## "Oh no! We forgot how to say... math... stuff!"

For the discussion of math. Duh.

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Xanthir
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### Re: "Oh no! We forgot how to say... math... stuff!"

quintopia wrote:
coolguy5678 wrote:|a...b| is the new [a,b] (the lines on either end are connected with the underline to make a sort of square bracket, and the lines on either end are about half-line-height)
a|...b| is (a,b]
|a...|b is [a,b)
a|...|b is (a,b)

I'd prefer it the other way actually. Make it so that |...| always indicates an open interval (since open sets are kind of the "fundamental" thing for analysis) and putting the end point on the outside makes it a closed set. In your formulation |...| has no consistent meaning (indeed, in the example a|...b|, it is unclear what sort of boundary your left | represents! I see the logic in having the location of the endpoints represent inclusion and exclusion, but for some reason I can't help but think about it this way...

I don't understand your objection. in a|...b|, the range includes B but not A, because B is inside the lines but A isn't.
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### Re: "Oh no! We forgot how to say... math... stuff!"

I understand that. And I'm probably alone in thinking that |a...|b is a more sensible way of writing that.

Xanthir
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### Re: "Oh no! We forgot how to say... math... stuff!"

Yes, because that doesn't make any sense. ^_^ The vertical lines represent the range itself, and then you indicate whether the closest specifiable number is inside or outside the range.
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quintopia
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### Re: "Oh no! We forgot how to say... math... stuff!"

In my version, the vertical bars represent the limits of an open set, and a number appearing outside the bars indicates a "cap" ...the limit point that is included to close that end of the interval.

DrZiro
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### Re: "Oh no! We forgot how to say... math... stuff!"

Oh! I would change so many things. Where to start?

For one thing, base 16.
It's easy to see the appeal of bases like 6, 12 or 30, where you can easily write fractions. But I think there are more important things than writing fractions.
If you know base 16, you practically know base 2 as well. That can be pretty neat sometimes, since it makes the connection between arithmetics and logic.
As the OP mentioned, base 6 is neat for finger counting - you can get to 35 with two hands. But 16 is even better; you use your fingers (except the thumb) as binary digits - one finger is a base 2 digit, one hand is a base 16 digit. Then you get to 255 with two hands. (Technically you could get further by using your thumbs too, but that's really uncomfortable for some numbers. It's good to have the thumb to hold the other fingers down.)
Base 8 is also close to binary, but since it's an odd power of two, that would be, well, odd. Also, we are quite capable of remembering 16 digits - lower bases are a waste of space.
Base 64 would sometimes be neat in geometry, since it's a square and a cube, but it's no big advantage. And 16 is at least better than most, since it's a square.
Another obvious advantage is for computers, where base conversion causes all sorts of annoyances.
And then we have orders of magnitude. It's a great tool for estimation that should be taught much more in schools. But it suppers from our choice of base. Because occasionally you want to be just a little more accurate, and perhaps use half orders of magnitude. But √10 is a little messy. With 16, you can basically go into binary orders of magnitude. That also has the advantage of better handling of additions. When you round things off to powers of ten, 10^x + 10^x = 10^x, which is sometimes inconvenient. In binary, that is not the case.
But probably the biggest advantage is in measurements. Suppose you have a piece of string that's one meter long, and you want to measure something that's half a meter. How do you do that? You fold the string in half, of course. And if you want 1/16 of a meter, you just fold it four times. Easy. But what if you want to get to 1/10? That's a great deal more difficult. And the same applies to all sorts of measurements - volume, mass, etc. For this reason, there was actually a movement in the 1700s to switch to base 16, and it could well have succeeded - many units back then were base 16 or something similar. Unfortunately the French philosophers thought that it would be better to change the units, partly because they thought 10 was a holy number, and that side ended up winning.

Then, I like prefix or postfix notation. (But for the love of maths, don't call it "reverse Polish notation" - that makes it sound so horribly strange and unnatural.) Postfix makes sense in normal maths, since you generally need the numbers before you can do calculations on them. But I would be inclined to prefer prefix, because once you give up the constraints of normal immutable maths and go into the sort of formulas used in programming, that becomes more natural - if f(x) starts with an instruction to set x to 5, you don't need to calculate x first. I suppose it might be just as easy to learn both prefix and postfix; it's just a matter of mirroring, after all.

Deveno
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### Re: "Oh no! We forgot how to say... math... stuff!"

i believe we should start over with an exhaustive investigation into semigroups, and then slowly move to more intricate structures. by the time we've recovered linear algebra, we'll be awesome!

MHD
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### Re: "Oh no! We forgot how to say... math... stuff!"

I personally think the current notation for anti-derivatives is stupid.
$\int f(x) dx$
That's not an anti-derivative, that's a weird squiggly line...
You see, derivatives are denoted:
$\frac{df(x)}{dx}$
which means "change in [imath]f(x)[/imath] divided by change in [imath]x[/imath]" so why not:
$df(x)\cdot dx$
or something? "change in [imath]f(x)[/imath] multiplied by change in [imath]x[/imath]"
That also serves to make it plainly obvious that they're the opposite of each other.

Also OTTO: base 16:
(spoilered for big images...)
Spoiler:

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### Re: "Oh no! We forgot how to say... math... stuff!"

MHD wrote:Also OTTO: base 16:
(spoilered for big images...)
Spoiler:

Interesting, but IMHO that system needs more redundancy to make it easier to read and to reduce errors. In rapid writing the distinction between the digits for 2 & 4 could be easily lost. One simple improvement would be to alternate the side that the bit components extend out from. Eg, the 1 & 4 bit strokes could extend to the left like they do in the current system and the 2 & 8 bit strokes could extend to the right.

FWIW, I created a simple bitmap hex font years ago (on the Amiga) using the traditional numeral & letter shapes, where the number of black pixels in a digit is proportional to the digit's value. In that system, the '0' digit was blank and the 'F' digit was a totally black rectangle, but all the other digits were quite recognizable. That font was a handy way of representing low precision data that lent itself to a 2D grid format.

Writing systems with more redundancy are easier for humans to process than systems that try to reduce the redundancy, especially for those of us who are afflicted with various forms of dyslexia. For example, compareTolkien's Tengwar writing system with the Devanagari system used to write Hindi & Sanskrit (most of the writing systems of other Indian languages are similar as they share a common ancestor with the Devanagari script). The main "stop" consonants in Tolkien's Tengwar writing system have much less redundancy than those in Devanagari, but the Devanagari script is easier to parse than Tengwar even though the number of shapes that you need to learn is greater.

I learned both of these scripts in my youth - Tengwar in my early teens, Devanagari in my early twenties. At first I thought Tengwar was superior to Devanagari due to the higher regularity and reduced redundancy, but I soon came to appreciate the benefits of the extra redundancy of Devanagari. However, I will admit that Tengwar is simpler to learn to write, since it doesn't have the added complexity of combined consonant forms.

http://en.wikipedia.org/wiki/Tengwar
http://en.wikipedia.org/wiki/Devanagari

Talith
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### Re: "Oh no! We forgot how to say... math... stuff!"

I like the notation used for integration because it makes it clear how it's a similar operation to summation which uses the capital sigma notation we all know and love. The fact that integration lines up with anti-differentiation is more of a happy coincidence than a foundational definition though (even if it is called the fundamental theorem of calculus).

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### Re: "Oh no! We forgot how to say... math... stuff!"

coolguy5678 wrote:
I'd also kill the idea of independent/dependent variables as used in calculus in favour of ([imath]\lambda x[/imath].<some expression in x>)1 functions or some similar notation. I'm fine with 2(x+1) = 2x+2 having an implicit [imath]\forall x[/imath]. 2 But then you start doing d/dx (2x+2) = 2 and that (2x+2) should by all means just be a single value, so how does the d/dx operator "know" the whole function? And then you start saying y = 2x+2, and now y is some magic value which changes depending on what you decide x is. And people think of f defined by f(x) = 2x+2 and f defined by f(y) = 2y+2 as different in some way, since the one is a "function of x" and the other is a "function of y", whatever that means. Wow, none of this paragraph makes any sense whatsoever. What I'm really saying is that the distinction between a function and a single value becomes blurred.

...

I agree with this point, and I like your idea for the change (though I think there are other options). It especially gets muddled when doing chain rule stuff: what exactly do you mean by d/dx f(4x + 3)? Do you mean evaluating the derivative of f at 4x + 3, or you mean taking the derivative of the function x -> f(4x + 3)? Are we taking the derivative of the function f, or of the whole expression considered as a function of f?

It's not a huge deal in real life, since as long as you're being reasonably careful anybody should be able to understand what you mean. But as long as we're reinventing all of math notation...
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### Re: "Oh no! We forgot how to say... math... stuff!"

See, I would teach everyone "duotrigesimal"! Also known as Base Thirty Two! As someone posted earlier, we can indeed easily count to 16 using four fingers. I have found, however, that by using my thumb as my "one" digit, I can easily count from zero to thirty one. As a matter of fact, I am currently engaged in counting to 33,554,432 (32^5), and it all started with binary finger counting, and Base Thirty Two. See, currently we simply use the five digits on our hand to count to 5 - but we COULD, with not much more difficulty at all, be using them to count to "two to the POWER of '5'", or thirty two! Here is a blog where I have been presenting this form of finger counting:

www.binfinco.blogspot.com

gmalivuk
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### Re: "Oh no! We forgot how to say... math... stuff!"

doGraeF wrote:I am currently engaged in counting to 33,554,432 (32^5), and it all started with binary finger counting
How's that supposed to work? You can keep track of 32 different numbers for each one of your five fingers?
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Yakk
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### Re: "Oh no! We forgot how to say... math... stuff!"

gmalivuk wrote:
doGraeF wrote:I am currently engaged in counting to 33,554,432 (32^5), and it all started with binary finger counting
How's that supposed to work? You can keep track of 32 different numbers for each one of your five fingers?
No silly. He just uses 5 hands. One on his left arm, one on his right arm, one on his left leg, one on his right leg.

...

Spoiler:
Oh come on. I'm not going to go there.

The other one would be formed through your 4 limbs and your head (or whatever other limbs -- I'd advise against that one, because it would probably be the MSB, and would have to stay up for a long time (ok, I went there)). Bend forward for a 1, neutral for a 0.

But seriously, I'd be tempted to use 1-distance coding (what is the name of that?) for my finger counting instead of naive binary. Make it so you only have to move 1 digit at a time that way.

You could extend this to 32^5 even.

Which brings up the idea -- instead of binary coding your digits, use error theory to set it up so that modest errors in the digits you draw are correctable, more serious ones are detectable, and even more serious ones tend to result in nearby values.

On top of that, exploit Benford's law and make the more common (lower) digits easier to write, and possibly have more redundancy/error checking between them. The difference between 1 and 2 is larger than the difference between 8 and 9 in many if not most practical applications.

I expect that natural use would cause decay however -- the redundant space between them would collapse.
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### Re: "Oh no! We forgot how to say... math... stuff!"

Yakk wrote:But seriously, I'd be tempted to use 1-distance coding (what is the name of that?) for my finger counting instead of naive binary. Make it so you only have to move 1 digit at a time that way.

http://en.wikipedia.org/wiki/Gray_code wrote:The reflected binary code, also known as Gray code after Frank Gray, is a binary numeral system where two successive values differ in only one bit. It is a non-weighted code.[citation needed]

The reflected binary code was originally designed to prevent spurious output from electromechanical switches. Today, Gray codes are widely used to facilitate error correction in digital communications such as digital terrestrial television and some cable TV systems.

Yakk wrote:You could extend this to 32^5 even.

Which brings up the idea -- instead of binary coding your digits, use error theory to set it up so that modest errors in the digits you draw are correctable, more serious ones are detectable, and even more serious ones tend to result in nearby values.

On top of that, exploit Benford's law and make the more common (lower) digits easier to write, and possibly have more redundancy/error checking between them. The difference between 1 and 2 is larger than the difference between 8 and 9 in many if not most practical applications.

I expect that natural use would cause decay however -- the redundant space between them would collapse.

You have some nice ideas there, Yakk.

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### Re: "Oh no! We forgot how to say... math... stuff!"

I'd most definitely change the names of "open" and "closed" sets.

gmalivuk
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### Re: "Oh no! We forgot how to say... math... stuff!"

Why?
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kalakuja
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### Re: "Oh no! We forgot how to say... math... stuff!"

7 and 1 look so alike on my chalkboard. 7 should have a line in the middle. Ugly wiki picture:
.
Number symbols should be fast to write and somehow look like the amount they present.

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### Re: "Oh no! We forgot how to say... math... stuff!"

ShaiDeshe wrote:I'd most definitely change the names of "open" and "closed" sets.

gmalivuk wrote:Why?

Well, I think you can probably guess why. Some people don't like the fact that "open" and "closed" aren't negations of each other -- "open" doesn't mean "not closed", and "closed" doesn't mean "not open". Many (most) sets are neither open nor closed.

I guess the question is then: Should we (a) use different words than "open" and "closed", or (b) emphasize more, when teaching topology or analysis, that "open" and "closed" are not negations of each other?

I like to emphasize that open sets contain none of their boundary points, closed sets contain all of their boundary points, and so there are many sets that contain some but not all of their boundary points.

I sympathize with ShaiDeshe to an extent, but what should we say rather than "open" and "closed"?

Maybe "totally open" and "totally closed" to emphasize that a set can be neither? Those are long. Maybe we could talk about "no-BP sets" and "all-BP sets" to emphasize that we're talking about containing either all or none of the boundary points? Those are maybe a little ugly.

I sympathize somewhat, but I'm not sure what other terminology we could use that's better.

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### Re: "Oh no! We forgot how to say... math... stuff!"

skullturf wrote:I sympathize with ShaiDeshe to an extent, but what should we say rather than "open" and "closed"?

There are options that harmonize well with other concepts we use.

For instance, in abstract topology we say that U is a neighborhood of x if U is an open set that contains x. Neighborhood is such a brilliantly intuitive word that we should just use it instead of open -- you know it means that every point in the set has sufficiently close neighbors.

For closed, how about "complete"? It's an overtaxed word across all the fields of math, but here it would do a better job of describing that a set contains all of its limit points. This brushes up against the idea of a complete metric space, but again that's harmonious to a degree. When we extend the rationals to the reals by adding points so that every Cauchy sequence has a limit point, we say that the reals are the completion of the rationals, so that would have some relation to when we would start using "completion" instead of "closure". (I don't see a reason to get rid of "interior", though.)

The cost is that the theorem that a set is a neighborhood if and only if its complement is complete and vice versa loses some of its zing, but I don't think I would miss it and perhaps students *should* be more surprised that the two definitions are related so intimately.

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### Re: "Oh no! We forgot how to say... math... stuff!"

I kinda like "neighborhood complete" and "limit complete".
That also doesn't distract people from the fact that some sets are both open and closed (like every subset of every discrete space).
It would also help us avoid stupid names like "clopen sets".

ShaiDeshe
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### Re: "Oh no! We forgot how to say... math... stuff!"

Come to think of it, "limit complete" is obviously misleading. "Boundary complete" works for me, though.

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### Re: "Oh no! We forgot how to say... math... stuff!"

I don't know, open and closed make plenty of sense to me. If a set contains none of its boundary, it is open, if it contains all of its boundary its closed. If it has no boundary points, its both closed and open, if it contains only some of its boundary then its neither closed nor open. I don't see any of the suggestions put forward being clearer or more intuitive than that.
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### Re: "Oh no! We forgot how to say... math... stuff!"

jestingrabbit wrote:I don't know, open and closed make plenty of sense to me. If a set contains none of its boundary, it is open, if it contains all of its boundary its closed. If it has no boundary points, its both closed and open, if it contains only some of its boundary then its neither closed nor open. I don't see any of the suggestions put forward being clearer or more intuitive than that.

That's because you've been jaded by excessive use of this cumbersome terminology.
After all, in everyday language "open" and "close" are negated terms. I'm assuming that made sense before abstract topology, as the terminology did appear back when norm spaces were the shizzle. Back then, though, a set couldn't be both open and closed (though it could be neither) so that made sense.
However, taking my first steps in abstract topology, I remember finding myself almost offended by how misleading this terminology is; especially given that it has nothing to do with any real life concepts of openness and closeness.

So yeah, you can get used to it, as we all eventually did. But given the chance, this would be the first convention I'd flush down the toilet of terminology-which-used-to-be-good-in-its-context-but-became-confusing-in-light-of-recent-developments. The second convention to go would be Baire categories, I mean, WTF?

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### Re: "Oh no! We forgot how to say... math... stuff!"

Even normed spaces have the empty set and the full space as clopen subsets.

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### Re: "Oh no! We forgot how to say... math... stuff!"

Open, closed, clopen, and ajar.
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### Re: "Oh no! We forgot how to say... math... stuff!"

Ajar?

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### Re: "Oh no! We forgot how to say... math... stuff!"

Not quite open, not quite closed.
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### Re: "Oh no! We forgot how to say... math... stuff!"

Ha! I actually kind of like "ajar"!

I don't know if it'll catch on in the wider math world, but I think I'm going to use that word next time I teach analysis or topology. Just to remind students that it's very, very common for a set to be neither open nor closed.

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### Re: "Oh no! We forgot how to say... math... stuff!"

Students should arrive at their first topology course having knowledge that closed and open aren't logical opposites, in that they should be given that terminology when they're given interval notation. That closed and open are opposite under set complements should be enough to appease the intuition.

And, in a course where you're going to be talking about complete metric spaces, overloading "complete" will be a lot more confusing than having closed and open not being logically opposite, something you should have got over in highschool.
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### Re: "Oh no! We forgot how to say... math... stuff!"

jestingrabbit wrote:And, in a course where you're going to be talking about complete metric spaces, overloading "complete" will be a lot more confusing[...]

Except that they essentially mean the same thing, which is that a sequence made of members of your set that "should" converge does converge to a member of your set. The difference is the exact notion of the "should" up there -- in a complete set you wouldn't have a convergent sequence that converges to a point outside the set and in a complete metric space you wouldn't have a sequence that satisfies the Cauchy criterion that fails to converge. In the same way, the completion of a set involves adding in the limit points and the completion of a metric space is adjoining new points representing the limits of the Cauchy sequences. I think it's every bit as natural an association.

As I see it, "open" and "closed" are nicer if you think that the thing that makes sets in a topology interesting are their behavior on their boundaries. In the first run through calculus and simple geometry, it may well be. But even when you start getting into real analysis and start having to seriously understand the Cantor set and countable dense subsets the boundary-centric intuition betrays you and you really would have been better off with a terminology that appeals to other intrinsic notions.

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### Re: "Oh no! We forgot how to say... math... stuff!"

Tirian wrote:
jestingrabbit wrote:And, in a course where you're going to be talking about complete metric spaces, overloading "complete" will be a lot more confusing[...]

Except that they essentially mean the same thing, which is that a sequence made of members of your set that "should" converge does converge to a member of your set. The difference is the exact notion of the "should" up there -- in a complete set you wouldn't have a convergent sequence that converges to a point outside the set and in a complete metric space you wouldn't have a sequence that satisfies the Cauchy criterion that fails to converge. In the same way, the completion of a set involves adding in the limit points and the completion of a metric space is adjoining new points representing the limits of the Cauchy sequences. I think it's every bit as natural an association.

Except that the proposal put forward was for "neighborhood complete" for open and "boundary complete" for closed, not just "complete" as a synonym for open.

imo terminology isn't going to be the determining factor of whether you grok some field of maths or not. No set of terminology is going to communicate all the nuance of such an abstract subject, and you shouldn't expect it to. So long as its not completely obfuscating or ridiculous (and open and closed really aren't that bad).
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### Re: "Oh no! We forgot how to say... math... stuff!"

Also, even though 'open' and 'closed' are not mutually exclusive, the two concepts are dual to each other in a very concrete sense. The terminology helps reflect this duality. Also, since the terms are used so frequently, replacing them with significantly longer words (ie 'neighborhood complete') would make the whole business clunkier. Imagine if we called Normal subgroups "conjugate-invariant subgroups". The latter describes it better than "normal", but is so unwieldy.
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### Re: "Oh no! We forgot how to say... math... stuff!"

I think the way we define indefinite integrals doesn't make much sense, we're sort of pretending that they equal other functions with arbitrary constants, when they describe entire sets of functions. So I think having

[imath]\int f(x) dx = \int f[/imath] = {F|F'=f} would make more sense.

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### Re: "Oh no! We forgot how to say... math... stuff!"

The thing I dislike is that indefinite integration and antidifferentiation are treated identically. This seems wrong to me, because the integral sign and all the proofs for the value of certain integrals all derive from the idea of summing values in which case the indefinite integral is the definite integral from 0 to x (for f(x)), now this is notationally awkward, but the current notation for indefinite integration fits this perfectly, where it doesn't is when you insert the constant of integration. This simply doesn't fit the notion of an integral being a sum and, as such, I would exclude it from indefinite integrals but leave it in for antiderivatives (which would need a new symbol).
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### Re: "Oh no! We forgot how to say... math... stuff!"

One thing that I have found annoying is that the radix point is a point at the bottom of the line, where it can easily be missed; this leads to "protections" like "0.34" in order to prevent that from being read as "34"; a number less that .1 must start with a "0", and it is rare to start a number greater than or equal to 1 with a "0". However, this "protection" does not prevent the radix point from being missed in a number like "3.4". Oddly, the semantically meaningless thousands separators are more visually distinct; they are commas. While swapping the role of commas and periods in these functions would be an improvement; I really think that a better solution would be to make the radix point even more distinct; one possibility is using the broken vertical bar symbol that is used for ASCII 124 ( | ) in some fonts; that I think is the kind of visual distinctness that the radix point needs to have. The thousands separators -- which we used commas for now -- can have the visual indistinctness of the point at the bottom of the line; indeed, that seems to me to be the perfect representation of them.

Lothar
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### Re: "Oh no! We forgot how to say... math... stuff!"

fagricipni wrote:One thing that I have found annoying is that the radix point is a point at the bottom of the line, where it can easily be missed; this leads to "protections" like "0.34" in order to prevent that from being read as "34"; a number less that .1 must start with a "0", and it is rare to start a number greater than or equal to 1 with a "0". However, this "protection" does not prevent the radix point from being missed in a number like "3.4". Oddly, the semantically meaningless thousands separators are more visually distinct; they are commas. While swapping the role of commas and periods in these functions would be an improvement; I really think that a better solution would be to make the radix point even more distinct; one possibility is using the broken vertical bar symbol that is used for ASCII 124 ( | ) in some fonts; that I think is the kind of visual distinctness that the radix point needs to have. The thousands separators -- which we used commas for now -- can have the visual indistinctness of the point at the bottom of the line; indeed, that seems to me to be the perfect representation of them.

Now that you point that out, I feel like the decimal point is just not in the right place. I feel like it would be more symmetric to mark the units place, rather than the space between 100 and 10-1. Perhaps something like

10 = 10
01 = .1
Always program as if the person who will be maintaining your program is a violent psychopath that knows where you live.

If you're not part of the solution, you're part of the precipitate.

1+1=3 for large values of 1.

pizzazz
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### Re: "Oh no! We forgot how to say... math... stuff!"

I would want fewer words beginning with "c":
complete
compact
connected
continuous
closed
countable

I'm pretty sure we used all of those but 1 in class today. It's kind of annoying coming up with all these different abberviations.

skullturf
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### Re: "Oh no! We forgot how to say... math... stuff!"

I find it a little annoying that the words "parallel" and "perpendicular" are both multisyllabic words starting with P. I often misspeak and say one when I mean the other. I often wish that they started with different letters.

I guess we could try to make the word "orthogonal" more popular.

Talith
Proved the Goldbach Conjecture
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### Re: "Oh no! We forgot how to say... math... stuff!"

For some reason I feel that perpendicular is the word you use in the 2d euclidean case, and orthogonal is what you use whenever two vectors have an inner product of 0, (ie it has a more general use). Certainly I don't think many people use perpendicular to compare vectors in dimensions other than 2.

moiraemachy
Posts: 190
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### Re: "Oh no! We forgot how to say... math... stuff!"

Sorry to necro this thread, but... I have to get this out of my chest. Index notation is reversed! It should be columns X rows (a row vector should be a n x 1 matrix) in order to obey cartesian coordinates.