Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

denisegaskins.com

Post reply on HN
Print view View original

401–410 of 425 posts

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#401

Post by aaplok »

alentist wrote: post #334
aaplok wrote: post #289

Earlier quoted context omitted.

Multiplication is literally repeated addition, as in "taking a number multiple times". It is in the name. The fact that the operation is so useful that it has been generalised to the point where that original meaning is eventually lost through more and more abstractions doesn't invalidate that, because with all the generalisation and abstraction, multiplication as repeated addition still works, and any generalisation…

Define π times e in terms of repeated addition. I wouldn't say multiplication is literally repeated addition. I'd say it reduces to repeated addition when the multiplier is a natural number.

I am afraid I did not make myself clear. I don't mean that multiplying by irrational numbers is repeated addition. What I mean is that multiplication by irrational (or even non-integers) was defined to generalize the earlier notion of multiplying by positive integers.

Also I mean "literally" literally :). Etymologically, Multiplication is the act of taking/creating many (multi in Latin) copies of something (as in "multiplying breads").

That these generalizations don't fit the concept of repeated addition doesn't erase the fact that the primitive concept meant precisely that. Whether we decide to hide that fact from children can be a conscious decision, but at least it should be deliberate.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#402

Post by eru »

alisonkisk wrote: post #113
eru wrote: post #95

Earlier quoted context omitted.

> There will be infinite addition involved, and algorithms are finite. People are just bit sloppy, and say algorithm when they mean something slightly different. See https://stackoverflow.com/questions/28841260/what-is-the-dif... and https://en.wikipedia.org/wiki/Corecursion Basically, you don't want an 'algorithm' here to produce the whole number. All you need is some scheme that will produce the next digit in finit…

Which is a technical way of saying "in real life people use finite rational or algebraic approximations for reals, so uncountability of reals and infinite precision aren't a problem".

Not quite.

Consider the following real number made of binary digits:

Enumerate all Turing machines and all possible inputs, iff the i-th machine/input combination holds, the i-th binary digit in our number is 0, otherwise 1.

This number is well-defined (once you fix your enumeration scheme).

But there's no finite algorithm to produce approximations in your sense.

What I was after were what's also called Computable numbers (https://en.wikipedia.org/wiki/Computable_number). But I used the more general term of co-recursion, that also applies to arbitrary other data-structures like infinite lists, or with some generalization, infinite event-loops where the important condition is that each run through the body of the loop only takes finite time.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#403

Post by simias »

Double_Cast wrote: post #292
simias wrote: post #82

Earlier quoted context omitted.

I agree that it's a generally pointless distinctions, although it might be useful in some cases such as number systems where the multiplication isn't commutative, or for a particular implementation where the distinction matters. After all if you were to code a multiplication that was implemented naively as a series of additions it'd be generally much faster todo 2x1000 than 1000x2. To return to TFA I think the author…

Suppose your friend Alice arranges your wedding. You ask her to arrange the lawn chairs in an arrangement of 2 rows and 3 columns. But she misinterprets your request as 3 rows and 2 columns. Oops. Now a particular family can't all sit in a single row without rearranging the chairs. If all you care about is the total number of chairs, the order of operands is irrelevant. but if you care about the structure, "2 x 3" ma…

I address that point specifically in my comment. What you say is that the units of the operands matter, which I agree, but the order doesn't. 2rows x 3chairs and 3chairs x 2rows is the same thing.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#404

Post by amelius »

kmill wrote: post #250
amelius wrote: post #244

By the way, why doesn't exponentiation have units in physics? E.g. why don't we ever have something like kg^s?

I think it would have to play well with Taylor series -- for example e^s would be 1 + s + s^2/2 + s^3/6 + s^4/24 + ... -- and that would be mixing units. That's not to say it's meaningless, but rather you'd have to accept measurements like 4s + 5s^2, with incompatible units hanging around. (There's something similar in geometric algebra, I think, where expressions contain terms of different dimensions added together.…

Interesting point. But I think that the Taylor series is just a mathematical tool, and as such it hides the units that exist in these expressions. From a physics point of view, every term in the Taylor series needs to be augmented with a unit that follows from the definition of the Taylor series.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#405

Post by andi999 »

drewcoo wrote: post #34
Agingcoder wrote: post #9

I don't understand. The author is mixing up physics (dimensional analysis) and maths, and trying to give the multiplicand a special role (I didn't even know there was a distinction between the two terms - to me they are both factors). This might be true in the physical world, but in the world of numbers, I think the distinction is irrelevant. Furthermore, being able to compute/define multiplication through repeated a…

Agreed. Multiplier and multiplicand are different words but the commutative property says their values can swap equivalently, so . . . what was the author's point again?

> (2 rows × 3 chairs/row) is not the same as (3 rows × 2 chairs/row), even though both sets contain 6 chairs.

The author has it all backwards. Basically that you can turn this arrangement by 90 degrees (and turning the chairs, if you do not modell them by points by 90 degrees) is the reason why multiplication is commutative. It is non trivial that 5+5+5+...5 (100 times) is the same as 100+100+100+100+100

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#406

Post by alentist »

codethief wrote: post #399
alentist wrote: post #365

Earlier quoted context omitted.

The question is not whether you can define it in terms of addition in some abstract way, but whether you can define it in terms of repeated addition , i.e. something of the form x + x + … + x.

> The question is not whether you can define it in terms of addition in some abstract way This is splitting hairs. The reals themselves are defined "in some abstract way". For instance, what does addition of reals even mean? How do you add two arbitrary real numbers? Exactly, by adding the elements of the Cauchy sequences. (And similarly for multiplication.) This is the definition of addition (multiplication) and the…

> This is splitting hairs.

Nope.

The question addressed by the article is precisely whether multiplication is repeated addition.

> adding the elements of the Cauchy sequences. (And similarly for multiplication.) This is the definition of addition (multiplication)

Nope.

See https://news.ycombinator.com/item?id=26336108.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#407

Post by eigenket »

alentist wrote: post #362
eigenket wrote: post #119

Earlier quoted context omitted.

You first define multiplication of natural numbers to be repeated addition, then define multiplication of rationals in terms of multiplication and addition of naturals, then define multiplication of reals in terms of multiplication of Cauchy sequences of rationals ;)

You are making a mistake that many students unfamiliar with abstract algebra make, which is to confuse a particular " encoding " or " implementation " with the algebraic structure . The concept of a real closed field [0] (and its categorical second-order version, the Dedekind-complete ordered field) stands on its own, without multiplication being defined in terms of repeated addition . It is completely independent of…

I think its easy to get too hung up on particular definitions, you're free to define the reals to be the isomophism class of Dedekind-complete ordered fields if you like, I'm free to define the reals by some construction (Cauchy sequences, Dedekind cuts or whatever). As long as the chosen definitions are isomorphic it doesn't matter at all.

In general in a field I have addition and a multiplicative identity 1, as well as the distributive property, which means that for all elements x satisfying x = 1 + 1 + .... 1 for some number of ones I can write a.x = a.(1 + 1 + .... 1) = a + a + .... a, so there is always a subset for which multiplication works like repeated addition.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#408

Post by alentist »

eigenket wrote: post #407
alentist wrote: post #362

Earlier quoted context omitted.

You are making a mistake that many students unfamiliar with abstract algebra make, which is to confuse a particular " encoding " or " implementation " with the algebraic structure . The concept of a real closed field [0] (and its categorical second-order version, the Dedekind-complete ordered field) stands on its own, without multiplication being defined in terms of repeated addition . It is completely independent of…

I think its easy to get too hung up on particular definitions, you're free to define the reals to be the isomophism class of Dedekind-complete ordered fields if you like, I'm free to define the reals by some construction (Cauchy sequences, Dedekind cuts or whatever). As long as the chosen definitions are isomorphic it doesn't matter at all. In general in a field I have addition and a multiplicative identity 1, as wel…

You are confusing a theory [0] with a model [1] of that theory. These are two different things.

> there is always a subset for which multiplication works like repeated addition

A subset, yes.

[0] https://en.wikipedia.org/wiki/Theory_(mathematical_logic)

[1] https://en.wikipedia.org/wiki/Model_theory

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#409

Post by eigenket »

alentist wrote: post #408
eigenket wrote: post #407

Earlier quoted context omitted.

I think its easy to get too hung up on particular definitions, you're free to define the reals to be the isomophism class of Dedekind-complete ordered fields if you like, I'm free to define the reals by some construction (Cauchy sequences, Dedekind cuts or whatever). As long as the chosen definitions are isomorphic it doesn't matter at all. In general in a field I have addition and a multiplicative identity 1, as wel…

You are confusing a theory [0] with a model [1] of that theory. These are two different things. > there is always a subset for which multiplication works like repeated addition A subset , yes. [0] https://en.wikipedia.org/wiki/Theory_(mathematical_logic) [1] https://en.wikipedia.org/wiki/Model_theory

Na, I'm not - I'm explaining how to "compute: \pi*\pi using repeated addition". Step 1 - find a convenient Cauchy sequence that converges to \pi.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#410

Post by alentist »

eigenket wrote: post #409
alentist wrote: post #408

Earlier quoted context omitted.

You are confusing a theory [0] with a model [1] of that theory. These are two different things. > there is always a subset for which multiplication works like repeated addition A subset , yes. [0] https://en.wikipedia.org/wiki/Theory_(mathematical_logic) [1] https://en.wikipedia.org/wiki/Model_theory

Na, I'm not - I'm explaining how to "compute: \pi*\pi using repeated addition". Step 1 - find a convenient Cauchy sequence that converges to \pi.

> Na, I'm not

Yes, you are. Cauchy sequences are a model of the real numbers (which are the Dedekind-complete ordered field).

> I'm explaining how to "compute: \pi*\pi using repeated addition"

How will you define π times π as a repeated addition, i.e. something of the form x + x + … + x?

Post reply on HN