratch

Chips & Chips · Oct 9, 2012

Ratch said:
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.

OK, you seem to take delight in taking my answers apart so why not another round. I am not really trying to argue, but just present another viewpoint. And I rechecked all your responses and did not see any "real world" example of a duplex or complex number. Did I miss something?

What does "modeling" have to do with complex numbers?

They are used as part of the model.

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.

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. If it does not then it must be modified or abandoned. But theories are never, ever to be considered absolute fact. They are only the best explanations that we have to date.

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.

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.

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.

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.

As long as the concept is correct.

Again, that is what I am saying. As long as it works correctly, we use it.

When has math ever stopped working? The advancement from Newton to Einstein was a physics addendum, not a math advancement.

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.

Isn't that stating the obvious?

Ratch

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.

Ratch · Oct 9, 2012

Chips and Chips,

And I rechecked all your responses and did not see any "real world" example of a duplex or complex number. Did I miss something?

No, I did not list any. I didn't think it was necessary to mention electronics, control theory, mathematics, physics, and anything where a sinusoidal phase is present between two quantities.

They are used as part of the model.

OK, fine. But your explanation of a model did not say that.

I don't see the difference. A theory and a model are essentially the same thing.

Well, I disagree. I already explained the difference. You can't call everything a model. A theory about something can be checked without any model being involved.

An assumption about how to explain experimental observations that must be tested. If it predicts further things well, it becomes stronger.

Theories become stronger because they explain happenings, not because they are predictive.

What do you mean by that.

Just what I said. Newton discovered the Law of Gravity. Specifically that mass attracts itself. He discovered laws that were always correct with respect to how that affected the dynamics of moving and stationary masses. He never advanced a theory of why mass attracts itself. He only published the results of this behavior, not why it existed in the first place.

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 does all this have to do with uses of complex numbers? Anyway, Newton did not formulate a theory of gravity. All this discourse about how science advances is off the mark with respect to the uses of complex numbers.

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.

Theories never turn into laws. See the last paragraph of this link. https://indianapublicmedia.org/amomentofscience/can-a-theory-evolve-into-a-law/

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.

Since j is defined to be an operator, and the definition includes what the operator does, there can only be one way at looking at it. How many ways can you look at the addition operator "+"? I never said it has a magical existence in reality. In fact, I said just the opposite.

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.

Then it's the physics that stops working, not the math.

Sometimes the obvious thing is also the most profound. So sometimes it does need to be stated.

Not to those who are familiar and have knowledge of the subject.

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.

Complex numbers are used to describe orthogonal quantities. Once described, calculations become routine. Predictions occur when the calculatons are set up and interpreted correctly.

Ratch

jeffrey samuel · Oct 9, 2012

This segment is heated up I guess

Then what exactly why did you bring the word model in that is drawing a lot of heat

And great thing of splitting our answers apart Ratch

Chips & Chips · Oct 9, 2012

I brought the word "model" into the discussion to try to bring a new way of viewing the topic to some of the people discussing it here. My background includes a fair amount of reading on the philosophy of science in addition to scientific and engineering training. I like to think that it is important to understand how science itself works. Too many working scientists do not have a good appreciation of this. Some do not have any.

jeffrey samuel · Oct 9, 2012

I find models both helpful and not so helpful in any analysis

But the implementation part here is not so easy as a model is not exactly a rotated image as in the case of the complex numbers

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