4 ms·

"It's like it's a different language!" (incredulously) Yes, it is a different language. It is built for expressing ideas about deduction and quantity clearly,

by wfo 10y ago

"It's like it's a different language!" (incredulously) Yes, it is a different language. It is built for expressing ideas about deduction and quantity clearly, flexibly, creatively, and it works beautifully. Saying 2^3 = 8 and log_2 8 = 3 sort of get at the same fact but not really, that's misunderstanding their purpose. They express that we are evaluating functions. You can express 8 - 2 = 6 or 6 + 2 = 8 "in the same way" (Crazy! How can we have two notations +, - when they are just inverses of each other?! We shouldn't give ourselves language to express both "the difference between 8 and 2 is 6", and "2 more than 6 is 8" because they happen to be rearrangements of the same equation!) but the equations are used to convey meaning in different ways in different contexts. The second example, the 8^(1/3) is not even equal to 2, it's equal to three values, two of them are imaginary. It's important to have notation for "the thing that when you cube it is equal to 8" distinct and understood so that when you begin understanding imaginary numbers (high school iirc) you have context. Then you can explain the definite article in that quoted sentence is actually inaccurate. What if he had selected an example with two real roots, like sqrt(4)? What should we put in the the "triangle of power"? The positive branch cut? Okay, now you have to explain that you're really doing a different operation now, that has multiple answers, but we are going to pick one of them and put it there, but we have to remember that it could be either. Which is best expressed using separate notation to explain the operation you are doing that isn't even a function. The fact of the matter is that these ideas are distinct and relating them is a separate, worthwhile exercise that helps understand the structure of exponentiation and its inverses. And in doing mathematics you will find that switching to equivalent but more informative or clean or applicable notation is one of the most valuable workhorses we have for solving simple problems.

apalmer 10y ago
cant upvote enough... the triangle notation is actually "wrong"
nemetroid 10y ago
> The second example, the 8^(1/3) is not even equal to 2, it's equal to three values No, the expression "8^(1/3)" is equal to 2 and only 2, even though there are three values of x for which it holds that x^3 = 8. > What if he had selected an example with two real roots, like sqrt(4)? The notation √a refers to the principal square root of a, which is defined to be positive. You might recall expressions such as "x = +/- sqrt(a)", which would be redundant if sqrt itself was a multivalued function. a^b, for rational b, is defined in a similar manner.
- wfo 10y ago
  It depends entirely on the context -- mathematical notation, as always, depends upon the level of instruction, textbook, context, mathematician etc. In algebraic geometry we take the radical ideal sqrt(I) for an ideal I which is most certainly a set of values. When you're teaching it for the first time (as is the context we are discussing right now) it's quite important to reinforce how many square roots there are. Context generally makes it clear (among people who already understand what is happening) which version you mean as is usually the case. Similarly with logs, though we have the nice log/Log distinction which I suppose has become standard-ish. There is literally no reason we select the positive branch other than notational convenience and one of the biggest mistakes I've seen with freshmen students trying to learn calculus is algebra mistakes like this -- taking a square root and only considering the positive branch. Because they were taught square root of 4 is 2. And that's true sometimes (when you are computing sqrt(x)), and not others (when you take the square root of something for the purpose of solving an equation). I've noticed some online resources recently are very careful about referring to sqrt(x) as the "principal square root function", something which is very good but that I have never once heard anyone say in real life, teacher or mathematician. Obnoxious irrelevant pedantry aside, context matters. You understand the concern here: simplifying notation masks the actual mathematics. These operations aren't identical or as simple as the video would have us believe, distinct notation exists for a reason -- to separate separated concepts.
  - nemetroid 10y ago
    > In algebraic geometry we take the radical ideal sqrt(I) for an ideal I which is most certainly a set of values. But that's just overloaded syntax, isn't it? We're concerned with real numbers here. > There is literally no reason we select the positive branch other than notational convenience Agreed, it's just a convention, though a very helpful one. > I've noticed some online resources recently are very careful about referring to sqrt(x) as the "principal square root function", something which is very good but that I have never once heard anyone say in real life, teacher or mathematician. I had to look up that one, since my maths education wasn't in English. I agree that it's nicer than what appears e.g. in my textbook: "the square root is always a positive number or zero", which conflicts wiwitthe definition that's on Wikipedia. It's probably the conflicting uses of "square root" to refer to two different things (all solutions, which is more relevant analytically, or the unique positive solution, which is more helpful for notation) that causes the kinds of freshman errors you mention. > You understand the concern here: simplifying notation masks the actual mathematics. My point is that it doesn't mask the actual mathematics any more than the regular notation already does. Which also is a problem, but not the one at hand.

by wfo 10y ago

apalmer 10y ago

cant upvote enough... the triangle notation is actually "wrong"

nemetroid 10y ago

> The second example, the 8^(1/3) is not even equal to 2, it's equal to three values No, the expression "8^(1/3)" is equal to 2 and only 2, even though there are three values of x for which it holds that x^3 = 8. > What if he had selected an example with two real roots, like sqrt(4)? The notation √a refers to the principal square root of a, which is defined to be positive. You might recall expressions such as "x = +/- sqrt(a)", which would be redundant if sqrt itself was a multivalued function. a^b, for rational b, is defined in a similar manner.

wfo 10y ago
It depends entirely on the context -- mathematical notation, as always, depends upon the level of instruction, textbook, context, mathematician etc. In algebraic geometry we take the radical ideal sqrt(I) for an ideal I which is most certainly a set of values. When you're teaching it for the first time (as is the context we are discussing right now) it's quite important to reinforce how many square roots there are. Context generally makes it clear (among people who already understand what is happening) which version you mean as is usually the case. Similarly with logs, though we have the nice log/Log distinction which I suppose has become standard-ish. There is literally no reason we select the positive branch other than notational convenience and one of the biggest mistakes I've seen with freshmen students trying to learn calculus is algebra mistakes like this -- taking a square root and only considering the positive branch. Because they were taught square root of 4 is 2. And that's true sometimes (when you are computing sqrt(x)), and not others (when you take the square root of something for the purpose of solving an equation). I've noticed some online resources recently are very careful about referring to sqrt(x) as the "principal square root function", something which is very good but that I have never once heard anyone say in real life, teacher or mathematician. Obnoxious irrelevant pedantry aside, context matters. You understand the concern here: simplifying notation masks the actual mathematics. These operations aren't identical or as simple as the video would have us believe, distinct notation exists for a reason -- to separate separated concepts.
- nemetroid 10y ago
  > In algebraic geometry we take the radical ideal sqrt(I) for an ideal I which is most certainly a set of values. But that's just overloaded syntax, isn't it? We're concerned with real numbers here. > There is literally no reason we select the positive branch other than notational convenience Agreed, it's just a convention, though a very helpful one. > I've noticed some online resources recently are very careful about referring to sqrt(x) as the "principal square root function", something which is very good but that I have never once heard anyone say in real life, teacher or mathematician. I had to look up that one, since my maths education wasn't in English. I agree that it's nicer than what appears e.g. in my textbook: "the square root is always a positive number or zero", which conflicts wiwitthe definition that's on Wikipedia. It's probably the conflicting uses of "square root" to refer to two different things (all solutions, which is more relevant analytically, or the unique positive solution, which is more helpful for notation) that causes the kinds of freshman errors you mention. > You understand the concern here: simplifying notation masks the actual mathematics. My point is that it doesn't mask the actual mathematics any more than the regular notation already does. Which also is a problem, but not the one at hand.