Talk:Multiplication

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Contents 1 Old article 2 error 3 Multiplication for non-integers 4 Add m to itself n times 5 Multiplication tricks 6 Multiplication in computers 7 Quaternions 8 Correction to Multiplying Fractions example 9 Multiplication of negative numbers 10 Times less 11 Narrow Scope?

[edit] Old article

For every assortment (unique or otherwise) of numbers there is a unique number called the product. Any two numbers in such an assortment can be replaced with their product without effecting any change in the product of the assortment. Any number of ones can be added or removed with no change in the product. Assortments with products other than zero contain only numbers other than zero.

The word multiplication also is used to refer to reproduction.

"Any two numbers in such an assortment can be replaced with their product without effecting any change in the product of the assortment."

This isn't a property of multiplication. This is a property of algebra that states for B = A, B can be substituted for A in any expression without effecting the value of the expression. The note on reproduction should probably be re-added, though.--BlackGriffen

No, no, you're both missing the point of the original article (which is not mine, but I know enough about group theory to understand it). The text above describes the meaning of "multiplication" in group theory, which is any operation (such as traditional multiplication, which the article now describes) that has the properties noted. The specific property mentioned above is not simple one-for-one substitution. Read it again: it's two-for-one substitution. For any collection of numbers, any two can be removed and replaced by the one number which is the product of those two numbers, and the product of the collection will stay the same. --LDC

Would it be too much trouble to motivate how multiplication gets defined for the rationals and reals? Also, the first paragraph is problematic in defining multiplication as repeated addition; how do you add 2.5 to itself 3.7 times? --Ryguasu 01:56 Feb 25, 2003 (UTC)

[edit] error

I think there is an error, when defining the infinite products from -oo to +oo as the sum of two limits, instead of the product. -Marçal

[edit] Multiplication for non-integers

Could anyone write about how to define multiplication for non-integers in the article? (Current, it says one can define multiplication for real numbers but does not say how.) Or more like is it impossible? --Taku 18:45, Apr 2, 2005 (UTC)

I will try to do so, or at least put an adequate link. -Unknown

The idea is that rationals are obtained from integers by localization, and reals are obtained from rationals as factor ring, idem for complex numbers from reals. Both operations involve an injective ring morphism, so that the result of multiplication remains the same for elements of the previous subset. MFH 17:42, 5 Apr 2005 (UTC)

In other words, one first defines the multiplication of integers, then the multiplication of rationals in the usual way, by multiplying the numerators in denominators. After that, if one wants to multiply to real numbers, one approximates them by rationals and multiplies the rationals instead. Of course, to make this rigurous, you create two sequences of rational numbers converging to the two real numbers, then the product of that pair of rationals converges to the product of the pair of reals.

I could go in more detail if necessary. Oleg Alexandrov 17:58, 5 Apr 2005 (UTC)

There is another way to address multiplication for the rationals which doesn't require localization. Define non-zero rationals to be pairs of non-zero integers and with the multiplication induced from Z(+)Z quotient the set by the equivalence relation (a,b)~(c,d) iff ad = bc. If you want a more algebraic sort of quotient, you can describe the multiplicative structure of Z\{0}(+)Z\{0} as a semigroup.

As for the reals in my opinion the best way to characterize them is as the fraction field of power series in one variable with coefficients in Z/pZ quotiented by (x-1/p). As we exploit the fraction field construction, we might as well use localization in defining the rationals. TJSwaine 12:10 3/24/2006

[edit] Add m to itself n times

I think in the Discussion section "Add m to itself n times" should actually read "Add m to itself n-1 times". Adding m to itself once is m+m. --Heycam 08:45, 9 Apr 2005 (UTC)

In my opinion n x m, which reads as n times m, should be interpreted as m+m+...+m, rather than n+n+...+n. I can imagine laying 5 times a pound on a table, but hardly laying a pound times 5. 130.89.220.52 21:19, 16 Apr 2005 (UTC)

[edit] Multiplication tricks

There are many tricks out there, we've all seen those mofo's who use their fingers to multiply things like primative abacii and envied them terribly. I recently learnt the trick to the nine times table one, there are many more; further there are many simplistic methods of dealing with even large number multiplication. If I were to do a summary of the nine times in a seperate article and link it through to here, would anyone else be willing to take on the challenge of equalising for all the geeky kids out there who went through the same mathmatically repressed childhoods that us non-finger-jedi-masters went through with me? :P Jachin 12:23, 11 May 2006 (UTC)

Does anyone know if this has a specific name, or even a wiki page? 64.179.161.110 22:57, 24 November 2006 (UTC)

[edit] Multiplication in computers

Not sure where to add this but shouldn't we have a link or say something about how computers mulitply? There are generally two ways of doing it. The simpler processors only add and let multiplication be done in macro instructions, the other - and today more common way - is to implement a multiplication unit. The algorithm such a unit might use may vary but often go along the lines of multiplying an n bit number A by another n bit number B to produce a 2n bit result. This is in the simplest form done by having a counter and repeat the same process n times. For example by having a 2n bit register with the high n bits called H and the low n bits called L and the least significant bit of L called L0 (a single bit) you start by setting L equal to A, H equal to 0 and then add B to H if L0 is 1 and do not add if L0 is 0. Then shift all bits of H:L down by one so old L0 is dropped and the bit that used to be bit 1 becomes L0 and the carry if any from adding B to H is made most significant bit of H. Then the process is repeated until you have done it n times and all bits of old A is gone and H:L contains the result of the multiplication. Schemes to make this go faster should probably also be mentioned.

The algorithms as described above can thus be written like this:

[Initialize] H := 0 L := A Carry (a single bit storage) := 0

repeat n times.

  if L0 then H := H + B, Carry := Carry from H + B operation.
  [shift down H:L]
  Shift down L, set Lmax (most significant bit) equal to H0.
  Shift down H, set Hmax equal to Carry.
  Carry := 0

end repeat -- H:L contains result of A * B with H holding the n most significant bits and L holding the n least significant bits.

This algorithm works for unsigned multiplication. For signed multiplication you add an initialization step prior to this multplication and also a step after it.

First, check signs of A and B and compute sign of result based on it. If A and B have the same sign, the result is positive while if A and B have different sign the result should be negative. Thus, a sign_result := sign(A) == sign(B).

Compute absolute values of A and B (this is done in parallell and can also be done in parallell with the sign computation above. I.e. if either A or B are negative, negate them.

Then perform the unsigned multiplication described above. If sign_result is negative you then negate the result of the multiplcation.

For sign magnitude reperesenation a negation above can be as simple as ignoring the sign bit but for 2 complement representation you essentially invert and add 1.

Multiplication of floating point numbers are also based on integer multiplication. You extract the exponent part and the mantissa separately and then you add the exponents and multiply the mantissa. In this case you typically preserve the more significant bits of the result and round or truncate the less significant bits. I.e. you only keep H and forget about L except perhaps for the most significant bit of L which is used in rounding.

As I said, these are the basic algorithms - various ways to optimize them exist so that a computer can multiply faster. One such optimization to bear in mind is a "tree structure" way of doing it. Multiply the n bits by n bits to produce a 2n bit result can be done by considering half of n (say n = 2m) and each value A and B can then be considered to be split in two - a high part and a low part.

A = AH * F + AL, B = BH * F + BL

Here AH, AL, BH and BL are each m bits (half of n) and F = 2**m.

Multiplying A and B is then the same as:

(AH * F + AL)(BH * F + BL) = AH*BH * F*F + (AH*BL + AL*BH)*F + AL*BL

Thus a m by m multiplication producing a 2m = n result can be used to implement a n by n multiplication producing a 2n bits result. This can be repeated with half of m etc until you get down to single bits.

A single bit multiplier - i.e. 1 bit multiplied by 1 bit producing a 2 bit result is easy enough. The high bit is always 0 and the low bit is simply AND of the two input bits.

By combining this into a tree structure you can then perform a very fast multiplication. The problem is that you use an awful lot of transistors, so a middle way can be found where you find a lower n which is done by the algorithm described earlier and then you can use the latter method to build up 2n, 4n, 8n etc multiplication circuits by combining the two methods.

Does anyone know of additional procedures? Can anyone describe the methods used in typical modern day computers such as Intel series or others?

I added some links to the See also section to articles on computer multiplication. As a whole they are no t that great. More is neded. Also see category:Computer arithmetic--agr 12:27, 1 December 2006 (UTC)

[edit] Quaternions

In the section on properties, should it be mentioned that commutativity doesn't hold over ?

I'm not sure either way. This article's scope isn't terribly clear to me. Noncommutativity is mentioned at Product (mathematics), at least. Melchoir 18:37, 29 June 2006 (UTC)

Never mind, I'm stupid; ArnoldReinhold already worked it in! Melchoir 18:41, 29 June 2006 (UTC)

[edit] Correction to Multiplying Fractions example

I made a change to the example of multiplying fractions in the introduction. It used to say "a/b × c/d = ac/bd" and I changes it to "a/b × c/d = (ac)/(bd)" meshach

Definitely the parentheses in the denominator should be there to avoid what would be ambiguity at best. Michael Hardy 00:42, 3 July 2006 (UTC)

[edit] Multiplication of negative numbers

Re: http://en.wikipedia.org/wiki/Multiplication the following proof is shown:

(−1) × (−1)
= (−1) × (−1) + (−2) + 2
= (−1) × (−1) + (−1) × 2 + 2
= (−1) × (−1 + 2) + 2
= (−1) × 1 + 2
= (−1) + 2
= 1

Try as I might, I cannot follow the transition from
(-1) x (-1) + (-1) x 2 + 2 ------------ Line 1
to
(-1) x (-1 + 2) + 2 -------------------- Line 2

Comments, please. -Mark jager 23:16, 19 November 2006 (UTC)

Addition distributes over multiplication: a (b+c) = ab + ac. EdC 19:27, 20 November 2006 (UTC)

OK I have some knowledge retained from highschool maths some 38 years ago,
and my maths may be hazy but I 'read' Line 1 above as

((-1) x (-1)) + (( -1 ) x 2) + 2

and I can see no way to extract a common factor so that Line 1 may be expressed as Line 2.
I am now as much intrigued as to why I cannot understand the proof as I am by the fact that 'two negatives make a positive' when multiplied.

The common factor is (-1). Here's how it goes:

((-1) x (-1)) + (( -1 ) x 2) = (-1) x ((-1) + 2)

Another way to see this is to replace (-1) by "a" every place it occurs in line 1:

(-1) x (-1) + (-1) x 2 + 2 ------------ Line 1

Becomes:

a x a + a x 2 + 2

which is equal to

a x (a + 2) +2

--agr 04:44, 21 November 2006 (UTC)

[edit] Times less

Has anyone come across this phrase? In googling 'times less' there's actually a high incidence of it, but I think it's actually a horrible misconception. You can have 'times more', which is, multiplied by whatever number precedes it. Ala, 10 times more is whatever it's being compared to, multiplied by 10. How do you have times less though? What would 10 times less be?

In actuality, 'times more' is an incorrect usage of 'times'. As in, I've 10 times the amount of cookies. 'times more' is really unneeded, so by extension, 'times less' should be as well. What does it mean? It can't mean division, because we would simply use fractions, such as 'a tenth'. So I'm thinking anyone who uses it is simply not understanding mathematical language.

For example, I read this in a newsletter:

Females have about 10 times less anabolic hormones in their bloodstream than men do.

Now, would that mean they have 10%? Why don't people write that? I don't understand it. Tyciol 23:35, 2 January 2007 (UTC)

Well, evidently it does; and it is established usage, so saying it's "incorrect" is a stretch. I guess people prefer the "n times less" usage because it's concise, if not very clear. –EdC 04:53, 3 January 2007 (UTC)

[edit] Narrow Scope?

It may be beyond the normal means of this encyclopedia, but it seems odd to me that this article only discusses multiplication in the sense of R x R -> R. Given that there are other multiplications that exist for other rings and what not, shouldn't this article discuss a multiplication in general? Or perhaps I'm just strange :) 67.142.130.18 04:11, 23 February 2007 (UTC)JSto

See Product (mathematics). Perhaps the links to that article could be improved. –EdC 04:31, 23 February 2007 (UTC)