Does that mean we should abandon them? Absolutely not. Encoding phase (or in a much more common subset, parity) is so absolutely useful it’s no wonder we bake 90° intervals (-, i) into our notations: they can be intuitively dealt with. It’s still somewhat easy to skip over the property, however; as a student at least I seem to need to backtrack over signs at least once an hour when working with anything rigorous enough. I wonder if 2-tuple notation, eg (+, 23) or (-i, x²), would be more intuitive by making parity/phase explicit rather than implicit.
Complex numbers are a little more nuanced, but no less useful. I imagine you could develop an alternative notation to make things more intuitive, but thankfully it’s generally taken for given nowadays that they’re intrinsic to how we’ve explored nature.
The most confusing thing about complex numbers is the language. First you're told negative numbers can't have roots, then you're told they so can too, but you have to call the roots "complex" or "imaginary."
This sets up cognitive dissonance which can be harder to deal with than the math. (What even is an "imaginary number"? What are those words supposed to mean?)
In reality complex numbers are a way of moving from the number line to a number circle. (Which eventually generalises to a 3-sphere when you get to quaternions.)
That's all they are. Instead of linear arithmetic - which is about combining magnitudes in one dimension - you can now do arithmetic that combines magnitudes with rotations.
The extra dimension makes it possible to solve equations with solutions that don't exist on the basic number line. It also makes it easier to do calculations that combine magnitude with phase - which includes pretty much anything that rotates or processes linear combinations of sine waves, and which a straight vector tuple can't handle.
If someone had told me this when I was learning complex numbers the cognitive dissonance wouldn't have hurt quite as much.
For my part, I do like to think of adjoining numbers onto an existing system, but that immediately becomes matrices.
So you decide to adjoin an ε such that ε² = 0. Your numbers are now vectors (a, b) and the action of ε is to map this to (b, 0) so that it is represented by the matrix
[ 0, 0 ]
[ 1, 0 ]
[ a, 0 ]
[ b, a ]
[ a, b ]
[ b, a + b ]
When you adjoin to the reals an i such that i² = -1 these matrices have the shape of the 2D scaled rotations, so that is what complex numbers just “are” to me. The representation as a tuple is to me the same as representing the above matrix as (a, b) or (a, b, a+b) to save time or space... The whole thing is a matrix but the entries are indeed redundant and so you don't need to store all of them at once.
So that's why I do not understand what you mean by these being separate concepts. Can you elaborate?
Arguably we might one day find out that the universe is discrete at which point we could begin to try to define the naturals as something that "exists", at least up to some maximum large number. But even then the numbers are probably still best thought of as just a helpful abstraction.
Basic complex numbers, on the other hand, just require me to expand what I accept as the solution of an equation. Note that I've already done this with fractions.
Fractions: Given integers a and b, ax + b = 0 has a meaningful solution
Complex numbers: The equation x^2 + 1 = 0 has a meaningful solution
if you're going to prove such a fundamental thing, can you please provide the axioms that we start from? I.e. "we know" that a - a = 0, multiplication is distributive, and a x - b = - a x b. These seem arbitrary properties and "equally" fundamental to -a x -b = ab. Either start from peano and prove everything along the way, or tell the reader your assumptions. Don't just divine things along the way.
EDIT: Assumptions are in the third paragraph of the post. I highly doubt they were there when I wrote the comment. Either way, my concern has been resolved.
And why start with Peano axioms? They seem like a bad starting point because it would take pages upon pages of proof and it won't easily extend to other algebraic structures like rings and fields.
I gave Peano as an example. I don't mind the assumptions, as long as they're reasonable and presented before the proof. Another comment pointed me to the fact that they were mentioned in an earlier paragraph, so my issue is resolved.
It is right there in the first section of the article.
"In this discussion, we assume that we already know some basic properties of arithmetic operations such as the distributive property of multiplication over subtraction, existence of the additive inverse of real numbers, etc."
Hi! I am the author of this blog post. The assumptions in the third paragraph were there at least since 16 Feb 2019. See https://github.com/susam/susam.in/commits/master/content/blo... for the change history of this post.
Also, this post conflates the unary negation operator with negative numbers. The two are not the same. In so far as this post constitutes a proof (which IMO it does not), it is a proof about the behavior of the negation operator.
A good question to ask is why we made this specific choice of definition. Why should multiplication be defined such that -2*-3 = 6? This is a question that the post does shed some light on. If we'd chosen some other definition of multiplication, a lot of the "intuitive" properties of multiplication that hold over the natural numbers (such as the distributivity of multiplication over addition and subtraction) would no longer be true over the integers.
Well, sure, if you change the definition of something, then it may end up having different properties. What's your point?
The OP thinks that his "proof" is showing why multiplying negative values yields a positive result. But the proof is a load of nonsense because it assumes facts like distributivity of multiplication over addition and subtraction. It is literally impossible to prove that $\forall a, b, c \in Z. (a - b) * c = (a * c - b * c)$ -- distributivity of multiplication over subtraction -- without having already defined the meaning of a * b for all integers! This leads to a circular reasoning loop that the OP's "proof" can't get out of.
The thing to realize is that multiplication is not some magic operation handed down to us by god. It is just a binary total function defined over the integers. What the OP is trying to confusedly get at is the following:
1. There is an intuitive definition of multiplication as repeated addition over natural numbers.
2. It is not clear what the corresponding definition of multiplication over negative numbers is.
3. If we want to define multiplication as a total function over the integers, we need to define what the result should be when multiplying negative integers.
4. Specifically, with (3), we are taught in school that the result of multiplying two negative numbers should be positive, but it is not clear why this seemingly arbitrary choice was made.
Unfortunately, the OP is going about this all backwards. One cannot prove what the OP wants to prove. What one can instead do is argue that the specific (but seemingly arbitrary) definition that one has chosen for multiplication is a "good" choice because it has the same properties (distributivity etc.) as multiplication over natural numbers. At its core, this is a stylistic appeal about the "naturalness" of the definition.
This is backward reasoning. The chosen definition of multiplication is not to keep things "intuitive". If you start with the field axioms, the chosen definition of multiplication is pretty much dictated by the axioms. If you choose another definition of multiplication, you would end with contradictions like 1 = 0 and such nonsense! And mathematicians abhor contradictions!
"Product of additive inverses of two elements is equal to the product of the two elements" is dictated by the field axioms in all fields.
-a is a standard way to represent additive inverse of an element in field.
The point about "unary negation operator" seems irrelevant.
In the real number field, additive inverse of a positive real number is indeed the negative of that number. The negative of that number is also obtained by the application of unary negation operator on the positive number.
The additive inverse of 3.14 is -3.14. Unary negation operator applied to 3.14 gives us -3.14. I don't see how conflating unary negation operator with negative numbers here is any issue here.
This is a fact that follows from the definition of +. But + needs to be defined before you can start making assumptions about what the additive inverse is. The set over which the field is defined (Z or R) already contains -3, -2 etc. and -3 * -2 or -3 + -2 needs to be defined when you're constructing the field. It then turns out that -3 is the additive inverse of 3. You can't use this when arguing about the definition of why applying * on negative 2 and negative 3 gives you the result positive 6. Because you need to define * over all members of the field before you construct a field in the first place.
All: if you notice fishy things (as a user did in this case), please let us know at hn@ycombinator.com. We catch a lot of abuse between software and moderation, but unfortunately not all. Vigilant users make a huge difference, and protecting the integrity of HN is a community effort.
(Please don't post insinuations about abuse in the threads, though, since most suspicions don't end up leading to real evidence. Send them to hn@ycombinator.com. This is in the site guidelines: https://news.ycombinator.com/newsguidelines.html)
What would we have noticed, in this case?
Here's one thing though: multiple accounts submitting, commenting, and promoting the same person's sites, articles, and (importantly) repos.
https://hn.algolia.com/?dateRange=all&page=0&prefix=false&qu...
-1*-1 =
-1*-1 + -1*1 + 1 =
-1*(-1 + 1) + 1 =
-1*0 + 1 =
1If number N is an arrow on the number line from 0 to N, then multiplying N by -1 flips the arrow with the result -N. Multiplying by -1 again would be another flip, taking you back to N. So a flip followed by a flip is same as no change (i.e. multiplicative identity 1).
e^[i(2nπ+π)] × e^[i(2nπ+π)] =
...
e^[2i(2πn+π)] = 1
I know there are precise definitions for fields and rings but can someone here give me some good examples of fields and rings? Being a non-mathematician, I find it easy to manipulate examples than manipulate definitions.
Are the set of integers a field? I guess not because the multiplicative inverse of 2 is not present in this set.
Is the set of integers a ring? I think, yes.
For prime p, is Z_p = {0, 1, ..., p - 1} a field? I think, yes.
Are there any non-numeric rings where product of negatives is positive?
For example, the integers mod n is a ring, so (-a) * (-b) = a * b holds, but it doesn't make sense to call a number mod n positive or negative, since -a mod n effectively means n - a mod n.
(posted an earlier version of this comment on susam.in.)
I thought the concept of "negative" was defined by reference to an operation. "Negative 5" is whatever value Q satisfies the equation 5 + Q = 0.
That definition immediately tells you that the negative of a negative is a positive. Once we know 5 + Q = 0, we ask what the negative of Q is. It's the value V such that Q + V = 0. But by the definition of Q (and the commutativity of addition), we already know V = 5.
Once you define negatives this way, it's trivial to show that negatives obey the standard ordering. But that ordering wasn't necessary in order to define them.
Summing up, the product of negatives is positive because negation is a kind of inversion (additive inversion), and two successive inversions always cancel in any context.
A more interesting example: If R is a ring, then R-valued square matrices of fixed size also give a ring, using addition and multiplication of matrices. Matrices aren't just positive, negative, or zero; they can have a mix of positive and negative entries. In these "matrix rings", the product of negatives isn't exactly positive, although I bet that somebody can make this more rigorous. (Come to think of it, this applies to the rings of polynomials, too.)
Contestents are put in a dome filled with gold and silver tickets being blown around by fans. For every gold ticket they collect, they get a point. For every silver ticket, they lose a point. If they collect enough points, they win a prize.
Sorting through the team's collection of tickets and throwing away a silver ticket (minus a -1) is just as good as adding another gold ticket (+1).
Not sure the kids these days are down with the crystal maze though. More loss to them - Richard O'Brien was a national treasure.
Create a video of your friend walking 3 metres. Now play the video 4 times. Your friend walks 12 metres in the video. Play the video in reverse 4 times. Your friend walks 12 metres backwards in the video.
Create another video of your friend walking backwards 3 metres. Now play the video 4 times. Your friend walks 12 metres backwards in the video. Play the video in reverse 4 times. Your friend walks 12 metres forwards in the video.
It also even helps explain division when you talk about it in terms of direction and how many times you have to move to get to 0. It also helps them understand why you can't divide by 0. e.g. 3/0 would be explained like "Starting at 3, how many times can you move 0 to get to 0," They can clearly see it's impossible, and it helps give them at least some basic intuition into it.
I've also found this extends to at least some properties of complex numbers as well, as you can easily extend from the number line to a coordinate plane.
Parallel to your statement: If both 1 * 0 = 0 and 2 * 0 = 0, then 0 / 0 has two solutions.
The first number represents the amount of something. If it's negative, you have a debt. The second number represents either a gain (if it's positive) or a loss (if it's negative).
From that point you can explain it to yourself using plain english. So, -4 * (-3) can be understood as "Lose a debt of 4, three times". If you have -4 * 3, you could be said to "gain a debt of 4 three times". 4 * -3 means (Lose 4 three times).
In the video Mathologer criticized exactly the kind of proofs like in this video. Just saying it's intuitive doesn't make it so. Fundamental things shouldn't be proven using a number of laws. They should be understood on the intuitive level and a proof is just to double check.
If you want an absolutely rigorous proof, you can view this Metamath proof: http://us.metamath.org/mpeuni/mulge0.html ; this has more far more steps, but is totally rigorous. It particular, its only axioms are those of classical logic and ZFC set theory (not even numbers are presumed, the system first proves "numbers exist and have these properties").
https://betterexplained.com/articles/rethinking-arithmetic-a...