Imaginary numbers... (last one, I promise)
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I'm slightly leaning into your statement, but to be fair to mathematicians, they can calculate sqrt(-1). It's sqrt(-1). It's a misunderstanding of what an imaginary number is to assume that it can be further simplified. sqrt(-1) is a 90 degree counter-clockwise rotation into the "imaginary" plane from 1 (which is a rotational plane, not a cartesian plane). That's why if you rotate it again (i.e. 1 * i * i) you arrive at -1. And if you rotate twice again you end up back at 1. They can be very useful for doing rotational math in a standard cartesian plane, for example, and avoid doing coordinate-system translations. This is a good article [^] I read awhile ago on the topic. I know in school my teachers never really explained them beyond "they just pop up so you need to know the rules", so I never developed any intuition on what they meant. A quote from the article, attributed to Carl Gauss:
If +1, -1, √-1 had not been called a positive, negative, imaginary (or even impossible) unit, but rather a direct, inverse, lateral unit, then there could hardly have been any talk of such obscurity.
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I'm slightly leaning into your statement, but to be fair to mathematicians, they can calculate sqrt(-1). It's sqrt(-1). It's a misunderstanding of what an imaginary number is to assume that it can be further simplified. sqrt(-1) is a 90 degree counter-clockwise rotation into the "imaginary" plane from 1 (which is a rotational plane, not a cartesian plane). That's why if you rotate it again (i.e. 1 * i * i) you arrive at -1. And if you rotate twice again you end up back at 1. They can be very useful for doing rotational math in a standard cartesian plane, for example, and avoid doing coordinate-system translations. This is a good article [^] I read awhile ago on the topic. I know in school my teachers never really explained them beyond "they just pop up so you need to know the rules", so I never developed any intuition on what they meant. A quote from the article, attributed to Carl Gauss:
If +1, -1, √-1 had not been called a positive, negative, imaginary (or even impossible) unit, but rather a direct, inverse, lateral unit, then there could hardly have been any talk of such obscurity.
I can further the development: see (-1)^(1/4) :-\ Imaginary numbers are actually from Italy. Solving the quintic, they came up, and everyone said they weren't real, but Cardano showed that you could actually plug in the answers and get zero out, so... They did give answers, but not a real solution that you could use in this world.
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I'm slightly leaning into your statement, but to be fair to mathematicians, they can calculate sqrt(-1). It's sqrt(-1). It's a misunderstanding of what an imaginary number is to assume that it can be further simplified. sqrt(-1) is a 90 degree counter-clockwise rotation into the "imaginary" plane from 1 (which is a rotational plane, not a cartesian plane). That's why if you rotate it again (i.e. 1 * i * i) you arrive at -1. And if you rotate twice again you end up back at 1. They can be very useful for doing rotational math in a standard cartesian plane, for example, and avoid doing coordinate-system translations. This is a good article [^] I read awhile ago on the topic. I know in school my teachers never really explained them beyond "they just pop up so you need to know the rules", so I never developed any intuition on what they meant. A quote from the article, attributed to Carl Gauss:
If +1, -1, √-1 had not been called a positive, negative, imaginary (or even impossible) unit, but rather a direct, inverse, lateral unit, then there could hardly have been any talk of such obscurity.
Sorry for that question, but at the moment I'm too lazy to think in deep for it... The sqrt of a positive number x has always two solution +/- (abs(sqrt(x))). How it is about the sqrt of a negative number? Are there also two solutions? Sorry again, to be that lazy and simply ask for it ;)
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Physicists do the same all the time.
"In testa che avete, Signor di Ceprano?" -- Rigoletto
CPallini wrote:
Physicists do the same all the time.
Presuppose they don't :laugh:
M.D.V. ;) If something has a solution... Why do we have to worry about?. If it has no solution... For what reason do we have to worry about? Help me to understand what I'm saying, and I'll explain it better to you Rating helpful answers is nice, but saying thanks can be even nicer.
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Sorry for that question, but at the moment I'm too lazy to think in deep for it... The sqrt of a positive number x has always two solution +/- (abs(sqrt(x))). How it is about the sqrt of a negative number? Are there also two solutions? Sorry again, to be that lazy and simply ask for it ;)
x^2 - 1 = 0 x = +/- sqrt(1) => x^2 = 1 x^2 + 1 = 0 x = +/- sqrt(-1) => x^2 = -1
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Sorry for that question, but at the moment I'm too lazy to think in deep for it... The sqrt of a positive number x has always two solution +/- (abs(sqrt(x))). How it is about the sqrt of a negative number? Are there also two solutions? Sorry again, to be that lazy and simply ask for it ;)
The reason for the +/- is not due to imaginary numbers, it's a byproduct of how square roots are defined in general. Squaring a number removes sign information, so while we can reverse the process to determine the magnitude of the original value, we can't regain the sign information* so we put +/- because we don't know. *: You can reason about and figure this out sometimes, but it relies on the specific problem and how it's constrained.
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I'm slightly leaning into your statement, but to be fair to mathematicians, they can calculate sqrt(-1). It's sqrt(-1). It's a misunderstanding of what an imaginary number is to assume that it can be further simplified. sqrt(-1) is a 90 degree counter-clockwise rotation into the "imaginary" plane from 1 (which is a rotational plane, not a cartesian plane). That's why if you rotate it again (i.e. 1 * i * i) you arrive at -1. And if you rotate twice again you end up back at 1. They can be very useful for doing rotational math in a standard cartesian plane, for example, and avoid doing coordinate-system translations. This is a good article [^] I read awhile ago on the topic. I know in school my teachers never really explained them beyond "they just pop up so you need to know the rules", so I never developed any intuition on what they meant. A quote from the article, attributed to Carl Gauss:
If +1, -1, √-1 had not been called a positive, negative, imaginary (or even impossible) unit, but rather a direct, inverse, lateral unit, then there could hardly have been any talk of such obscurity.
Jon McKee wrote:
I know in school my teachers never really explained them beyond
Unfortunately there are very many people that do not understand proofs. And even for those that do (or claim so) they don't understand what assumptions and definitions really mean for those proofs.
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Mathematicians are special. They cannot calculate sqrt(-1) and instead of giving up math, they simply state that we have discovered a new number. I swear, you cannot win against these people...
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One of my math professors had said that "all numbers are imaginary". For example, I can show you two oranges, or two cats, but I cannot show just "two". So, all the numbers we deal with, are in our head, mind. We write it, of course, as 2, but that is just a representation of "two" in my written language. The same 2 is written as ೨ in my language Kannada, and as २ in Devanagari script. But none of them are the concept called "two"; the concept called "two" is imagined in my mind, it is imaginary. This was the logic of my professor. Stated otherwise, I cannot see, hear, touch, smell or taste "two" or "three". None of the numbers is tangible in that sense. The concept of "imaginary number i" takes that imagination abstraction to the next level.
You missed the joke.....
ed
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I can further the development: see (-1)^(1/4) :-\ Imaginary numbers are actually from Italy. Solving the quintic, they came up, and everyone said they weren't real, but Cardano showed that you could actually plug in the answers and get zero out, so... They did give answers, but not a real solution that you could use in this world.
Kenneth Haugland wrote:
Imaginary numbers are actually from Italy. Solving the quintic, they came up, and everyone said they weren't real, but Cardano showed that you could actually plug in the answers and get zero out, so... They did give answers, but not a real solution that you could use in this world.
Actually, it was to solve the cubic. The only way to solve a real cubic that has 3 real solutions is to have an intermediate quadratic solution of complex conjugates. The quintic cannot be solved, and that was proven by Abel, who refined the proof that Ruffini had done (and which was a mess). Later on, Galois used Abel's work to come up with the idea of permutation groups, from which spring the discipline of Group Theory. Arnold came up a with proof that does not rely on Group Theory, but instead relies on the complex plane and functions that wind around the plane.
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Kenneth Haugland wrote:
Imaginary numbers are actually from Italy. Solving the quintic, they came up, and everyone said they weren't real, but Cardano showed that you could actually plug in the answers and get zero out, so... They did give answers, but not a real solution that you could use in this world.
Actually, it was to solve the cubic. The only way to solve a real cubic that has 3 real solutions is to have an intermediate quadratic solution of complex conjugates. The quintic cannot be solved, and that was proven by Abel, who refined the proof that Ruffini had done (and which was a mess). Later on, Galois used Abel's work to come up with the idea of permutation groups, from which spring the discipline of Group Theory. Arnold came up a with proof that does not rely on Group Theory, but instead relies on the complex plane and functions that wind around the plane.
My comment got autotuned/autocorrected on me, but yes. Between the proofs and the Renaissance Italians, there was also the paper from Lagrange that showed the similarities in how to get the solutions to these two equations, cubic and quartic, which is basically the start of group theory. I wanted to study complex numbers in detail but never got around to it. I mean, I know the basics, but there are a lot of neat and cool theorems you can use from them to solve real-world problems.