221–230 of 425 posts
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It looks exactly like scalar multiplication when your matrices are diagonal.
Then: all matrices look like the identity matrix if applied to a suitably rescaled eigenvector.
The Jordan-Chevalley decomposition makes the difference precise -- the nilpotent part (off diagonal 1's in the jordan normal form)
The HN crowd is way above me when it comes to math, but I'm going to chime in here w/ a question, come what may. I read about this a couple of years ago when I was doing basic arithmetic instruction at home to supplement my 5 y/o daughter's schooling. We were just doing basic numeracy activities at the time-- playing with physical representations of a number line, etc. I wanted to have some idea of how to approach mu…
Or putting it another way, I doubt that considering repeated addition gets in the way of understanding scaling.
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> there is Keith Devlin involved.... I am sure he enjoys it a lot to be in a position where he can write pedantic falsehoods while at the same time being taken seriously I don't think his argument (or the similar argument being made in the subject article of this thread) for not teaching students that multiplication is repeated addition is a "pedantic falsehood". I think he has a valid point: that there are downsides…
> teaching students one thing, and then later coming back and saying "well, that thing you were taught before isn't actually correct..." My kids are learning multiplication. I honestly don't feel like they're being lied to, or confused about what they are being taught. Yes, they start with addition and subtraction as crutches (e.g. `9 x N` is initially taught as `10 x N - N`). It's certainly not how I learned (I had…
That isn't a crutch, that is how large numbers can be multiplied in one's head!
90 * 21 is (90 * 20) + 90
Right after removing that extra 90, now you have 9 * 2 * 10 * 10 + 90, which is easy to do mentally.
Heck if you ask me what 9 * 14 is, I'm going to do 10 * 14 - 14.
Did I have to memorize 0x0 through 12x12? Yup. But now days I have a few key mid points memorized and I'll add my way from there. I don't remember 7 * 6 but I know its 7 * 7 - 7.
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How would you see multiplication by pi as repeated addition? By taking 3 times as usual and then another .14 approximately. E.g. for 2, it is 3+3+28/100, roughly 6.28. (edit: if you’re confused by /100, read it as “two places next to a point”) If you’re asking how to do that exactly , first tell how do you even add pi to some number exactly? Let’s start with one, 1 + pi:
> By taking 3 times as usual and then another .14 approximately First, "taking 3 .14 times" doesn't make sense, at least not in the elementary school student's understanding of "repeated addition". Second, pi is irrational, as you evidently realize since you say "approximately". There is no way to add 3 pi times. > If you’re asking how to do that exactly, first tell how do you even add pi to some number exactly? Let’…
Let’s make it clear: you cannot add pi or take pi times at all. You only can add to factors to the left of it (same as a+bi). That’s why you cannot multiply by pi by repeated addition, not because the numeric method is wrong.
Wouldn't it be better to just admit up front that multiplication is not repeated addition, although the two are similar for some types of numbers?
For generalized reasoning, yes. For teaching, not sure. Which is better, a student who knows that mul is rep-add, or a student who gave up and doesn’t know even that? When they learn, they ask themselves why and it’s a rabbit hole. You should stop somewhere before turning into maths professor, if that’s not your goal.
The HN crowd is way above me when it comes to math, but I'm going to chime in here w/ a question, come what may. I read about this a couple of years ago when I was doing basic arithmetic instruction at home to supplement my 5 y/o daughter's schooling. We were just doing basic numeracy activities at the time-- playing with physical representations of a number line, etc. I wanted to have some idea of how to approach mu…
I suspect that the problem isn't introducing a given model, rather the problem is then not moving on to additional models. Or putting it another way, I doubt that considering repeated addition gets in the way of understanding scaling.
The gist of my statement was something like: Adding repeatedly is a fine method to solve this kind of problem but it doesn't for all classes of numbers. We can't say, like we can w/ addition and subtraction, that multiplication implies a specific direction of movement (towards or away from zero) on the number line.
When I was an early college student I started to become curious about the prime numbers. Why are some numbers in particular prime, and others aren't? I started with a thought experiment:
What if it were that 2*2 = 5 ?
Pretty quickly I realized that the relationship between addition and multiplication could not remain the same, because the distributive property guarantees that 2*2=4
2*2 = (1+1) * (1+1) = 1+1+1+1 = 4
1. Associative
2. Commutative
3. Multiplicative Identity
4. Increasing ( If i,j > 1, then m(i,j) > i,j )
5. Bigger number, bigger product ( If j > i, then m(j,k) > m(i,k) )
6. Uniqueness of Prime Factorization
An easy example is the odd numbers.
1,2,3,4,5, ... n
1,3,5,7,9, ... odd(n)
odd(n) = 2n-1
inverse_odd(x) = (x+1)/2
m(i,j) = inverse_odd( odd(i) * odd(j) ) = 2ij - i - j + 1
It turns out that this definition is really a (nice) subset of something known as the Beurling Integers. The Beurling Integers are neat, because you can basically choose whatever distribution of prime factorizations you want (following the rules 4, 5, 6 above) and find a sequence of real numbers that satisfies that distribution. The catch is that we had to sever the ties of addition and multiplication.
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If you define the reals axiomatically, you still need an existence proof. Which will involve addition and multiplication algorithms.
Well, no, not really. The standard definition of the reals is as the unique nontrivial totally-ordered, Dedekind-complete, Archimedean field up to isomorphism. So what you would really need is a uniqueness proof, with addition and multiplications "provided" by the hypothesis.
I've been teaching my 5 year old multiplication and division for the last couple of weeks. I'm at a loss how you would teach it without explaining that it is repeated addition. For example, the other day I asked her how many fingers and toes the three kids at the table had, and she came up with "20 fingers and toes each times 3 kids means there are 60 fingers and toes." I think the units are intuitive in most cases a…
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> At any rate we have to impose that n·0=0, which can't be writen cleverly as "repeated addition" and worked up backwards. n * 0 = n * (1 + (-1)) = n + (-n) = 0. ----- The 0-element in a Galois Field works identically btw. In GF(5), the 0 element is 5 (5 mod 5 == 0). n * (5) == n * 0 == n * (1 + (-1)) == n * (1 + 4) == 0 mod5. For example, if we take "n" == 2, 2 * 5 == 10 mod 5 == 0. 2 * (1 + 4) == 2 + 8 == 2 + 3 (mo…
Repeated addition that ain't... It boggles my mind that you've been studying Galois theory yet somehow try to reduce the algebraic structures associated to two binary operations to playing with one of them.
Nonetheless, in a ring (and all fields are rings), multiplication must and always is related to addition, through the distributed property (which I argue, the distributed property IS the mathematical term for "repeated addition").
Without the distributed property, you have no ring. You at best only have a group. Therefore, all multiplication operators ever defined (or more precisely, all rings) must have multiplication related to addition: (A * (B+C) == AB+AC)
Ah yes, there is Keith Devlin involved.... I am sure he enjoys it a lot to be in a position where he can write pedantic falsehoods while at the same time being taken seriously. You know, most, if not all, mathematical objects of significance can be defined in many, many different ways. In fact, this is the hallmark of an mathematical object of significance: it keeps popping up in many context and can therefore be def…
> there is Keith Devlin involved.... I am sure he enjoys it a lot to be in a position where he can write pedantic falsehoods while at the same time being taken seriously I don't think his argument (or the similar argument being made in the subject article of this thread) for not teaching students that multiplication is repeated addition is a "pedantic falsehood". I think he has a valid point: that there are downsides…
But there is a way to teach mathematics from axioms that is intuitive: geometry. Euclid's Elements was the key mathematics textbook for about 2000 years.
Of course, you simply can't cover as much ground if you must derive everything.
But I do wonder if the collapse in public discourse is partly because of faith-based mathematics. i.e. taught as: it's true because we tell you it's true, not because you can see for yourself that it's true.