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Can 1/3 and 1/3 = 2/6? It seemed so

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Re: Can 1/3 and 1/3 = 2/6? It seemed so

#91

Post by Spivak »

grodeni wrote: post #70
Spivak wrote: post #46

Earlier quoted context omitted.

I think it's important to know that this really isn't true. Maths at all times is a subjective language. Maths notation is imprecise "intentionally confused", or just ad-hoc defined all the time. When you see something like. f(x) = summation(x^n, n=0, 10) We conveniently ignore that this polynomial is defined at x=0 despite 0^0 not making any sense by ad-hoc defining 0^0 = 1 in this context.

actually 0^0 is usually chosen equal to 1... which probably still supports your argument.

Corrected! Thank you!

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#92

Post by billfruit »

wcarey wrote: post #4

This is a lovely (edit: having been in similar shoes, also terrifying-in-the-moment) example of a broader problem in teaching mathematics: the language we use to describe mathematical reasoning is a natural language, like English or Latin, and therefore full of the sorts of bizarre irregularities you'd find in a natural language. Mathematics is also a language about rigorously and precisely defined objects. The conce…

Another is sin^2(x) meaning (sin(x))^2, but by a more intuitive reading it should mean sin(sin(x)).

I don't know what exactly is being gained by the usual notation: surely better clarity should be preferred over the time/effort saved in writing the extra pair of paranthesis, and I would prefer it be written as (sin(x))^2 always.

Thing is mathematics, and mathematical pedagogy seems hardly concerned with such rampant notation confusion plaguing much of maths. Perhaps moving to some type of machine-readable notation will be better for consistency and avoiding of much notational confusion.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#93

Post by lisper »

DavidSJ wrote: post #48
twic wrote: post #18

Earlier quoted context omitted.

> x^{-1} means \frac{1}{x}, but f^{-1}(x) doesn't mean \frac{1}{f(x)}. That notation for inverse functions is truly appalling. I don't know how the first mathematician to think of that didn't immediately discard it as nonsensical and misleading.

They’re both inverses: one with respect to multiplication, the other with respect to function composition. In abstract algebra, we observe that there are many types of “products”: multiplication, addition, function composition, composition of rotations, matrix multiplication, etc. A common, unified “power” notation for repeatedly taking the product of a single element, or of its inverse, has some value. There is some…

> one with respect to multiplication, the other with respect to function composition

And, of course, multiplication is a function so one is just a specific instance of the other.

Infix notation is another blight on the mathematical notation landscape. The amount of human effort that has been put into figuring out how to parse a+b*c is staggering. All of this confusion could have been avoided if we'd just started with s-expressions in the first place.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#94

Post by CydeWeys »

jpindar wrote: post #61
CydeWeys wrote: post #30

Earlier quoted context omitted.

A lot of people are at home right now trying to teach their kids who are also at home.

I'm surprised I haven't been seeing more discussions about education recently on HN and everywhere else.

I suspect we skew young enough on here that most of us don't (yet?) have kids.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#95

Post by Spivak »

ColinWright wrote: post #68
Spivak wrote: post #46

Earlier quoted context omitted.

I think it's important to know that this really isn't true. Maths at all times is a subjective language. Maths notation is imprecise "intentionally confused", or just ad-hoc defined all the time. When you see something like. f(x) = summation(x^n, n=0, 10) We conveniently ignore that this polynomial is defined at x=0 despite 0^0 not making any sense by ad-hoc defining 0^0 = 1 in this context.

> We conveniently ignore that this polynomial is defined at x=0 despite 0^0 not making any sense by ad-hoc defining 0^0 = 0 in this context. What do you think the polynomial is? I ask, because in all situations similar to this that I've encountered it made sense to define 0^0 as 1, not as 0. If you genuinely have a case where 0^0 = 0 makes consistent sense then I'd be interested in understanding it. So, what do you t…

Sorry you're right, it should be 0^0 = 1.

An example where 0^0 = 0 occurs when dealing with areas. The measure of the real line in the plane is zero but it's also a rectangle with sides (0, inf) and we define the area of a rectangle to be l*w.

You usually see this written as inf × 0 = 0 but you sometimes you see the interpretation as 1/0 × 0 = 0^0 = 0. And you know this is a an ad-hoc definition because you're not allowed to algebraically manipulate it at all.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#96

Post by andred14 »

Apologies but I don't see the mystery here?

A fraction describes a proportion ie. the amount of something(s) in relation to another amount of something(s) thus, if we double all of the things in a fraction the proportion and hence the fraction is equivalent.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#97

Post by CydeWeys »

wcarey wrote: post #77
uryga wrote: post #73

Earlier quoted context omitted.

> (1,2) + (2,4) - can't be done. are you talking about points or open intervals? the notation is ambiguous ;) but addition of points is fine if you look at them as vectors > a^2 + a^2 - fine. a^2 + a^3 - nope. a^2 + b^2 - nope. what's wrong with the 2nd and 3rd ones? the 3rd one looks like half of Pythagoras theorem which is widely considered "fine" afaik ;)

Indeed. And if they're cartesian vectors, you're good. But if that second number is an angle measured in radians, you use yet a different set of rules for the addition. The conversation with 16 year olds when you explain that their previous teacher who told them that you couldn't add points wasn't lying, but was, perhaps oversimplifying things to make their life easier, is a fun and fraught one. I've had to reason ki…

How did I go all of my life so far without hearing the super useful word "monomial"? It's such an obvious concept to have a meaningful name, and yet I don't recall anyone ever having said that word.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#99

Post by tsimionescu »

p4bl0 wrote: post #63
forbiddenvoid wrote: post #40

The units are missing, and I think that's a key factor here. Both of the equations up on the board at the end are correct because they are counting different things. This is a huge miss if you use a numeric only approach to fractions. The student came up and wrote 1/3 + 1/3 = 2/6. What they meant by that is 1/3 (of the students at a table) + 1/3 (of the students at a different table) = 2/6 (of the students at those t…

Yup, it's a typing problem. Do it with different things in the two sets and it becomes clearer: 1/3 apples + 1/3 oranges = 2/6 fruits . This is another instance of situations where teaching compsci or at least programming could help teaching maths rather than the other way around as it is traditionally thought.

It's not really a typing problem, it's a problem of references. You can't add 1/3 apples and 1/3 oranges. However, you can add 1/3 * x people with 1/3 * y people and get 2/6 * z people, if x people +y people =z people.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#100

Post by chpmrc »

forbiddenvoid wrote: post #40

The units are missing, and I think that's a key factor here. Both of the equations up on the board at the end are correct because they are counting different things. This is a huge miss if you use a numeric only approach to fractions. The student came up and wrote 1/3 + 1/3 = 2/6. What they meant by that is 1/3 (of the students at a table) + 1/3 (of the students at a different table) = 2/6 (of the students at those t…

Exactly what I was thinking. Maybe it could be formalized this way? (with t = 1 table)

1/3 * (1 table) + 1/3 * (1 table) = 2/6 * (2 tables)

or

1/3 * t + 1/3 * t = 2/6 * 2 * t

but 2/6 * 2 = 2/3 so you get

1/3 * t + 1/3 * t = 2/3 * t

which is correct.

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