Talk:Multiplication/Archive 1

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)

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
 * Perhaps there should be another class of numbers called "unrepresentable numbers," which are real (or complex!) numbers that are defined that there is no way, in a finite amount of space, for them to be represented. All representable numbers can be multiplied together. Indeed123 22:12, 8 May 2007 (UTC)


 * There is a notion of representable numbers -- see Computable numbers, although this may not be what you are thinking about. Unfortunately with this representation you can only decide when two numbers (as represented by programs) are different and may not be able to positively determine when they are the same.  They are, however, closed under multiplication. TooMuchMath 22:49, 9 May 2007 (UTC)


 * It seems a mistake, to me, to flatly define multiplication with multiplication of whole numbers, since very early in grade school that will break on fractions. As this seems to have been mentioned, but not addressed, I'll take a shot at it. Pete St.John (talk) 23:27, 28 November 2007 (UTC)


 * I added some words to the intro; and a new section for multiplication of different kinds of numbers. I don't think this wins the $100 prize :-) but I hope it helps a bit. Pete St.John (talk) 00:38, 29 November 2007 (UTC)

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

I corrected the 2 x 3 x 5 = 15 and the 2 x 2 x 2 x 2 x 2 = 16 examples please revert them if I did it wrong Markr9 (talk) 17:52, 6 September 2008 (UTC)

Quaternions
In the section on properties, should it be mentioned that commutativity doesn't hold over $$\mathbb{H}$$?
 * 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)

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)

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)

Hello. Concerning the proofs for {1} -1*x = -x and {2} -1*-1 = 1:
 * in {2} ... = (−1)·(−1) + (−2) + 2 = (−1)·(−1) + (−1)·2 + 2 = ...
 * Here we use {1} for (-2) -> (-1)·2, but {1} can be applied just at the beginning: -1*-1 = -(-1) = 1


 * in {1} ... = 1·x - 2·x = -x
 * Can we do this directly? I think that this involves {1} itself, leading to incorrect proof. —Preceding unsigned comment added by 85.130.107.83 (talk) 20:17, 31 January 2008 (UTC)


 * Good catch. To fill the gaps in the proof of {1} would have taken several more lines; I have replaced it by a proof along different lines. I have also used the above as a much simpler proof of {2). --Lambiam 07:35, 1 February 2008 (UTC)


 * The problem with the proof you gave is that it uses distributive law for subtraction which itself uses (-1)x = -x. 85.130.107.83 (talk) —Preceding comment was added at 15:07, 2 February 2008 (UTC)

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)

Yes, I agree that it is improper usage of the term "times" but it is standard usage. However, I have wondered that same question before as well. In my experiences with the phrase, Females have about 10 times less anabolic hormones in their bloodstream than men do. means, Males have about 10 times the anabolic hormones in their bloodstream as females do. "Times less" appears to be just a more convenient albeit incorrect way of expressing the inverse. Zrs 12 (talk) 00:57, 31 January 2008 (UTC)

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)

"terms"
The section on infinite products calls the things being multiplied "terms". This is correct for summation, but surely it is wrong when we are multiplying? What do we call these components? Factors or multiplicands, maybe? nadav 08:30, 12 April 2007 (UTC)

multiplication
the answer is the product —Preceding unsigned comment added by 72.211.208.77 (talk) 03:35, 9 October 2007 (UTC)

Egyption multiplication
I'll leave a note on Silly Rabbit's talk page regarding his changes. Pete St.John (talk) 17:22, 23 January 2008 (UTC)


 * Why not discuss it here? --Lambiam 23:06, 23 January 2008 (UTC)


 * because my remarks include the context of a previous edit revert (that had a happy ending). Of course if there are any content questions sure, they belong here. But I think it's a hastiness question. And surely, surely, I'm hasty sometimes myself. Pete St.John (talk) 23:11, 23 January 2008 (UTC)

As the original author of the passage in question, I am in a rather unique position to comment on the author's intent. To compute 13 &times; 21, you must double 21 three times. So, starting with 21 &times; 1 = 21 (which doesn't count as a "doubling"). Then, doubling again, 21 &times; 2 = 42. Then, again, 21 &times; 4 = 42 &times; 2 &= 84. Finally, 21 &times; 8 = 84 &times; 2 = 168. In other words, you need to go through all these steps. You are invited, of course, to write out an algorithm, and see that in fact all the intermediate stages are necessary to give a complete illustration of the technique. Silly rabbit (talk) 01:54, 24 January 2008 (UTC)


 * Please see also the article Ancient Egyptian multiplication, of which this section is only a summary. Note that all intermediate powers of two are included in the calculation, even if they have no direct relevance to the final result. Silly rabbit (talk) 02:04, 24 January 2008 (UTC)


 * In performing the algorithm (the ancient way), yes, the doubling that makes 42 would have been performed, in order to get the next doubling, 84. Then some of those intermediate calculations (the ones corresponding to 1, 4, and 8, which add up to 13) would be summands for the final calculation. It's fine to express it that way if it's clear. So for example:

1. Decompose 13 into a sum of powers of two, getting 1, 4, and 8.

2. Double 21 repeatedly, getting 21 (the vacous doubling, corresponding to the summand 1 which is 2 to the power zero), 42, 84, 168;

3. Choose the intermediate results that correspond to the addends 1,4, and 8; that would be 21, 84, and 168;

4. Add up those results, getting 273, which indeed is 13 times 21.
 * My objection to the latest wording is purely pedagogical. If we just distinguish the "doublings" which are intermediate results (2*21=42, computed so as to get 4*21 by doubling 42) from those which produce summands used in the final reckoning, then it's fine. Also, it's confusing to say "...doesn't count as doubling" but then "...doubling again..."; think how that reads to a schoolchild. You can't double "again" if you hadn't doubled previously. So I would call 1*21 the "null" doubling; it's indeed a doubling, it's just a doubling zero times (that is, 2^0). Pete St.John (talk) 17:51, 24 January 2008 (UTC)


 * OK, let me back off on this. The wording as it existed (and now, modulo the character chosen for multiplication) is quite good, and I shouldn't have reverted it. Pete St.John (talk) 22:23, 24 January 2008 (UTC)

A Question
I'm not sure that I understand the operators $$\sum$$ and $$\prod$$ so could someone tell me if these equations are correct?

$$\sum_{n=m}^k\frac{n}{2}=\frac{m}{2}+\frac{m+1}{2}+\frac{m+2}{2}+...\frac{k-1}{2}+\frac{k}{2}$$ $$\prod_{n=m}^k\frac{n}{2}=\frac{m}{2}\cdot\frac{m+1}{2}\cdot\frac{m+2}{2}\cdot...\frac{k-1}{2}\cdot\frac{k}{2}$$

(sorry- signed Zrs 12 (talk) 00:42, 31 January 2008 (UTC))


 * I'm not sure this is the best place to ask, but yes, sure, you have it right. The Sigma is short for "summa" (summation, addition) and the Pi is short for Product (multiplication). Incidentally, the funky streched out S used for integration is a germanic 'S' and also comes from "summation" (a definite integral is the limit of a sequence of sums). (Incidentally, if that had been signed, I would have moved it to the talk page...) Pete St.John (talk) 20:21, 30 January 2008 (UTC)

Your understanding of the math is right, but for typesetting style I'd rather see


 * $$\sum_{n=m}^k \frac{n}{2} = \frac{m}{2} + \frac{m+1}{2} + \frac{m+2}{2} + \cdots +\frac{k-1}{2} + \frac{k}{2}$$

instead of


 * $$\sum_{n=m}^k \frac{n}{2} = \frac{m}{2} + \frac{m+1}{2} + \frac{m+2}{2} + ... \frac{k-1}{2} + \frac{k}{2}$$

(with \cdots instead of "...", and a "+" BOTH before and after the \cdots). Michael Hardy (talk) 20:33, 24 February 2008 (UTC)

TeX typesetting
If you're going to contribute to this article with mathematics, please take the time to mark it up correctly in the TeX language (this link may help). Otherwise it not only looks poor but can be difficult to read. Thank you! Stephen Shaw (talk) 19:21, 8 May 2008 (UTC)


 * For inline expressions, HTML markup is often preferable to LaTeX-style markup. See also Manual of style (mathematics). Neither looks good on all platforms, unfortunately. --Lambiam 10:30, 9 May 2008 (UTC)


 * I was mainly referring to things like writing a2 instead of $$a_2$$, which are two entirely different things when written on paper. Since the manual of style states that mark-up changes of simple expressions from HTML to LaTeX or vice-versa are condoned provided the entire article is consistent, it is therefore necessary that any further mathematics posted to this article be marked-up in LaTeX, to keep everything consistent. Stephen Shaw (talk) 20:18, 9 May 2008 (UTC)

Terminology
I am looking for the terminology. Is it "factor times factor equals product" or is it called "multiplier," or "coefficient" etc. --Tattoe (talk) 09:45, 23 June 2008 (UTC)
 * "Coefficient" has a different implication (i.e., that a variable, not a constant, is being multiplied by it. "Factor" or (inelegantly but accurately) "multiplicand" is fine. Bongomatic (talk) 09:52, 23 June 2008 (UTC)

"Presumptuous"??
So it's "presumptuous" to say that most people learn multiplication in elementary school (see the recent edit history)?? Will we next read that it's "presumptuous" to say that most elementary schools teach reading and the most people learn to tie their shoes before they're 40 years old? Michael Hardy (talk) 03:36, 18 November 2008 (UTC)


 * I removed it because I thought it might seem offputting or patronising to people coming here who have not had any sort of mathematical education. Kind of like: "Most people already know all this, so if you're looking here then you must be pretty dumb". And for those who have learned it at elementary school, well, they don't need to be told. But if you feel especially strongly that it should be included then please put it back in. Matt 21:15, 18 November 2008 (UTC). —Preceding unsigned comment added by 86.133.55.73 (talk)

I don't want to be offputting or patronising, but unless things have changed FAR more than I thought since I was in school, it would be extraordinarily unusual not to learn what multiplication is in elementary school. It's much simpler than learning to read and write, and if someone's looking at this page, they must have done that. Michael Hardy (talk) 18:25, 29 November 2008 (UTC)

Could someone check this?
This is beyond my level of expertise, but on the face of it, it does not appear to make much sense:


 * Multiplication for some types of "numbers" may have corresponding division, without inverses; in an integral domain $$x$$ may have no inverse "$$\frac{1}{x}$$" but $$\frac{x}{y}$$ may be defined. In a division ring there are inverses but they are not commutative (since $$\left(\frac{1}{x}\right)\left(\frac{1}{y}\right)$$ is not the same as $$\left(\frac{1}{y}\right)\left(\frac{1}{x}\right)$$, $$\frac{x}{y}$$ may be ambiguous).

Why would 1/x not existing be expected to cause problems for x/y? For example, 1/0 is undefined yet 0/1 is fine.

Why would (1/x)(1/y) not being commutative be a problem for x/y? If the non-commutative example was x(1/y) then it would seem to make more sense.

Matt 21:22, 18 November 2008 (UTC). —Preceding unsigned comment added by 86.133.55.73 (talk)

Capital pi notation
Hi. What means :
 * $$ \frac{F_n(x,c)}{\prod_{m} G_m(x,c)}, n:m $$

I do not know how to read Pi notation here.

it is from paper Andrey Morozov : Universal Mandelbrot Set as a Model of Phase Transition Theory

--Adam majewski (talk) 15:44, 21 November 2008 (UTC)

[question and answers]. Maybe it can be included in article ? --Adam majewski (talk) 08:13, 29 November 2008 (UTC)
 * You might get a quicker reply to this at Reference_desk/Mathematics. —Preceding unsigned comment added by 86.134.53.253 (talk) 12:43, 27 November 2008 (UTC)
 * You are right. thx.

Assessment comment
Substituted at 15:25, 1 May 2016 (UTC)

Symbol usage
Which symbol is more common - the cross or the dot? I was hoping that'd be answered in the article. As it is now I can only assume the list of common notations is ordered by which is most common, in which case the answer would be the cross. Since I received a telling off from one of my German maths teachers once for using the cross and defending it as the more-in-use symbol, being told that's bogus, I'm a bit curious what, exactly, is true. Trivial question, I realise, and it's not life-altering, but I do wonder who was 'right'. :) Maybe that could be added to the article. -pinkgothic 14:34, 9 July 2007 (UTC)


 * I dunno... I haven't used the cross since the beginning of 7th or 8th grade. Looks kinda stupid, besides being easily confused with "x" as a variable-- of course, the dot looks like a period just as badly. When you think about it, it's a matter of personal preference, but it seems to say "I go through so many multiplications that I have to use a shortcut-- therefore I'm smarter and am a mathematician!!!" when you use the dot.75.36.45.94 04:18, 10 August 2007 (UTC)


 * In my experience (with numerals) the dot is used more commonly. (It seems to me that the cross is used more commonly in elementary school when variables are out of the question).  However, as the article states, when multiplying variables, they should be juxtaposed (i.e. xy).

Zrs 12 (talk) 00:53, 31 January 2008 (UTC)


 * I'm facing the same problem: decide what to use in a global company. My idea is to follow ISO. They use the cross, but only if a sign is really necessary. —Preceding unsigned comment added by 163.157.254.25 (talk) 14:00, 14 March 2008 (UTC)


 * Mathematicians normally don't use either of these symbols in their mathematical writings, but instead juxtaposition; they write 2xe−2x, and not 2×x×e−2×x or 2·x·e−2·x. The exception is when this results in something that is unclear or ambiguous, as for products of numerals, in which case the more common solution is to use centred dots, as in 5! = 1·2·3·4·5 (and not 5! = 12345). When the subject matter is not itself a mathematical topic, and the target audience is not assumed to be familiar with the conventions of mathematicians, the cross symbol may be used in formulas. See for examples Percentage. --Lambiam 21:43, 15 March 2008 (UTC)


 * I live in Sweden and I've been taught to only use the centre dot ·. (The only exception seems to be when giving the dimensions of a rectangle, e.g. a screen resolution of 640 x 480, where an x is used rather than a cross.) This is what the swedish article has to say on the topic:
 * The multiplication sign is a dot · placed at the same height as the plus + and - minus sign, alternatively a cross ×. However the cross should be avoided since it also has a different meaning, namely the cross product.
 * Looking through some of the american math and computer science literature on my shelf I notice they tend to use the dot as well. My impression though is that neither sign is automatically wrong, they're simply used in different contexts. The cross is used in simple everyday maths and the dot is used in more "scientific" context. Some countries (like Sweden) have decided to promote only one standard. (The same goes for division by the way. When I was in elementary school the horizontal bar was the only way to write a division/fraction. Neither / nor ÷ were accepted.)
 * I believe both symbols should be give the same status in the article, at least until the topic has been discussed further. In the Wikipedia Manual of Style both seem to be given the same status (with the consequence that both might be mixed within an article - just take a look at this one).
 * Another thing that might be worth mentioning in the article is the relation to the signs used for the dot and cross products. As long as ordinary multiplication is represented by a dot "v · w" there's no problem telling it from the cross product "v × w". However, sometimes an author might find the need for a separate sign for the dot product. Here I've seen two solutions, one is the dingbat "v•w", the other is the more lengthy "(v|w)". Tasnu Arakun (talk) 02:53, 12 April 2008 (UTC)


 * Use of the × symbol is common in primary school, when the students are learning how to multiply numbers, like in the tables of multiplication, something they have to unlearn if they continue to study mathematics. I don't know about elementary education in Sweden; perhaps they use centred dots from the start. --Lambiam 17:06, 12 April 2008 (UTC)


 * Very nice discussion! I too have problems in reading some articles in English Wikipedia, because of that abundant use of "x" for mere multiplications. At first read, I thought what do Factorials have to do with vectors? ("5x4x3...") But then I got to know that they use the "x" for multiplication here. I don't like this at all; because this is the English Wikipedia and this language is mostly spoken in the U.S. and in the UK, so we should (!) stick to the dot. My 2c -andy 92.229.70.233 (talk) 01:40, 27 January 2009 (UTC)

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)


 * The algorithm used in computers is the same as used by Ancient Egyptian multiplication and its explanation is much simpler. --82.141.61.150 12:16, 20 May 2007 (UTC)


 * It's essentially the same, but I think the Egyptians used big-endian multiplication and most hardware implementations are little-endian.  However, it is certainly a point worth making.  Silly rabbit 12:27, 20 May 2007 (UTC)

if 1 = O and 2 = OO as object counts how much is .05 = ? multiplication has several uses, not just as size values. auctioneers and banktellers when counting 20 dollar bills or 10's use multiplication speed counting still today. An auctioneer does so when he counts 5,10,15,20 do i have a 25. I believe a bjorked. (point) decimal system is a large culprit of your problem. cheers on figuring that out as i have yet to find a mathematician that can and can tell what various decimal values represent as physical objects. math was made for doing inventory and counting stuff.

The fact that on the front page of wiki has incorrectly states that 3 times 4 means 4+4+4=12 is a prime example. 3x4 means 3+3+3+3=4 the 4 is the number of times you add the 3. but oh well. here we go again lol. —Preceding unsigned comment added by 75.128.44.182 (talk) 09:06, 7 August 2009 (UTC)

What about "·" as a multiplication sign?
The article mentions the "×" as a multiplication sign. But what about "·", is that not considered to be a multiplication sign? The multiplication sign article mentions it but seems to say it is only used in non-Anglophone countries. --Kri (talk) 20:45, 26 August 2010 (UTC)

Continuous Product / Multiplical
What about adding the "Continuous Product" (aka "Multiplical") to this article? It is the continuous equivalent to the product sequence (or the multiplicative equivalent to the integral) and very useful esp. when working with probabilities. See for more detailed information (exact definition, formulas, examplary usage).

I will give a preliminary definition here and suggest to work on it and include it in section 3 of this article and also in Product (mathematics):

Let $$f \colon \mathbf{R} \to \mathbf{R}$$ fulfill certain boundedness and positivity conditions on an interval [a,b]. The continuous product (a.k.a. "mulitplical") can then be defined in a Riemannian sense as

$$	\prod_a^b f(x) {}^{dx} = \lim_{n \rightarrow \infty} \prod_{i=1}^n f \bigl( a + i \cdot \Delta x(n) \bigr)^{\Delta x(n)} \quad\text{with}\quad \Delta x(n) := \frac{b-a}{n} $$

if the limit exists. An alternative definition is derived from the discrete equivalence $$\prod_i f_i = \exp ( \sum_i \ln f_i )$$ and is given by

$$	\prod_a^b f(x) {}^{dx} = \exp \left( \int_a^b \ln f(x) dx \right). $$

The definition of the continuous product is the continuous equivalent of the indexed product operator and the "product-wise" equivalent to the integration:

$$	\begin{array}{c|cc} & \text{additive} & \text{multiplicative} \\ \hline \text{discrete} & \sum_{i=a}^b f(i) & \prod_{i=a}^b f(i) \\ \text{continuous} & \int_a^b f(x) dx & \prod_a^b f(x) {}^{dx} \end{array} $$

The most common notations for the continuous product sign seem to be the Pi-like symbol also used for product sequences and $$\mathcal{P}$$. I have chosen to use the former one because it is less disturbing for readers not familiar with the concept, because it's more intuitive, and because $$\mathcal{P}$$ is already used for other things (like the power set or individual variables of mathematical texts). In the long run, a distinct, new symbol for the continuous product would be desirable (just like the integral sign for the continuous summation).

Keilandreas (talk) 10:55, 26 March 2010 (UTC)


 * I suppose a see also to Product integral would be okay. Dmcq (talk) 18:38, 26 March 2010 (UTC)


 * Thanks for pointing me to this article - I had no idea it existed! I created a link in product (which is given in this article's see also list). If others think it is appropriate to directly insert a link here, you have my vote on this. --Keilandreas (talk) 04:22, 26 October 2010 (UTC)

External Link for Free Math Flash Cards
I'm writing to request that we add an external link to Free Math Flash Cards. Free Math Flash Cards is more than a web site, or even a computer program. It's a process. Free Math Flash Cards has everything students need to memorize the multiplication table. With Free Math Flash Cards, students know the answer to every problem from 1 X 1 through 12 X 12 like they know their own name. They don't have to think about it, they just know all the answers. Most students memorize the multiplication table in less than four hours with Free Math Flash Cards. The average is 2.5 hours.

Students do need help with math. According to the National Center for Educational Statistics (NCES) indicator list 61% of fourth graders and 66% of eight graders are not proficient in math.

The biggest challenge fourth grade students face is memorizing the multiplication table. If you can count you can add and subtract. Division is easy if you know the multiplication table because division is just the inverse of multiplication. Thus memorizing the multiplication table is the biggest challenge fourth grade students face. It's also important that older students memorize the multiplication table. For example it's difficult to focus on the Pythagorean theorem when you don't know the multiplication table. Also, if you do know the multiplication table you can finish your homework faster with fewer mistakes.

Here is how Free Math Flash Cards works.

First students should review some basic rules. It's very important that the student knows all these rules before they begin to study. They are all easy, but if the student does not know one or two rules the process will be much more difficult.

Then complete one virtual deck of Free Math Flash Cards every day for a week.

Free Math Flash Cards builds a virtual deck of cards in computer memory. It presents a card, or math problem, and the student types their answer. If they answer quickly and correctly, it removes that card from the deck and they will not see that card again. However, if they struggle with the problem it puts the card back in the virtual deck and they will see the same card again. When they correctly and quickly answer all questions, it starts over.

If you have any questions please send me an email at support@smartflashcards.net.

Thanks for your consideration.

Jarhead422 (talk) 00:25, 17 January 2011 (UTC)


 * I can see you are concerned but this is an encyclopaedia, that site does not provide encyclopedic content. The purpose of Wikipedia is to inform not to teach. Dmcq (talk) 12:25, 17 January 2011 (UTC)

Dear DMCQ,

I see your point, but while teaching and the encyclopedia are not exactly the same thing, they are closely related. The definition of an encyclopedia is "a book or set of books giving information on many subjects or on many aspects of one subject and typically arranged alphabetically". One definition of the word "teach" is  give information about or instruction in (a subject or skill).

Since they are not the same, I'm suggesting an External Link to Free Math Flash Cards not including it in the article. However, I believe Free Math Flash Cards is the best method to memorize the multiplication table. Perhaps Wikipedia should have an article about Free Math Flash Cards. What do you think?

It also seems you did not visit Free Math Flash Cards. Because it does contains information about multiplication as you mention above on the help pages.

The most important reason to add an external link to Free Math Flash Cards is to help the users of Wikipedia. I believe most people who view this article are parents teachers or students who need help with multiplication. The Wikipedia Multiplication page does a wonderful job of defining multiplication. The next logical step for your users is to provide methods for them to actually memorize the multiplication table.

Jarhead422 (talk) 03:09, 19 January 2011 (UTC)


 * Go to External links/Noticeboard to debate this sort of stuff if you want to but I believe I am applying the policy correctly and it is the consensus of how editors think what should be done about links like this. I had a look at that help page and it failed on two counts, it didn't provide extra information and it is a blog. Blogs are expressly not linked except in special circumstances see WP:EL. There really would need to be a secondary source that talked about the problems of teaching mathematics plus some review showing how this software was good at solving these problems so the relevance to the article could be shown. At the moment it is jut software some made up and is trying to promote on Wikipedia. Dmcq (talk) 09:20, 19 January 2011 (UTC)

You have external links to Chinese abacus and cut the knot. I added Free Math Flash Cards and it was removed. I got a message to go to this talk page and discuss the matter. Now you tell me it's not the right place to discuss it.

My help pages are a blog because I invite user participation, however the external link I propose here is not a blog.

The proposed link does provide further information on a new learning method.

You are right, I want to promote a site AND help people. The latter is more important.

How do I request a review? Who would conduct it? Jarhead422 (talk) 16:01, 19 January 2011 (UTC)


 * This is an encyclopaedia. Those other external links you mentioned gave extra information of the type an encyclopaedia would give about multiplication. For instance the abacus one was not a free abacus and course in using the abacus so you could improve your technique. It described an abacus and how multiplication was accomplished on an abacus. External links/Noticeboard has editors who have checked a whole lot of links and argued the case one way or the other and I would be happy to defer to their opinion about the applicability of the policy. Even if the site was marginal and had some extra information about multiplication I think it would have grave problems because it asks people to download an executable program, see WP:ELNO #8. Dmcq (talk) 16:25, 19 January 2011 (UTC)

The executable program is only one of three options and it has been approved by Apple. The other two options are web based and there is nothing to download. Jarhead422 (talk) 16:28, 19 January 2011 (UTC)

Multiplication tricks
There are many tricks out there, we've all seen those mofo's who use their fingers to multiply things like primitive abaci 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 separate 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 mathematically repressed childhoods that we 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)

Basically, when you do multiplication, multiplication was originally used by repeatedly adding/counting objects in groups. It was a way of quickly counting single objects, a form of speed counting. This is why counting was done on abacuses and still is.

So if you want to quickly learn to multiply you can either understand the code clearly or simply repeatedly count and add objects in groups.

x means added not times and a multiplication problem such as 4x3=12 is correctly pronounced adding four three times equals six or four added three times equals six (or four counted three times equals six added=counted=totaled).

in a problem such as 2x3=6, 3 is the number of times you add 2, 2x3 is shorthand or initialing for 2+2+2=6, like writing your initials, the multiplication code was designed to reduce writing times and reduce counting operations to speed in counting by acting as a memory aid. thus you no longer have to do the math for 2x8=16, ie adding 2 eight times to equal sixteen, you simply memorize and use the initialing code. just like if benjamin franklin wrote his initials they would be b.f., a quick way of writing his name, the multiplication code is a quick way of writing longhand repeated addition, designed to reduce writing times and save space on the paper and to act as a memory aid to get rid of the times needed to do longhand repeated addition.

you can appreciate and see this directly when multiplying 2x10=20 vs doing old school repeated addition wich is quicker to write? 2x10=20 or 2+2+2+2+2+2+2+2+2+2=20 and you can appreciate the code is a time saver and space saver. writing letters and numbers i believe are forms of initialing for pictographs, designed to reduce writing times for pictographic drawing and are also a form of map making.

2x3=6 also means two objects in three rows equals six total using an abbacus. it also means what is the 3rd multiple of 2? 6 is listing multiples of 2 in order 2 4 6, 6 is the third.

I believe programmers have not correctly understood this and computers/calculators are actually doing the math operations which is why multiplication is done so poorly on modern computers and hangs/lags many computers/calculators, the code was designed to get rid of the adding operations to make multiplication counting faster.

in other words a modern calculator is not doing 2x100=200, its actually doing the math which takes time and lags out the machine.

multiplication is also a way of predicting and assessing sizes as x2=twice the size or amount, x3 means three times the size or amount and so on.

multiplication is not rocket science, mathematicians need to stop making the simple more and more complicated, multiplication was used as a quick, simple and easy way of counting. It's basic addition, not rocket science.

I believe I am noticing errors by learning math by rote and memorization rather than understanding and the removal of traditional counting out of western schools (removal of abacuses, etc). hope this helps and not to be redundant as I do not know if my other essay will be read by the author of this. —Preceding unsigned comment added by 75.128.44.182 (talk) 07:14, 3 August 2009 (UTC)


 * What you're suggesting is a good way to introduce someone to the concept of multiplication, but multiplication is about scaling a number, not just repeated addition. While multiplication may have originated as a method of "speed counting", as you say, that definition does not capture the full meaning of multiplication.


 * As a simple example, you may be able to represent 2x3 as 2+2+2, but how would you represent 2x3.5? 2+2+2+(2x0.5)? What does it mean to count something 0.5 times? Without the concept of scale behind multiplication, we would be unable to proceed.


 * Moreover, the convention of saying "four times three" to represent 4x3 reflects the concept of scale: in the same way that one might say "twice three" as verbal shorthand for "twice as much as three" to denote the scaling of 3 by a factor of 2, "four times three" or "four times as much as three" denotes the scaling of 3 by a factor of 4. So in fact, saying "four three times" has just the opposite meaning of scaling 4 by a factor of 3. A subtle, but important distinction.


 * Imagine a situation, where children have to memorize "times tables" for non-positive integers.   — Preceding unsigned comment added by Anuja.g (talk • contribs) 09:20, 5 April 2011 (UTC)


 * That said, I agree with you that the memorization of "times tables" and "multiplication tricks" is somewhat regressive. In fact, the only place they are found in common application, outside of education (unfortunately), is in computer science and in nonstandard algebraic systems. That's why they are not and should not be in this article. 74.178.45.239 (talk) 18:06, 3 August 2009 (UTC)

Removed Indus Valley section
I removed the Indus valley section with its bit about slate tablets and lattice multiplication because as User:Naliboki said there seems no evidence of that at all and it looked like fantasy. They do seem to have had some maths but I couldn't see anything like this. Dmcq (talk) 18:04, 12 April 2011 (UTC)


 * I agree. It seems to have been made up. There is no evidence of such tablets or lattice multiplications.--Indian Chronicles (talk) 11:33, 13 April 2011 (UTC)

Incredibly basic capital pi notation identities
Hi guys! I'm working with the capital pi notation in one of my classes, and it would have helped me if some stupidly obvious identities were listed here, so that I could confirm that I didn't just make them up. An example is:

$$\prod_{i=0}^{n} x^{y_i} = x^{\sum_{i=0}^{n} y_i}$$

Which is obviously just a generalization of $$x^m x^n = x^{m+n}$$ to any number of factors.

Or perhaps it should be put in Exponentiation, because that's where the simple identity is listed? I'm not sure.

Thanks!

129.10.244.112 (talk) 04:06, 13 April 2012 (UTC)
 * Did a book list it as special in some way? If not then we should not be listing it. Basically we'd need a citation. Dmcq (talk) 06:08, 13 April 2012 (UTC)
 * Which policy is it which dictates that? There are a grand total of three citations in this page, only one of which seems to have anything to do with actual mathematical equations and such. By your metric, pretty much this entire article should be scrapped. I'm pretty sure a citation is not needed for basic mathematical truths - as I said above, this a simple generalization of an already listed equality (although it's listed in Exponentiation, as I noted above). It is not some grand new achievement which would violate WP:OR. 129.10.244.84 (talk) 07:58, 17 April 2012 (UTC)


 * If you are disputing that the stuff in the article is WP:Verifiable then you're free to put cn against the bits you dispute. Please give a reason why other than that just that there isn't a citation. We shouldn't be sticking stuff we thought up ourselves over breakfast into articles, that's what WP:OR is about. Dmcq (talk) 10:20, 17 April 2012 (UTC)

Definition is not clear.
The opening sentence uses the word "scaling" without explaining or defining it - nor is it linked. If it is being used as a technical word in mathematics, it should be cleared up; if not, it should be replaced with either a clear technical word or a lay term that means something to a non-mathematician user. Certainly the introduction to an article on such a common and basic term as this should be understandable to non-mathematicians without their scratching their heads trying to figure out what "scaling" means [and probably coming up the answer that it is something like multiplication]. Kdammers (talk) 13:09, 29 April 2012 (UTC)
 * Can some-one respond?Kdammers (talk) 02:45, 9 January 2013 (UTC)

Error in definition.
Multiplication by zero, doesn´t give a property, it gives a one to many relationship. Within multiplication, neither multiplication by zero nor division by zero is allowable, which for what are the real numbers, implies that no multiplication by zero nor infinity is allowable, neither giving a one to one relationship. Both zero and infinity are delimitations, the maximum limits which you never reach (see limit: the limit towards).

Verbatim paraphrasing of what is in error, does not make it truth, that makes that a repetitive error.

For the real numbers between zero and infinity, the mirror point of inflection for the operator set op(+,-) is zero, the mirror point of inflection for the operator set op(*,/) is one, and not zero, which in what is reality (physics), bounds all real numbers to be between the numbers 0 and 1 (IE: R (0<x<1)). When you expand these numbers, holding zero fixed (which is another error of writ, the mean_average not being zero), you obtain R (0<x<∞). Notice that that is the < sign, and not the ≤ sign.

For those lovely prima donna´s with immediate wishes to place, Original, here and there, ´if you are not quite capable of thought, then you aren´t quite capable of defining an origin´. My response to that is, so sorry you are an orphan, the good news being, that that was a multiplication process, and that therefore you are NOT quite the zero that some would make you out to be, multiplication solely being definied for numbers greater than zero, with resultants greater than zero. — Preceding unsigned comment added by 186.94.187.76 (talk) 12:43, 26 February 2013 (UTC)

Nonsense. As usual. — Arthur Rubin (talk) 20:57, 26 February 2013 (UTC)

Moving graphic
Does the graphic "explaining" multiplication as scaling make any sense to anyone else? It's confusing to me. 211.225.33.104 (talk) 09:27, 9 July 2014 (UTC)

unexplained image
The rectangle next to the text on the Indians inventing modern multiplication is without explanation (other than something about 4and 5 being reversed). An encyclopedia should explain, not present a mysterious illustration with no explanation. Multiplying various digits, I couldn't figure out what was going on. A little help here, please! 211.225.33.104 (talk) 09:39, 9 July 2014 (UTC)

Advanced Capital Pi Notation
Would have hoped to see also some explanation on more advanced notations like in Sigma_notation and maybe even further. For example, separated using a comma next or under to the start value one can apparently add conditions that have to hold true as well. See for example the following.

On Lagrange_polynomial you find:

which is the same as the following (found on German WP):

or:

or:

or:

Correct? The last two forms are the least compact but in my opinion also most concise ones. When I first came across the second form, it took me a while to realize what was meant because the always effective condition is next to the starting index. That's why I hoped that this article here might shine a bit light on whether there are any plausible "rules" on that. --Unverbluemt (talk) 12:02, 28 August 2014 (UTC)
 * I have to admit I don't like #2 or #3 at all, and #4 looks ambiguous (especially if j can be 0.) But this seems more an MOS:MATH question than something which belongs in the article.  I see no reason why something like Summation should not be imported here.  — Arthur Rubin  (talk) 14:03, 28 August 2014 (UTC)
 * By the way, I believe there's a typo on 5. Shouldn't it be

\prod_{m \in \{e \mid 0\le e\le k,e\neq j\}} \frac{x-x_m}{x_j-x_m} $$
 * ? — Arthur Rubin  (talk) 14:06, 28 August 2014 (UTC)
 * Thanks, you're right, that was a typo. And I don't like #2 and I guess I going to take it to the German WP page, but since I've also seen #2 as well as #3 it the scripts of one of my classes I guess it seems to be a common notation form. I just wondered whether there exists any guideline, not WP-specific but in general. But I guess it doesn't. :) --Unverbluemt (talk) 18:06, 28 August 2014 (UTC)

"The sum can be rewritten using the distributive property as the product, and the whole-number product  can be written as " Then it goes on to give an example with a value   (that my copy function can't copy hither) that does not allow for distribution. Distributive law requires the distributed value to be in the same location in both sums, e.g. ab+ac. For ab+ca to be converted to a(b+c), the commutative law has to be first applied (if possible, e.g., with multiplication, but not with powers, i.e., ca=ac, but c^a rarely equals a^c).Kdammers (talk) 00:01, 6 March 2015 (UTC)

3 times 4
I've always thought 3 times 4 meant 4+4+4 as in "three times a lady". For the other meaning as in 3+3+3+3 I'd have to put in a pause after the 3 as in 3, times 4. This is in line with "Think of a number, multiply it by 4". Is there anything on the relative frequency of the two?, perhaps some elementary teaching gives a guide somewhere. Dmcq (talk) 09:06, 23 October 2009 (UTC)

I just did a quickl survey looking at the first few entries that came up with a google of "multiplication by repeated addition" and I got:

3*4 = 4+4+4 (Wikipedia) aaamath hartfordprimaryschool youtube

3*4 = 3+3+3+3 multiplication homeschoolmath grahamwroe

both teachingideas lessonplanspage cumbriagridforlearning

Neither: It Ain't No Repeated Addition maa devlin numberwarrior

Both or neither globaledresources

So there ain't no consensus that I can see. I'm rather surprised there's such an even match of divergence of opinion. Perhaps even there's some room in the article about it isn't repeated addition. Dmcq (talk) 17:33, 23 October 2009 (UTC)
 * There was a big spiel here by Nightgamer360 saying 3×4 IS REPEATED ADDITION. Nightgamer360 removed it and put in another one below. Dmcq (talk) 23:44, 12 November 2009 (UTC)


 * Interesting. Wrong, but interesting.  Your "reading" of multiplication as a "shorthand for repeated addition" is just wrong, although it might be an alternate definition in some contexts.  Perhaps if you could supply a source that 3 &times; 4 is 3 + 3 + 3 + 3, rather than 4 + 4 + 4, it might justify some of your comments.  As it stands, it shouldn't be in the article.  — Arthur Rubin  (talk) 05:57, 12 November 2009 (UTC)


 * Well certainly even saying that 3 miles × 4 miles is repeated addition adding up to 12 square miles is rather a long stretch. I'll have a look at The development of the concept of multiplication as it's something that can be cited. Dmcq (talk) 12:03, 12 November 2009 (UTC)

It is not wrong and real mathematicians do math, they dont argue with people for the sake of arguing.

—Preceding unsigned comment added by 68.190.230.129 (talk) 22:55, 12 November 2009 (UTC)

—Preceding unsigned comment added by Nightgamer360 (talk • contribs) 19:58, 12 November 2009 (UTC)


 * I sympathize with your desire for truth but wikipedia cannot accept contributions based on truth, only on verifiability. That means you need to show a book or reputable journal where this is all explained. Your own explanation counts as original research which is also not allowed. That;s why I was looking at a citable publication I could read about this in. I can't put in anything like this otherwise, it would just get reverted and quite rightly so. Dmcq (talk) 23:44, 12 November 2009 (UTC)

youve confused popularism with education then and are supporting the masses. verifiable proof at math.com http://math.about.com/library/blm.htm

Multiplication - Often referred to as 'fast adding'. Multiplication is the repeated addition of the same number 4x3 is the same as saying 3+3+3+3. —Preceding unsigned comment added by Nightgamer360 (talk • contribs) 00:26, 13 November 2009 (UTC)

Its also of note wich is why we have become so mindless as a society and perhaps the internet will prove the undoing of the educational system simply because where a few had a voice in the beggining, individuals are drowned out by the millions of masses and by a few elitists attempting vainly to control a system that is uncontrollable, much like mother nature. Its sad and humorous that you have not noticed that that is a direct quote from math.com and its actualy the statement of the reverse of 3x4=12 is the same as saying 3+3+3+3 not 4x3 wich is 4+4+4=12. You miss entirely what multiplication is used for assesing size precisely whats with the 13x14 miles nonsense? how about asking/calculating how many in a like how many days in a year?

365 x number of years you wish to find number of days in = number of days in a given number of years. example 365x1=365 days in one year because one year is 365 days added one time how many days in 2 years? 365x2=730

How many inches in a given number of feet? 12 x number of feet = number of inches in any given number of feet 12x1=12 inches in one foot because one foot is 12 inches added one time

how many inches in 2 feet? 12x2=24 inches in 2 feet

how many in 3 feet? 12x3=36

how many seconds are thier in any given number of minutes?

60 x minutes= number of seconds in any given number of minutes

60x1=60 seconds in 1 minute 60x2=120 seconds in 2 minutes 60x3=180 seconds in 3 minutes.and so on

same calculation used for finding number of minutes in an hour

60 x h = number of minutes in any given number of hours 60x1=60 minutes in 1 hour 60x2=120 minutes in 2 hours 60x3=180 minutes in 3 hours 60x4=240 minutes in 4 hours

5x1=5 correctly pronounced adding five one time equals five short for 5=5 5x2=10 correctly pronounced adding five two times equals ten shorthand initialing for 5+5=10 5x3=15 correctly pronounced adding five three times equals fifteen 5x4=20 correctly pronounced adding five four times equals twenty short for 5+5+5+5=20 7x3=21 correctly pronounced adding seven three times equals twenty one

the proof is in the math, simply add the first amount the number of times of the second (multiplier) will produce the result. (says so at math.com guys). This is not inccorect as its stated at math.com of all places multiplication is used as repeated addition and speed addition, its a form of speed counting. Unbeleivable.

the multiplication code is designed as a means of initialing.writing old school longhand repeated addition, the code is quicker to write as numbers get bigger, saving time writing and space on paper as well as saving time calculating by acting as a memory aid. the multiplication technique is designed to count beads on an abbacus quicker as its quicker to count in groups than to count one at a time.repeated amounts are also used because the same reason einstien wore the same cloths all the time, its quick and easy and you dont have put much thought in it, you can develope a standard for doing something an you can predict the results. you can use multiplication to scale numbers but thats if your simply into doing pure mathematics instead of applied mathematics wich in my opinion is what math and science need to be based on. To my heckler on here if you want to argue for the sake of argueing your going to accomplish nothing but arguing. This isnt some Spiel, its how classic mathematics was taught and why it was used for doing inventory for farming and monetary systems and actual counting, not just scaling numbers.

Im hoping this at least makes it clearer for students, and teachers will stop saying times in the middle as 7x3 seven times three is improperly said as it confuses students for those that read this discussion. its correctly said seven added three times, your students arent stupid, they dont recognize its basic addition, much like the gents reading this who are established educators, math.com doesnt lie fellas.

thiers an old saying dont through your pearls of wisdom at pigs, they will only trample on them and tear you to pieces. more proof that institutions are truly for the institutionalized.

multiplication was used for several purposes, the code itself was for initialing and shortening traditional longhand repeated addition. —Preceding unsigned comment added by Nightgame (talk • contribs) 07:35, 17 November 2009 (UTC)


 * I haven't 'confused popularism' etc or anything like that. I have simply quoted the principles by which this encyclopaedia is being built. Contributions that don't follow those policies get deleted. If you write a book or an article for a mathematics magazine saying what you've said here then perhaps it can be cited. Otherwise not. Dmcq (talk) 01:08, 13 November 2009 (UTC)

aparantly you have and have an ego complex to boot. i believe thats a misdiognosis and you arent going to be doing anything valid for the scientific or educational commuinity by supporting wiki's agenda. Its not supporting masses, its supporting the ideal of no thinking allowed by individuals and supporting elitism. you can only quote written and verified material by other people wich means your supporting a status quo and a social system that supports elitism as those people your quoting from will have to be already established. True education is done by individuals going against the grain and being original not by being mindless puppets of society. I think any real individual who wants to contribute to wiki needs to think again as unless you can "quote other reasources" your not going to wich means if your not established. thats the educational system at its worst and why wiki wont contribute anything relevant other than "parrot education". wiki doesnt like organizations like scientology comming in and re-chaing things based on thier elitism and tactics, its sad and hypocritical that wiki supports "mindless education". I think I and other mathematicians obviously need to publish elswhere. Real mathematicians,scientists and educationalists think for themselves, they dont quote from books. Typical mindless american educationalism and why we barely go anywhere. Newton wouldnt have gotten anywhere if he would have simply held hands with the educational system and quoted aristotle as countless others. Aparently you need to read more about educational history and less time in dictionaries.


 * I think you're getting the point. If Wikipedia had been around at the time of Newton then it would not have been able to say anything about his theories until after he published his Principia and other people said something about it. Wikipedia does not involve itself in original research, it is only after itr is published and other people comment on it that it becomes suitable for inclusion. Wikipedia is the wrong place if you have ideas of your own rather than just reporting what other people have written about. Dmcq (talk) 10:08, 17 November 2009 (UTC)

I found the wikipedia in different languages give different definitions, it may just depends on the multiplication habit but not the harsh fixed definition. Mansasakura (talk) 06:12, 4 November 2015 (UTC)

Multiplicand, multiplier
In a product the first factor is usually called multiplicand and the second called multiplier. The article calls both multiplicand. --Xypron (talk) 13:14, 13 November 2010 (UTC)


 * They are both mentioned in the Notation and terminology section though I'm not sure what saying there is correct. I don't believe there is any general rule about what order they occur in, which is related to the bit in the leader saying 'There are differences amongst educationalists which number should normally be considered as the number of copies or whether multiplication should even be introduced as repeated addition'. If you could find something that discusses this or defines the terms well that would be good. Dmcq (talk) 20:46, 13 November 2010 (UTC)


 * Agree; there does not appear to be any consensus. I find that a great many people are not even aware that there are other people who don't conceptualize it the way they do. Un sch  ool  07:24, 10 November 2015 (UTC)

3 multiplied by 4
Then text was changed to say that 3 multiplied by 4 was 4 times 3 and I changed it back. I understand the reasoning but I think we really need a citation showing people do this reversal as it isn't altogether obvious and could be confusing. Dmcq (talk) 11:59, 30 October 2015 (UTC)
 * Thank you for your concerns. I added a reference: "With multiplication you have a multiplicand (written second) multiplied by a multiplier (written first)." (by Keith Devlin) --Neo-Jay (talk) 15:24, 30 October 2015 (UTC)
 * Thanks. Dmcq (talk) 20:23, 30 October 2015 (UTC)
 * There are many other references. See definitions of "multiplicand". E.g., http://mathworld.wolfram.com/Multiplicand.html, https://www.mathsisfun.com/definitions/multiplicand.html, https://en.wiktionary.org/wiki/multiplicand, http://www.britannica.com/topic/multiplicand 209.20.29.248 (talk) 07:39, 1 November 2015 (UTC)
 * I was querying the swapping around when the words 'multiplied by' are used. We needed something that was pretty obvious rather than required any reasoning. We weren't discussing the word 'multiplicand'. Was there something you wanted to say about multiplicand?. Dmcq (talk) 13:30, 1 November 2015 (UTC)

I believe that older arithmetic texts used the order: multiplicand x multiplier. Here's a reference: http://www.crewtonramoneshouseofmath.com/multiplicand-and-multiplier.html TomMathCA (talk) 22:53, 1 November 2015 (UTC)
 * I think the Devlin link and the link above to, http://mathworld.wolfram.com/Multiplicand.html are much more reliable. Plus that one seems to me to be trying to argue against something so it implies they thought they were in a minority. We'd need something of the same order to say otherwise. Dmcq (talk) 12:38, 3 November 2015 (UTC)
 * Both of these contradictory references are blogs and as such we should toss them both and find some reliable secondary sources. Having said that, I do agree with 's assessment of these sources. Devlin is a highly respected mathematician and popularizer of mathematics and his blog is a monthly column carried by the Mathematical Association of America and thus is implicitly sanctioned by that organization (btw, that doesn't mean that they bless everything he writes, just that he is held in high regard for his opinions). Crewton Ramone is who exactly? There is a large credibility gap here and unfortunately has just promoted them to equal footing. We should not be taking sides, if in fact there are sides here, and should await the results of a more serious search for sources. Mention has been made that calculator manufacturers are also divided on this terminology. I would like to see some references for that statement. Bill Cherowitzo  (talk) 19:08, 9 November 2015 (UTC)
 * Bill, your point about Devlin's value as a source, versus dome bloggers, is obviously valid. But if you read the entirety of Devlin's article, you will note that even he acknowledges that this is his personal impression about multiplication. He is not claiming to give an authoritative opinion.
 * As to the calculators, I do not have any online sources at the moment, but you can see for yourself by comparing a TI model to some others. Last time I checked, the TI models differed even from the calculator that comes with Windows. Un  sch  ool  07:13, 10 November 2015 (UTC)
 * Devlin's personal impression refers to his view of multiplication as scaling instead of repeated addition (the thrust of this article) and I would say that this has nothing to do with the terminology issue. As a mathematician and respected writer of mathematics, he would use the appropriate terms and would signal his readers if he was using terms in a non-standard way. It is also important for his argument that he get the terminology right since in his view the multiplier is to be seen as a scaling factor. It is this interpretation, not the terminology, which is his personal view. Bill Cherowitzo (talk) 17:46, 10 November 2015 (UTC)


 * If we had one reasonable reference for the opposite point of view we could say that some consider it to be the other way round and have what is there at the moment as the mainstream. I just removed an edit where someone was complaining about children being marked down in school for getting this order wrong! Jeez talk about wanting to hammer pegs into holes. Dmcq (talk) 12:43, 13 January 2016 (UTC)


 * It looks like removed the Devlin cite in the lead and put it as secondary to one saying the other way round in the notation section. The lead should summarize the text. I'll put the Devlin cite back in the lead and put in a statement citing the other for that others sometimes consider them to be other way round to the other cite, though I'd prefer a much better citation for it. Dmcq (talk) 14:02, 13 January 2016 (UTC)

This needs to be reconciled with the hyperoperation 'multiplication' in the Hyperoperation page https://en.wikipedia.org/wiki/Hyperoperation#Examples, where a x b is a repetitively added b times. 12:14, 16 September 2017 (UTC+10:00) — Preceding unsigned comment added by 124.190.82.101 (talk)

when thinking of multiplication as repeated addition
The part "when thinking of multiplication as repeated addition" in the intro should be deleted, or moved to a less prominent place, because it refers to a non-standard point of view. Wikipedia is not an appropriate place to promote our own ideas, we should simply write what is standard material. MvH (talk) 04:02, 22 January 2016 (UTC)MvH

So basically, that sentence should read like this: "Multiplying two whole numbers is equivalent to adding as many copies of one of them (multiplicand) as the value of the other one (multiplier)." MvH (talk) 04:05, 22 January 2016 (UTC)MvH


 * I'm not keen. It cuts out scaling and array view and the dimenensional analysis view and the citation directly says they don't like it being thought of that way. At an elementary level the dimensional analysis view of [4 bags] by [3 apples per bag] is more common I'd have thought. Dmcq (talk) 11:41, 22 January 2016 (UTC)


 * I'm not proposing to cut out scaling all together. What I object to is having a prominent link to this page Multiplication_and_repeated_addition which promotes a non-standard point of view.  The header gives an example where 3 times 4 is explained as repeated addition. That's good.  It also gives an example where 3 times 2 is explained as scaling. That's also good. Both examples should be kept.  The header contains a prominent link to a page about math education with a non-standard (advocated by 1 person) point of view.  That view should not be advocated in the header of this page, we should remove that link.  MvH (talk) 16:16, 22 January 2016 (UTC)MvH


 * What evidence have you that multiplication considered as repeated addition is the most common idea? In set theory it would be the cardinality of the cartesian product of two sets and that I would think is the closest to the general idea of what the multiplication of whole numbers is. If you look at Product (mathematics) you'll see the array or Cartesian product view as the example for whole numbers. In fact why have we the separate articles, they seem to overlap in a rather uneven manner. Dmcq (talk) 18:48, 22 January 2016 (UTC)


 * Repeated addition is how children are taught multiplication in elementary schools. If you need proof, I can show you their homework and their textbooks. It is true that multiplication and exponentiation for non-negative integers can be defined directly in terms of set theory (you can define exponentiation without using multiplication or addition).  However, that is not the first method that kids are taught.    MvH (talk) 20:25, 22 January 2016 (UTC)MvH


 * Hopefully children are taught a meaning for multiplication first rather than a method to get the result. And the usual sort of thing they are taught is three bags of four apples. The repeated addition is a way of calculating the result quicker than counting up all the apples. The use in calculating how much to pay overall for five apples if each costs a certain amount usually comes later. Saying it is repeated addition without saying why anyone would do that is pretty bad and close to just telling them to memorize meaningless tables. Try doing a Google of what you are thinking if and you'll see the debate about defining multiplication is very real and it is not settled. We should keep it as the first definition because that's what dictionaries say, but we are supposed to reflect the major views, and it is by no means anywhere near being an unquestioned view. Dmcq (talk) 22:42, 22 January 2016 (UTC)


 * Here's a good example of a child not being taught any sort of idea of what multiplication is  Dmcq (talk) 15:55, 26 January 2016 (UTC)

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)


 * We say "add m to itself n times" because there are n summands, not n pairwise additions. "Add up n instances of m" might be better (as defining an algorithm) but is less natural language. Pete St.John (talk) 17:22, 29 November 2007 (UTC)


 * I completely disagree with the bizarre notion that multiplication is repeated addition - it is simply a different operation in its own right. What is true is that repeated addition is a special case of multiplication. And I think that the best way to show multiplication is not by showing groups of objects, but as a rectangle. It does not matter which number is the "number of copies" - the role of both numbers is identical - both numbers are like sides of the rectangle. Also, to get the area of a rectangle in cm2, you multiply one side by another, in cm. No matter how many times you add cm together, you will never get cm2! Showing multiplication that way makes sense in general. For example, if you multiply one line segment by another, and you do it by taking a certain number of line segments of one length, then it won't work, because you don't know the length of each segment in units; and the number of units depends on the units themselves too. But if you make a rectangle with sides the same length as those line segments, then it will be their product. Majopius (talk) 18:25, 7 February 2010 (UTC)


 * Agree with Majopius, multiplication is not repeated addition. Where is the addition sign in 1 x 1 = 1? There isn't one. And what about (-pi) x (-e)? Read Keith Devlin on this. Jzimba (talk) 20:53, 6 January 2015 (UTC)


 * Per the above, I rewrote the introduction to avoid characterizing multiplication as repeated addition.Jzimba (talk) 13:18, 19 February 2015 (UTC)


 * Strongly disagree. You simply deleted the explanation (what then is the purpose of the article?), then inserted some properties that were already given elsewhere in the article. MvH (talk) 02:15, 12 April 2015 (UTC)MvH


 * What I deleted was not an explanation, because it was incorrect. It was not even correct for whole numbers. It is not correct to say that "The multiplication of two whole numbers is equivalent to adding as many copies of one of them, as the value of the other one." To see this, let's try it for 1 x 1 and see what happens. "The multiplication of 1 and 1 is equivalent to adding as many copies of 1, as the value of 1." Addition is a binary operation; you can't "add one 1." Also, even if we somehow gloss over this point, it is a strong editorial choice to introduce multiplication using only concepts that apply to (not quite all) whole-number cases, without even mentioning that these concepts are not applicable for a product such as Pi * Sqrt[2]. The relationship between multiplication and addition is *fully captured* by the distributive property a(b+c) = ab + ac; to say more than this about the relationship is to court error. But I desist; have it your way! Jzimba (talk) 15:53, 21 April 2015 (UTC)


 * Thanks people, for diminishing the value of another Wikipedia article.
 * Pretty much every thorough introduction to algebra will show you how the formal definition of multiplication (of natural, whole, rational, real, complex numbers) starts by first defining it as repeated addition for natural numbers (in spite of the various linguistic misdirections above: I suggest that you read the actual definitions instead of criticising your own misinterpretations of them), after which the definition is extended to more general cases. Rather than pointing out this valuable insight (and perhaps improving whatever the original text was), the article now treats multiplication as something that has to be treated as a god-given miracle, or whatever you want to call it.
 * Extra points for justifying this by going back in time at least 400 years to suggest that numbers must have a geometric interpretation and cannot be dimenionless. 49.197.99.186 (talk) 01:57, 16 August 2016 (UTC)
 * After that rhetoric could you come down and look at the article and explain what you think wrong? Preferably by pointing to specific parts and suggesting changes? Thanks.Dmcq (talk) 10:13, 16 August 2016 (UTC)


 * Coming down and apologising: I focussed too much on the critisisms above to see that the article hadn't been compromised by them. 100% my mistake. 49.197.99.186 (talk) 11:59, 16 August 2016 (UTC)

Animation
Does the animation of 2X3 make sense to anyone? why does the two drop down? How does the value six get justified or explained? I have no idea what this animation is trying to convey (other than, obviously, THAT 2X3=6, but not HOW or WHY). Kdammers (talk) 15:27, 8 November 2017 (UTC)
 * It seems to be very similar to the preceding image, with 2 and 3 commuted, and the blue line from 2 to 4 lacking). For these reasons, I would agree if you remove it. D.Lazard (talk) 15:47, 8 November 2017 (UTC)

Unhelpful graphics
Boy, am I glad I learned multiplication in grammar school instead of trying to learn it from the graphics in this article. Except for the bags of balls, all they do is confuse. If I didn't already know I know how to do multiplication, those other graphic would make me feel that I had no idea how to do it. Putting in graphics without explanations is less than helpful. (P.S.: Don't tell me to correct them if I don't like them -- I don't understand them, so I can't very well correct them.) 67.209.131.253 (talk) 02:04, 27 July 2020 (UTC)