Some authors might prefer -- \ / -- 0 Ass opposed to n -- \ / -- i = 1
Sometimes we even leave it as "i" instead of "0 < i < n" or "i = 1 to n". Sometimes the range of the summation doesn't even need to be determined in intermediate steps.
For mathematicians, = does not mean equality
71–80 of 206 posts
Re: For mathematicians, = does not mean equality
#72Earlier quoted context omitted.
The usual notation for modular arithmetic uses three dashes, not two, to denote congruence.
I probably should have said "common" notation; it's certainly what I learned. I think you're right in that many authors prefer ≡ to emphasize that it is an equivalence rather than equality relation. Source that both are in use: https://math.stackexchange.com/questions/196081/the-right-wa...
Yeah, sorry, I didn’t mean to split hairs. They’re definitely both in use. I do plenty of work in Zp and I feel bad every time I don’t write \equiv or the \pmod for that matter :)
Re: For mathematicians, = does not mean equality
#73Minor note: > A = { n^2 : n = 1, 2, …, 100 } > In Python, or interpreting the expression literally, the value of n would be a tuple, producing a type error. (In Javascript, it produces 2.[link] How could it be Javascript if it didn’t?) That linked JS code uses ^, which is xor, not pow. Math.pow([], 2) = NaN. Or maybe that was the joke and it flew completely over my head.
I suspect that's the joke. Making fun of how Javascript parses or evaluates expressions is a favorite passtime in programmer circles.
See, for example, https://www.destroyallsoftware.com/talks/wat
Re: For mathematicians, = does not mean equality
#74Think about the equation x + 3 = 1 Typically we write that the solution is "x = –2". This to me is the most abusive form of usage for "=" in mathematics. The solution to the equation is –2. The solution to the equation x = –2 is also –2. Solving the equation x = –2 is very easy. We can solve it just by looking at the equation. What we are really doing when solving an equation is transforming the original equation int…
> x is a variable that can assume a myriad of values.
No it can’t. x in this case is a bound variable. It’s not that it just so happens to take on -2, but that it is already bounded to -2 to make the statement true.
It really says: There exists some x in Z such that x + 3 = 1. What is that x, or what is a proof of the statement? The answer is in the form of a logical implication.
Re: For mathematicians, = does not mean equality
#75Earlier quoted context omitted.
At least in my university, in one of the first Algebra courses about integers, factorization and congruence the usual notation is 3 * 3 ≡ 1 (mod 4) but in the next year Algebra course about groups and crazy algebraic structures the notation in the group Z_4 is just 3 * 3 = 1 and everyone understand that you are working in Z_4 (and the * and = symbols are "overloaded" (but no one call them "overloaded")).
Sure; I think we’re in violent agreement here, it’s absolutely the case that people write the simpler version when the meaning is clear from context. I’ve definitely done that a bunch.
Piling on with a bit more pedantry, my experience is a bit different.
In my current ring theory course, we have indeed written things like 3 * 3 = 1 when working in |F_5 (not sure that notation is going to work as well as I hope, looks alright in the app I use), but it's not the equality symbol is overloaded, but the numbers themselves. Rather than using = to mean numeric equality and equality w.r.t. equivalence classes, we just use the numbers themselves as shorthand for their equivalence classes.
Re: For mathematicians, = does not mean equality
#76Earlier quoted context omitted.
OK. Downvotes happily accepted; but I think we all operate in "peephole" mode now; the cost of tracing sources back to the original is significant. There is no karma to be had from complaining about HN's software, and yet it is so very inadequate.
That it was a quote from the article seemed clear in the post in which the quote was presented, so while I agree that it can be difficult to clearly present quotes in some situations on HN and a good blockquote formatting facility would be preferable, I don't think that's really a problem here.
As I read the comments on this post, the top reply contains multiple mathematical symbols which aren't rendering on my recent Android phone. You can't pretend that HN has a huge proactive team working on these issues.
I'm loathe to infer that you think that the problem is "me" because I think the underlying question is whether the comments should be a friendly place for people who do not have time or inclination to read the article. You may see that as an appalling, lazy, degeneration in discourse; the reality is that reading the comments without wading through a blog post is a valid tactic. If clearer methods of quoting were available the two or three of us involved here would have wasted less time.
The "engineer-forward" alternative in which everyone strives to speak from a totalizing position of authority is just, frankly (as a technical person myself) unattainable.
Re: For mathematicians, = does not mean equality
#77There are few more examples that come to mind, like statements about intervals (π = 3.14 ± 0.01) and the usual notation for modular arithmetic; 3 * 3 = 1 (mod 4). Oh, and the wonderful notation for integrals, ∫ 2x dx = x² + C
Well the (mod n) notation is just an implicit homomorphism, phi(3*3) = phi(1) where phi : I -> Z4. But during lectures it was pretty common for professors to just drop all the greek and squiggly lines and commonly write things like 9 = 1, and use '=' with whatever arcane RST relation which happened to be relevant at the time.
Re: For mathematicians, = does not mean equality
#78(I assume this was inspired by https://news.ycombinator.com/item?id=16803874 ) The use of ‘=’ for assignment in programming languages comes, not directly from mathematics, but indirectly from the use of mathematics in science and engineering. As an example, consider the formula for kinetic energy, commonly written 𝑚𝑣² 𝐾 = ─── 2 Why isn't it written 2 K = m v ², which expresses the same mathematical equality in a s…
Because I started programming before taking maths at school, I didn't properly appreciate equality for a while.
Sure, algebra was fine, a(x+y)=ax+ay can go either way; but not ratios and other relationships.
What helped me was was geometry, where you can see it's just a relationship. All the components move together; one part isn't priviledged as the result.
e.g. you enlarge a circle. It doesn't make sense to ask whether the radius made the circumference bigger, or the circumference made the radius bigger.
Re: For mathematicians, = does not mean equality
#79Think about the equation x + 3 = 1 Typically we write that the solution is "x = –2". This to me is the most abusive form of usage for "=" in mathematics. The solution to the equation is –2. The solution to the equation x = –2 is also –2. Solving the equation x = –2 is very easy. We can solve it just by looking at the equation. What we are really doing when solving an equation is transforming the original equation int…
> x is a variable that can assume a myriad of values. No it can’t. x in this case is a bound variable. It’s not that it just so happens to take on -2, but that it is already bounded to -2 to make the statement true. It really says: There exists some x in Z such that x + 3 = 1. What is that x, or what is a proof of the statement? The answer is in the form of a logical implication.
Let’s assume we are talking about Q, the rationals. x+3 is an element of Q[x]. This element of Q[x] defines a natural map from Q to Q. The equation x+3 = -2 is equivalent to finding the pre-image of -2 of this natural map.
x is actually just, in the language of computer science, syntactic sugar. In reality x+3 is really the infinite tuple
(3, 1, 0, 0, .....)
Re: For mathematicians, = does not mean equality
#80Think about the equation x + 3 = 1 Typically we write that the solution is "x = –2". This to me is the most abusive form of usage for "=" in mathematics. The solution to the equation is –2. The solution to the equation x = –2 is also –2. Solving the equation x = –2 is very easy. We can solve it just by looking at the equation. What we are really doing when solving an equation is transforming the original equation int…
I disagree. If you are saying 'the solution's is -2, you have to be clear what the problem is. This becomes clearer when you have a problem with multiple variables. Then saying 'x=-2, y=3', makes clear the value each variable is taking in the solution.
Solutions to equations in multiple variables are ordered tuples. For instance,
x y + 2 = 0
has infinitely many solutions. One of them is (1, -2).