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ElwoodBlues · M
Well, 8.732891741295967 is the [i]cube[/i] root of evil don't forget!!!
Oh, right, you're gonna want the other two cube roots😂🤣
-4.366445870647983+7.562906096461629i
4.366445870647983-7.562906096461629i
This shows the 3 cube roots of 1 where x is the real axis and y is the "imaginary" axis.
![]()
Oh, right, you're gonna want the other two cube roots😂🤣
-4.366445870647983+7.562906096461629i
4.366445870647983-7.562906096461629i
This shows the 3 cube roots of 1 where x is the real axis and y is the "imaginary" axis.
helenS · 36-40, F
@ElwoodBlues What's i^(i)?
ElwoodBlues · M
@helenS So the question is: once we have complex numbers, are we done? Short answer: yes. Long answer:
Let's start with counting numbers. We can add them and or multiply them and still get a counting number. Those operations are "closed" for the set of counting numbers. Great!
But what about subtraction? Oops, gotta invent negative numbers. Now we have the set of integers.
But what about division? Oops, gotta invent "rational" numbers, those that can be expressed as ratios. BTW, the rational numbers are still countable, meaning you can form a 1:1 correspondence between rationals and counting numbers.
Unfortunately, multiplication implies squaring, cubing, etc - in fact all finite algebraic expressions. No problem, invent the "irrationals" which include all roots, sums of roots, fractions of roots, etc. But what about the square root of -1, etc? I'll get to that right after trancendentals.
Solutions to certain important problems are best expressed as waves (sin & cos) or exponentials or logs. If you try to write those as algebraic expressions, you get an infinite series. The solution numbers in general are trancendentals, and are NOT the roots of any merely algebraic expression.
Trancendentals + all the others are the real numbers. The reals form a continuum, and the reals are uncountable. A countable infinity is known as Aleph Null; the first uncountable infinity is known as Aleph One. Don't ask me about other Alephs!!
As long as you avoid sqrt(-1), log(-1) arcsin(2) and a few nasty things like that, all your operations are closed on the set of reals. <satisfied sigh>
However, if we DO allow all those nasty operations (electrical engineers need a bunch of them), we only have to add complex numbers. All the operations, including the nasty ones, are closed on the set of complex numbers. Some people call it the complex plane, because a complex number can be seen as a point on a plane with real & I axes. Now we're done! Kinda.
A guy named Hamilton invented a kind of object called a quaternion which had 4 components, one real and three different kinds of "complex." And quaternions are useful for representing rotations and orientations in 3-dimensional space; frequently used in computer graphics and in satellite control. But you don't NEED all three flavors of complex. You can represent the same thing as vectors with special rules (or as a direction vector (axis) plus a rotation (angle)). Somebody else invented octonions with 8 elements, but again, you don't NEED eight flavors of complex.
[sep][sep][sep]
tl;dr all the operations are closed on the set of complex numbers AKA the complex plane.
Let's start with counting numbers. We can add them and or multiply them and still get a counting number. Those operations are "closed" for the set of counting numbers. Great!
But what about subtraction? Oops, gotta invent negative numbers. Now we have the set of integers.
But what about division? Oops, gotta invent "rational" numbers, those that can be expressed as ratios. BTW, the rational numbers are still countable, meaning you can form a 1:1 correspondence between rationals and counting numbers.
Unfortunately, multiplication implies squaring, cubing, etc - in fact all finite algebraic expressions. No problem, invent the "irrationals" which include all roots, sums of roots, fractions of roots, etc. But what about the square root of -1, etc? I'll get to that right after trancendentals.
Solutions to certain important problems are best expressed as waves (sin & cos) or exponentials or logs. If you try to write those as algebraic expressions, you get an infinite series. The solution numbers in general are trancendentals, and are NOT the roots of any merely algebraic expression.
Trancendentals + all the others are the real numbers. The reals form a continuum, and the reals are uncountable. A countable infinity is known as Aleph Null; the first uncountable infinity is known as Aleph One. Don't ask me about other Alephs!!
As long as you avoid sqrt(-1), log(-1) arcsin(2) and a few nasty things like that, all your operations are closed on the set of reals. <satisfied sigh>
However, if we DO allow all those nasty operations (electrical engineers need a bunch of them), we only have to add complex numbers. All the operations, including the nasty ones, are closed on the set of complex numbers. Some people call it the complex plane, because a complex number can be seen as a point on a plane with real & I axes. Now we're done! Kinda.
A guy named Hamilton invented a kind of object called a quaternion which had 4 components, one real and three different kinds of "complex." And quaternions are useful for representing rotations and orientations in 3-dimensional space; frequently used in computer graphics and in satellite control. But you don't NEED all three flavors of complex. You can represent the same thing as vectors with special rules (or as a direction vector (axis) plus a rotation (angle)). Somebody else invented octonions with 8 elements, but again, you don't NEED eight flavors of complex.
[sep][sep][sep]
tl;dr all the operations are closed on the set of complex numbers AKA the complex plane.
helenS · 36-40, F
@ElwoodBlues Thank you – I should like to add infinitesimals to your list of number types. Yes I know they became unfashionable in the late 19th century, and were replaced by Cauchy's limit concept, but at least in physics they are still running strong.
If you increase the temperature by an infinitesimal amount dT, from T_0 to T_0 + dT, the internal energy U will increase by dU, and dU/dT is the heat capacity. That's much clearer than saying dU/dT is the limit of ΔU/ΔT if ΔT approaches Zero.
Infinitesimals! Give me infinitesimals!
If you increase the temperature by an infinitesimal amount dT, from T_0 to T_0 + dT, the internal energy U will increase by dU, and dU/dT is the heat capacity. That's much clearer than saying dU/dT is the limit of ΔU/ΔT if ΔT approaches Zero.
Infinitesimals! Give me infinitesimals!
ElwoodBlues · M
@helenS I've got no objection really. I'm fine with infinitesimals; I'm just not sure either way whether infinitesimals are separate category of numbers. I don't even know if they're different from the delta & epsilon in D-E proofs which are very much still a thing.
helenS · 36-40, F
@ElwoodBlues Yes they are very different from reals. Infinitesimals are an extension of the real number system, just like complex numbers.
Look at this equation:
[i]dy = k * dx[/i]
The product of the infinitesimal [i]dx[/i] and a real number [i]k[/i] will always be infinitesimal, however big [i]k[/i] is. The product will never be >1, so [i]dx[/i] isn't even Archimedean.
Look at this equation:
[i]dy = k * dx[/i]
The product of the infinitesimal [i]dx[/i] and a real number [i]k[/i] will always be infinitesimal, however big [i]k[/i] is. The product will never be >1, so [i]dx[/i] isn't even Archimedean.
helenS · 36-40, F
@ElwoodBlues It's amazing how easy it is to see if a given real function is continuous at x=x0, using hyperreals: if x is increased by an infinitesimal amount dx, then y(x) will also change by an infinitesimal quantity dy.
Check with y=x^2 if you want to see a trivial example:
y(x+dx) = (x + dx)^2 = x^2 + 2x*dx + (dx)^2;
y(x+dx) – y(x) = 2x*dx + (dx)^2.
2x*dx + (dx)^2 is, of course, infinitesimal, so the function is continuous for all x∈ℝ.
Compare that with Cauchy/Weierstrass epsilontics!
I've never read anything about the "surreals", but I read an introductory textbook about the hyperreals some time ago, and it clicked immediately. Now I'm no longer ashamed to see dy/dx as a fraction, instead of a limit.
Hey it's nice to talk with someone on SW about maths! 🌷
Check with y=x^2 if you want to see a trivial example:
y(x+dx) = (x + dx)^2 = x^2 + 2x*dx + (dx)^2;
y(x+dx) – y(x) = 2x*dx + (dx)^2.
2x*dx + (dx)^2 is, of course, infinitesimal, so the function is continuous for all x∈ℝ.
Compare that with Cauchy/Weierstrass epsilontics!
I've never read anything about the "surreals", but I read an introductory textbook about the hyperreals some time ago, and it clicked immediately. Now I'm no longer ashamed to see dy/dx as a fraction, instead of a limit.
Hey it's nice to talk with someone on SW about maths! 🌷