## Negative and complex numbers

Beyond the boundaries of established science an avalanche of exotic ideas compete for our attention. Experts tell us that these ideas should not be permitted to take up the time of working scientists, and for the most part they are surely correct. But what about the gems in the rubble pile? By what ground-rules might we bring extraordinary new possibilities to light?

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### Negative and complex numbers

Negative numbers are a human convention. These do not exist in nature. The "number" line, when it goes below 0 it's a convention. Nobody can have -3 apples, unless by convention we say he owes someone 3 apples.

https://en.wikipedia.org/wiki/Negative_number

-(-2) = 2
Why?
Because by convention the - in front says we are getting rid of (-2), meaning we are getting rid of a debt of 2, meaning we gain 2.

Also (-2) * (-3) = 6 to preserve the distributive property of multiplication.

But (+2)*(+3) does not equal (-6). In this way the number line is not symetric.

Then the square root of (-1) is i, by convention.
And this i seems to be useful in some areas:
https://en.wikipedia.org/wiki/Complex_n ... plications

(-1) = a debt of 1. What quantity multiplied with itself gives us a debt of 1? Or a debt of 100 = (-100)? It's not -1 or -10 because of the rules established above. The answer is i or 10i, by convention. Debt is a concept, it's not found or at least not visible in nature.
I find it amazing that this human established convention: i, can be used for equations that model the real world.
Maybe "debt" does exist in nature, and it's fundamental for making the Universe move.
Roshi

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### Re: Negative and complex numbers

An interesting video:

Visualizing the Riemann hypothesis and analytic continuation

I understand someone decided to use i to represent sqrt(-1). Because there are no real numbers that do that. So they used this "imaginary" number i and said "i that has this property".
And what does this have to do with nature?

Apparently it has something to do because of this:

http://www.math.toronto.edu/mathnet/que ... nlife.html
Another example is electromagnetism. Rather than trying to describe an electromagnetic field by two real quantities (electric field strength and magnetic field strength), it is best described as a single complex number, of which the electric and magnetic components are simply the real and imaginary parts.

Even if I don't understand anything about it. To me, the use of imaginary numbers: "let's say we have this magic number i that has this property, and then we have numbers in the shape of a+bi, real+imaginary, and let's use them in engineering", sounds like witchcraft, and it's amazing it works.

Are imaginary numbers vital to these calculations or just make them easier?

Why do they represent imaginary numbers on an axe perpendicular to the real numbers axe? Two perpendicular axes mean coordinates on a plane. What does this representation have to do with reality?
Roshi

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### Re: Negative and complex numbers

To describe a process that oscillates, such as an AC electrical circuit, how else would you describe or identify opposite polarity voltage or current flow?
Maol

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### Re: Negative and complex numbers

`
Maol politely states it as a question, but he’s absolutely right; the concept is essential to the representation of electricity concurrent in space and time.
Perhaps a more acceptable term would be counter-space, which also provides an algebraic means of comprehending the existence of inductance and capacitance simultaneously in an electric circuit. [Something the early Quaternion Math attempted at the cost of great confusion and which Maxwell’s Displacement Currents hint at, but is much better presented by versor expressions, (as opposed to common vector expressions)].

For a living author, see the works of Eric P Dollard, e.g. Four-Quadrant Representation of Electricity
seasmith

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### Re: Negative and complex numbers

I was gonna say, ' like a tank circuit ', but not sure if that would have resonated <-- shameless pun
Maol

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### Re: Negative and complex numbers

Maol wrote:To describe a process that oscillates, such as an AC electrical circuit, how else would you describe or identify opposite polarity voltage or current flow?

I have no idea what you are talking about. I only learned physics in high-school, and there were no complex numbers there.

What does 3+2i mean in reality? What does 2i mean in reality? Give me a quantity of 2i if you can...

http://visualizingmathsandphysics.blogs ... in_18.html
Lets take a number of 3 + 4i 3 ------> is actually the horizontal component of the force. 4 -------> is actually the vertical component of the force. Complex numbers come into place whenever one force gets divided into two or more components due to inclination or whatever other reason. There are more that one way an object can be inclined and thus more than one way these forces get divided into two. The i, j and k planes are a resultant of this.

Ok. But why the "+" sign between them? Between two entities that cannot be added, and in fact are coordinates on 2 axes. Perhaps it's not a real "+", and it has another meaning, and it's somehow easier to use these a+bi notation instead of X,Y coordinates.
Roshi

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### Re: Negative and complex numbers

Perhaps I sound stupid but if a number is not a quantity then what is it? What is sqrt(-1) and why is it in math?
Roshi

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### Re: Negative and complex numbers

This topic brings up a Buckminster Fuller memory.
In one of his early books "Bucky" talked about the i symbol in math as a kind of fudge factor;
especially as math relates to nature. As I recall, the i appears when the 90* math is used to model nature which He said was built on 60*. He claimed that his 60* math did away with the need for the i.
This might be a bit off what he actually said.
It has been 50 years.
Jack
jacmac

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### Re: Negative and complex numbers

I watched this:

What I understand from it: Complex numbers are a notation, to represent "higher dimensions", and they can also be written as matrices. So a+bi is in fact a matrix. But the notation a+bi is used perhaps to keep things simpler.

All these problems with square roots of negative numbers come because of multiplicaton.
Addition is cool, substraction give us negative numbers (these were a mystery for a long time), then multiplication comes and the reverse of multiplication does not always work.

So, i appeared because some math operations forced us into these higher dimensions.
Roshi

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### Re: Negative and complex numbers

2 x 3 = 6

A debt of 2 multiplied 3 times = a debt of 6:
(-2) x 3 = - 6

But:
(-2) x (-3) = 6 ?

Two debts multiplied:
(-2) x (-3) = -((-2) x 3) = -(-6) = 6 Subtract a debt of 6 , meaning we gain 6.

What about sqrt(-1)? Maybe this should be written as -sqrt(1) = -1? Or not applied at all, except on positive numbers, because it cannot function on negatives. Instead of inventing an imaginary "quantity". Some say "it's real, it's on an imaginary axis that is perpendicular to the real number line". Why? And why the "+" between a+bi ? And a+bi does not equal an area, as two coordinates on perpendicular axes should.

If we need to calculate something in 2D or 3D we can create a coordinate system and apply Pythagora, with real numbers.

We see that the only way to get a negative number (a debt) is by multiplying a positive and a negative number.
Meaning -1 = 1 x (-1) = - (1 x 1). A debt multiplied by itself, the debt will become a gain. (-1) x (-1) = 1. sqrt (-1) should not be an allowed operation.

As I see it, this "imaginary number" i is outside the reality of physics. All lengths and quantities are positive. A negative number, a debt, appears only if we humans do some math operations and create this concept. And it can be understood, and it works. I can't understand i.

Look at cos (-1) = pi. Because it's defined on a real physical object, a circle.
sqrt(-1) is defined on nothing, it's just "let's imagine this number i exists that satisfies this equation sqrt(-1) = i. And i is in fact not a number, or a quantity or a length. Not cool
Roshi

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### Re: Negative and complex numbers

Sorry, I mean cos(pi) = -1.

Here:

From the video:
"The brilliant thing about mathematicians: when they are on their way to a mathemathical discovery they don't let a little thing like a number not existing stop them"

Most cool. That's how we should treat all unsolvable problems, by inventing "imaginary" stuff.
But I found this answer that satisfies me:
https://electronics.stackexchange.com/q ... hase-of-ac
Therefore: Yes - using complex mathematics for describing sinusoidal signals has no direct physical relevance. It is just to "make analyses easier".

As an example: Introducing Euler`s famous formula for sinus signals into the Fourier series leads to negative frequencies (symmetrical to positive frequencies). Hence, the question arises: Do negative frequencies exist in reality? The answer is NO! It is just a helpful mathematical tool
Roshi

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### Re: Negative and complex numbers

All things being equal and real, a negative is only a result of the difference between 2 positives. There is no negative in nature, and the number line is only a 1d mind picture so it can not be used as evidence of anything real.
interstellar filaments conducted electricity having currents as high as 10 thousand billion amperes
Cargo

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### Re: Negative and complex numbers

The number line positive side can be any length or quantity in reality. But the real world may have a limit to how much it can be divided.
Then there are the "irrational numbers" like pi. A perfect circle cannot exist, either it's circumference or radius would be "irrational" meaning - never of a fixed value. But as I said - reality may be made up of "discrete units", so a length can't have an infinity of decimals.

https://en.wikipedia.org/wiki/Planck_units
The term "Planck scale" refers to magnitudes of space, time, energy and other units, beyond (or below) which the predictions of the Standard Model, quantum field theory and general relativity are no longer reconcilable, and quantum effects of gravity are expected to dominate. This region may be characterized by energies around 1.22×1019 GeV (the Planck energy), time intervals around 5.39×10−44 s (the Planck time) and lengths around 1.62×10−35 m (the Planck length). At the Planck scale, current models are not expected to be a useful guide to the cosmos, and physicists no longer have any scientific model whatsoever to suggest how the physical universe behaves. The best known example is represented by the conditions in the first 10−43 seconds of our universe after the Big Bang, approximately 13.8 billion years ago.

"The best known" example... Yes, it's very well imagined.

The negative side of the line can represent the concept of debt, or two forces pulling in opposite directions.
Roshi

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