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breckenridgeback t1_j69pjlm wrote

Sure, in the same way that, when you don't know about fractions, you can say 3 divided by 2 isn't possible, because there's no integer that, when multiplied by 2, gives you 3.

But you don't do that, because inventing the idea of a number 3/2 is pretty useful, and lets you do many calculations that you couldn't do without it. In particular, you get ideas like "3/2 is the same kind of no-solution-is-possible as 6/4 is".

Imaginary numbers are the same kind of thing. They aren't "impossible", and "imaginary" is just a name (you could call them the "Bob numbers" if you wanted to). They are a perfectly well-defined algebraic object. They're only imaginary in the sense that they don't fit onto the number line, and human intuition about numbers tends to come from ideas like length that happen to be real-valued.

But in practice, imaginary numbers show up all the time in descriptions of the world around us. In particular, they're critically important in quantum mechanics, where the fact that the states of particles aren't real-valued is essential to the theory (many quantum-mechanical phenomena actively require this). They also turn out to produce very compact representations of things like periodic behavior, as with a spring or a pendulum. And relativity tells us that the relationship between space and time is the same kind of thing as the relationship between the number "3" and the number "3i".

You could do all of this with objects that are just pairs of real numbers (a,b), defined in such a way that (a,b) "times" (c,d) = (ad + bc, ac - bd) and (a,b) "plus" (c,d) = (a + c, b + d). But the object you've invented has exactly the same properties as imaginary numbers do, so why add the extra complexity?

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betelguese_42 t1_j6c2mwk wrote

TLDR: The name imaginary numbers is bad and people would find it less futile if it was called Bob numbers

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Flaky-Proof5511 t1_j6cbvrs wrote

Didn't the YouTuber 3Blue1Brown suggested rotation numbers as a more user friendly name ?

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Magnetic_Syncopation t1_j6d9syu wrote

Another commenter mentioned that you can substitute calculations in electronic circuits that would normally use imaginary numbers instead with vectors. But it's a lot more work doing that, whereas imaginary numbers make calculations easier to do.

It seems like imaginary numbers provide a way of rotating between magnitudes in a sine/cosine way

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lunatickoala t1_j6chnor wrote

Imaginary number is a bad name for them... and intentionally so. It was meant to be derogatory. Mathematics has a history of people not liking new developments because people think of math as a very logical and objective thing and those developments can fly in the face of what they believe.

There's an apocryphal story that someone in the Cult of Pythagoras proved that the square root of two is irrational and they were so outraged by the idea that a number could be irrational that they took him to sea in a boat and returned without him. They believed that all numbers could be expressed as a ratio of two integers and an irrational number by definition is one that can't be expressed as a ratio.

The Ancient Greeks also didn't believe in the idea of zero or negative numbers and both were very controversial in the Western world for many centuries afterwards. In math today, the standard form for polynomials to put all the coefficients on one side and set it to zero because it's really useful. For example, Ax^2 + Bx + C = 0 for the quadratic polynomial where B and C are allowed to be zero and A/B/C are all allowed to be negative. But up until I think the 1500s, Western mathematicians didn't have a standard form but a family of forms specifically to avoid zeroes and negative numbers. Ax^2 + Bx = C, Ax^2 = Bx + C, Ax^2 = C, Ax^2 = Bx, etc.

Imaginary numbers first saw real use in the cubic equation because the people who found it realized that in some cases it involved taking the square root of a negative number, which people believed to be nonsensical. However, the cubic equation worked because the imaginary numbers cancelled each other out. Thus, they were called imaginary because people didn't think they were "real" and were only a mathematical trick that happened to work out and not something that's meaningful.

To get a feel for what it was probably like when irrational numbers, zero, negative numbers, and imaginary numbers were first introduced, look at the comments whenever 1+2+3+4+5+6+... = -1/12 comes up.

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otherestScott t1_j6d03us wrote

It’s not a bad name honestly, because “real numbers” also means something important in math and it excludes imaginary numbers

If we called them Bob numbers, you’d either have to change the name of “real numbers” to “non-Bob numbers” which isn’t great, or you would feel a lot less natural to exclude them.

It’s the same as rational vs irrational numbers. It’s not that pi or the square root of 2 is overly emotional, it’s just a convenient binary for mathematicians to use.

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Panceltic t1_j6cp5zt wrote

The same issue with grammatical gender. Instead of “masculine, feminine and neuter nouns” we could call them “Type A, B and C” and avoid many pointless discussions.

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pmruiz4 t1_j6cm5cp wrote

Electrical engineer here, which is 90% dealing with imaginary numbers. On that comment on using pairs of real numbers instead of imaginary numbers:

My uncle had (and gave me) a very old book on electrical engineering that described all of the equations and theory without ever mentioning imaginary numbers - they used vectors instead, so they had to tap into vector calculations for even the most basic operations.

Needless to say I never really read it as it was borderline impossible to follow once you knew how easy it would be to do the same with imaginary numbers instead...

edit. typos...

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MaxRoofer t1_j6ckf36 wrote

“In practice, it Happens all the time in the world around us, in theory”.

You may be correct I’m not smart enough but that’s sort of how you explained it.

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sveinb t1_j6cn15f wrote

Excellent description. I’d maybe save quantum mechanics to last though, because many people have the same feeling of doubt about qm as op has about imaginary numbers. Let’s just say that periodic phenomena are much easier to describe using complex numbers. To give a sense of why that is you can say it’s because these need to shift energy back and forth between two different forms of energy, which is reflected in the two different kinds of numbers that make up a complex number - the real part and the imaginary part. A pendulum, for example, converts potential energy to kinetic energy and back again. Quantum mechanics is just one example of periodic phenomena.

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Frednotbob t1_j6b027e wrote

Would something like '2.5 people per square mile' be considered an imaginary number? Since you can't have .5 of a person (well, you can, but that involves investigations and awkward questions XD)...

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breckenridgeback t1_j6b12wf wrote

No, "imaginary" has a specific technical meaning in mathematics. It doesn't just mean "constructed for some purpose". It means a member of a particular set of numbers that solve equations of the form x^2 = a, for some negative number a. For example, 2i is an imaginary number that solves the equation x^2 = -4.

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Frednotbob t1_j6b369s wrote

Ah, okay. Thanks :)

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Future_Club1171 t1_j6bmjid wrote

For basics i is basically the square root of negative 1, this lets it do a couple useful things cause when dealing with control systems you can help avoid real zeros by tweaking the controls to create zeros at imaginary points ( which for a quadratic is always in a A +- root(B) form). So instead of stalling out it will occilate. For a simplified case, imaginary numbers let us make a cruise control that won’t just stop working on a highway.

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Krixwell t1_j6cdt5e wrote

This comment took a sharp turn from "Explain Like I'm 5" to "Explain Like I'm 5i+30".

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Future_Club1171 t1_j6cy1ut wrote

Imaginary numbers get like that. Since well most there uses are in higher maths.

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skeeve87 t1_j6ca81v wrote

As a control systems engineer......what?

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Future_Club1171 t1_j6cys20 wrote

It’s been awhile for me, not a control engineer but did my studies in mechanical engineering. So can’t speak for what you use in your day to day, but from what I remember controls (the math behind it) is tied with laplus transformation and the whole 1/t. Basically from those classes for feedback controls the zeros in laplus will say how it behaves in real life. And one way to avoid real zeros is forcing imaginary ones.

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skeeve87 t1_j6dr80v wrote

Ohhh Laplace transformations. I graduated electrical engineering, we used it a ton in school.

I haven't used it since class.

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Future_Club1171 t1_j6dysd4 wrote

Given what I’ve heard of most engineers, because we have software designed to solve all that stuff most of the stuff in class we don’t use ourselves lol. Though understanding the theory behind it still helps. Specially when something goes wrong you know what to look for.

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