4 ms·

This is the first time I see how the famous equation e^{iɸ} = cos ɸ + i sin ɸ can be discovered from a simple assumption of what raising to the power i cou

by tromp 3y ago

This is the first time I see how the famous equation e^{i*ɸ} = cos ɸ + i * sin ɸ can be discovered from a simple assumption of what raising to the power i could mean. Namely, that for very small x, e^x is close to 1+x, not just for real numbers x, but also for complex numbers x. PS: how do I write superscripts in HN markup?

mjburgess 3y ago
A lot of the issues around complex numbers can be intuitively resolved by drawing a unit circle around 0 on a numberline, and reasoning backwards to a definition of i. Rather than reasoning forwards from sqrt(-1)
- deleted 3y ago
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- dboreham 3y ago
  Puzzled expression because that's how I thought complex numbers were taught. (Second generation EE)
  - acuozzo 3y ago
    In many American schools Mathematics is taught as if it were composed solely of magic recipes. Students need only to memorize the patterns described in the red boxes in the textbook to pass.
  - mjburgess 3y ago
    It is often taught by its historical development, so first as an alleged imaginary solution to an equation with no real roots. This has it backwards as far as intuition is concerned. The geometric interpretation is the obvious one, and the use of `i` in finding zeros is a special case.
deleted 3y ago
[deleted]
ajkjk 3y ago
It's a lot easier to see if you imagine i to be a symbol R that rotates vectors in the plane, since e^(i ɸ) = cos ɸ + i sin ɸ doesn't use the additive property of i at all. (Well... I guess it does in a sense, because all it means to be additive is that it's linear, so you can add up all the complex-valued terms in e^(i ɸ) and give their coefficients the name sin().) Figuring out what e^(iɸ) means requires figuring out "What `i` is" and "What e^x means on non-numbers" at the same time. That is, it requires you to perform two intellectual jumps at once instead of one at a time. No wonder it is so confusing. If these concepts are made sufficiently simple, I imagine that we could live in a world where we also teach e^(a d/dx) f(x) = f(x + a) in high school.
- ChainOfFools 3y ago
  Pardon my mathematical ignorance here, but I've always been curious as to whether negative numbers are the only alternate set that extends from the origin of zero, or whether they are merely the only set that is diametrically opposed to their positive counterparts, and there are in fact an infinite number of these lines extending from the origin at zero, radiating from it in all directions, and what we think of as the positive and negative numbers are simply one pair of rays extending from this origin arbitrarily chosen as our base units.
  - ajkjk 3y ago
    I often think that a lot of math would work out more easily if we _only_ used polar coordinates and regarded the negative numbers as a "separate number line" rather than a continuation of the positive numbers. (In particular, if you know about delta functions... a lot of weirdness around x=0 goes away if you write everything in terms of r \sgn (r) and take the derivatives of both terms. e.g. This gives "for free" the fact that the divergence of 1/r^2 is 4 pi delta(r).) I have heard of systems in which one sticks more lines out from 0 than just the positive and negative numbers. At some level that's what R^2 is, with four copies of the positive number line, but I don't see a strong reason why in principle you couldn't have an odd number of lines, which would correspond to... uh... R^1.5. But you have to define how these lines rotate into each other, and it is gonna be weird.
  - 3y ago

This is the first time I see how the famous equation e^{i*ɸ} = cos ɸ + i * sin ɸ can be discovered from a simple assumption of what raising to the power i cou

by tromp 3y ago

mjburgess 3y ago

A lot of the issues around complex numbers can be intuitively resolved by drawing a unit circle around 0 on a numberline, and reasoning backwards to a definition of i. Rather than reasoning forwards from sqrt(-1)

deleted 3y ago
[deleted]
dboreham 3y ago
Puzzled expression because that's how I thought complex numbers were taught. (Second generation EE)
- acuozzo 3y ago
  In many American schools Mathematics is taught as if it were composed solely of magic recipes. Students need only to memorize the patterns described in the red boxes in the textbook to pass.
- mjburgess 3y ago
  It is often taught by its historical development, so first as an alleged imaginary solution to an equation with no real roots. This has it backwards as far as intuition is concerned. The geometric interpretation is the obvious one, and the use of `i` in finding zeros is a special case.

deleted 3y ago

[deleted]

ajkjk 3y ago

It's a lot easier to see if you imagine i to be a symbol R that rotates vectors in the plane, since e^(i ɸ) = cos ɸ + i sin ɸ doesn't use the additive property of i at all. (Well... I guess it does in a sense, because all it means to be additive is that it's linear, so you can add up all the complex-valued terms in e^(i ɸ) and give their coefficients the name sin().) Figuring out what e^(iɸ) means requires figuring out "What `i` is" and "What e^x means on non-numbers" at the same time. That is, it requires you to perform two intellectual jumps at once instead of one at a time. No wonder it is so confusing. If these concepts are made sufficiently simple, I imagine that we could live in a world where we also teach e^(a d/dx) f(x) = f(x + a) in high school.

ChainOfFools 3y ago
Pardon my mathematical ignorance here, but I've always been curious as to whether negative numbers are the only alternate set that extends from the origin of zero, or whether they are merely the only set that is diametrically opposed to their positive counterparts, and there are in fact an infinite number of these lines extending from the origin at zero, radiating from it in all directions, and what we think of as the positive and negative numbers are simply one pair of rays extending from this origin arbitrarily chosen as our base units.
- ajkjk 3y ago
  I often think that a lot of math would work out more easily if we _only_ used polar coordinates and regarded the negative numbers as a "separate number line" rather than a continuation of the positive numbers. (In particular, if you know about delta functions... a lot of weirdness around x=0 goes away if you write everything in terms of r \sgn (r) and take the derivatives of both terms. e.g. This gives "for free" the fact that the divergence of 1/r^2 is 4 pi delta(r).) I have heard of systems in which one sticks more lines out from 0 than just the positive and negative numbers. At some level that's what R^2 is, with four copies of the positive number line, but I don't see a strong reason why in principle you couldn't have an odd number of lines, which would correspond to... uh... R^1.5. But you have to define how these lines rotate into each other, and it is gonna be weird.
- 3y ago