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Every time you're tempted to freak out because complex numbers, reflect that you should also be freaking out about negative numbers. Those don't exist either.

andoando 3y ago
Negative numbers are very natural to me. Math isnt just about counting real objects, but also transformations through space/time. The notion of duality or things going in reverse is extremely natural. -5 is just the inverse of 5, such that if you go 5 and go back 5, you end up at the same place.
- anyfoo 3y ago
  Complex numbers you just go sideways as well, and if you multiply you not only stretch or squish your path to the origin, you also change your angle to it by adding both angles together. Exponentiation is then doing this continuously, so you end up walking in a circle. That’s it.
anyfoo 3y ago
Besides, they’re really not that scary once you get to know them well. Take a deep dive into digital signal processing for example, and you become well comfortable with them, and understand how “the real world” fits into them quite neatly.
- woopsn 3y ago
  Or differential equations -- that grueling and irrelevant subject physical scientists waste so much time on. :)
mr_mitm 3y ago
What does it even mean for a number to exist? In math we are happy if the things we talk about are well defined. Numbers are just mathematical objects which satisfy a few interesting properties that lead to very rich structures we can study. Some of those structures map onto concepts in nature, which makes them actually useful. Instead of real and imaginary or complex we could have chosen entitirely different adjectives. They're completely arbitrary. They don't matter.
crazygringo 3y ago
There are straightforward interpretations of negative numbers that map intuitively to regular everyday situations, such as a distance in an opposite direction, moving in an opposite direction, or an amount of money owed. On the other hand, there is no straightforward intuitive mapping for complex numbers. Geometrically, they arise out of the spiral motion needed to provide a continuous solution to an oscillating function such as y=(-1)^x, but there is no regular everyday situation that mirrors that. Algebraically, they arise in solving certain polynomials, but in practical situations they are generally only a required intermediate calculation, and in practical (engineering) usage it is usually only the real roots that have any meaning. Complex equations are used greatly for modeling waveforms, but that's mainly just because it's more mathematically convenient than dealing with a bunch of sin() functions. Not because waveforms are inherently complex/imaginary -- they're not. So the idea that negative numbers are just as unintuitive as complex numbers, is an idea I think should be firmly rejected. Negative numbers make easy, intuitive, real-world sense in a way that complex numbers simply don't. There's still an ongoing philosophical debate as to the "reality" of complex numbers, and the formulation of QM plays a part in that debate. But it doesn't answer it -- similar to waveforms, we can argue whether the math behind QM is "essentially" complex, or if we use complex representations merely for convenience. It's entirely possible to express QM without complex numbers at all, obviously.
- xelxebar 3y ago
  > On the other hand, there is no straightforward intuitive mapping for complex numbers. Au contraire! Complex numbers are just rotations, translations, and scalings in the plane. In the same sense that real numbers capture 1D rigid transformations and scaling, complex numbers capture 2D rigid transformations and scaling. When you treat negative numbers as "distance in an opposite direction", you're more generally treating real numbers as a transformation of the 1D line, i.e. translation (addition) or scaling (multiplication) by some amount. Try thinking about the same but for the 2D plane and complex numbers. Addition is 2D translation and you'll see that multiplication corresponds to a rotation and a scaling. The first section of Needham's "Visual Complex Analysis" walks you through this perspective in detail, if you're interested and also want lots of good exercises.
xpl 3y ago
> negative numbers. Those don't exist either. 1. Do numbers exist? Have you seen a number in your life? 2. What does exist, can you name one thing? Because everything isn't what it seems.

andoando 3y ago

Negative numbers are very natural to me. Math isnt just about counting real objects, but also transformations through space/time. The notion of duality or things going in reverse is extremely natural. -5 is just the inverse of 5, such that if you go 5 and go back 5, you end up at the same place.

anyfoo 3y ago
Complex numbers you just go sideways as well, and if you multiply you not only stretch or squish your path to the origin, you also change your angle to it by adding both angles together. Exponentiation is then doing this continuously, so you end up walking in a circle. That’s it.

anyfoo 3y ago

Besides, they’re really not that scary once you get to know them well. Take a deep dive into digital signal processing for example, and you become well comfortable with them, and understand how “the real world” fits into them quite neatly.

woopsn 3y ago
Or differential equations -- that grueling and irrelevant subject physical scientists waste so much time on. :)

mr_mitm 3y ago

What does it even mean for a number to exist? In math we are happy if the things we talk about are well defined. Numbers are just mathematical objects which satisfy a few interesting properties that lead to very rich structures we can study. Some of those structures map onto concepts in nature, which makes them actually useful. Instead of real and imaginary or complex we could have chosen entitirely different adjectives. They're completely arbitrary. They don't matter.

crazygringo 3y ago

There are straightforward interpretations of negative numbers that map intuitively to regular everyday situations, such as a distance in an opposite direction, moving in an opposite direction, or an amount of money owed. On the other hand, there is no straightforward intuitive mapping for complex numbers. Geometrically, they arise out of the spiral motion needed to provide a continuous solution to an oscillating function such as y=(-1)^x, but there is no regular everyday situation that mirrors that. Algebraically, they arise in solving certain polynomials, but in practical situations they are generally only a required intermediate calculation, and in practical (engineering) usage it is usually only the real roots that have any meaning. Complex equations are used greatly for modeling waveforms, but that's mainly just because it's more mathematically convenient than dealing with a bunch of sin() functions. Not because waveforms are inherently complex/imaginary -- they're not. So the idea that negative numbers are just as unintuitive as complex numbers, is an idea I think should be firmly rejected. Negative numbers make easy, intuitive, real-world sense in a way that complex numbers simply don't. There's still an ongoing philosophical debate as to the "reality" of complex numbers, and the formulation of QM plays a part in that debate. But it doesn't answer it -- similar to waveforms, we can argue whether the math behind QM is "essentially" complex, or if we use complex representations merely for convenience. It's entirely possible to express QM without complex numbers at all, obviously.

xelxebar 3y ago
> On the other hand, there is no straightforward intuitive mapping for complex numbers. Au contraire! Complex numbers are just rotations, translations, and scalings in the plane. In the same sense that real numbers capture 1D rigid transformations and scaling, complex numbers capture 2D rigid transformations and scaling. When you treat negative numbers as "distance in an opposite direction", you're more generally treating real numbers as a transformation of the 1D line, i.e. translation (addition) or scaling (multiplication) by some amount. Try thinking about the same but for the 2D plane and complex numbers. Addition is 2D translation and you'll see that multiplication corresponds to a rotation and a scaling. The first section of Needham's "Visual Complex Analysis" walks you through this perspective in detail, if you're interested and also want lots of good exercises.

xpl 3y ago

> negative numbers. Those don't exist either. 1. Do numbers exist? Have you seen a number in your life? 2. What does exist, can you name one thing? Because everything isn't what it seems.