Defining MultiplicationDate: 08/16/2002 at 19:22:52 From: Cindy Miller Subject: Defining multiplication I just read an article in Phi Delta Kappan (Feb. 2001) by Deirdre Dempsey and John Marshall titled "Dear Verity: Why Are All the Dictionaries Wrong?" They state that Euclid and some dictionaries define multiplication equations like 3x4 to mean "the number 3 four times." I was taught, I teach, and math textbooks say that 3x4 means 3 lots/groups of 4. An array would look like **** **** **** Because multiplication is commutative, it may not matter symbolically; however, as the authors mentioned in the article, taking 4 pills a day for 21 days is a lot different from taking 21 pills a day for 4 days. If I want my students to know what multiplication "is," what is the correct explanation? Date: 08/16/2002 at 23:41:38 From: Doctor Peterson Subject: Re: Defining multiplication Hi, Cindy. Here is an answer I gave to a related question: Multiplicand, Multiplier http://mathforum.org/library/drmath/view/58567.html Your question is really not a math question, but a linguistic question, even though it involves symbols. Since, as you recognize, 3*4 and 4*3 have the same value, there is no mathematical value in the distinction. You are only asking about the underlying image when we say "three times four" in English. (It may have been different in Euclid's Greek.) And I submit that it can be reasonably interpreted either way. My first thought when analyzing the phrase is that originally '3 times' meant to repeat what follows three times; that is, it would mean the same as '4, taken 3 times'. So 3 times 4 means 4+4+4. But that seems awkward, considering the way we tend to say the phrase now. We might read through a calculation, saying "3, times 4 is 12, plus 2 is 14, ..." where each operation acts on the first number. we're really saying "3 multiplied by 4." Taken that way, we start with 3 and multiply it by 4, meaning that we repeat it 4 times. So 3*4 = 3+3+3+3. The difference here is entirely in the grammar: is '3 times' an adjective phrase modifying '4', or is 'times' a preposition equivalent to 'multiplied by', and 'times 4' a modifier of '3'? When we introduce children to multiplication, it's reasonable to start with one specific meaning, just so they have a concrete image to start with. But we can almost immediately point out that **** **** **** can be seen equally well as 3 rows of 4, or as 4 columns of 3. Once you've drawn the figure, or once you've written 3*4, no one can really tell which interpretation you meant. The distinction has been abstracted out of the problem. And that is a good thing, not bad: in mathematics the ability to work abstractly and forget what the concrete problem was is a major advantage, because we can ignore details that don't affect the result, and rearrange the work to make it easier. If I took 4 pills a day for 21 days, and now I want ONLY to know how many pills I took, I can call it 4*21 (I just put the numbers in the order I saw them, without thinking about which is the multiplicand) and re-model the problem as 4 sets of 21, allowing me to add 21+21+21+21, which is easier than adding 21 4's. I can switch freely among different models, because I know that the outcome is equivalent. So I never bother to define whether 4*21 means 4 groups of 21 or 21 groups of 4, because that distinction would only be a hindrance. And if I were doing a calculation for which it made a difference whether I took 4 or 21 pills a day, such as finding the concentration of medication in my blood, I wouldn't be multiplying, because that would be the wrong abstraction! So "what is multiplication?" It is a commutative operation that can be modeled in two symmetrical ways as repeated addition (when applied to whole numbers). Does that make sense? - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 08/16/2002 at 23:14:46 From: Doctor Tom Subject: Re: Defining multiplication My understanding is that multiplication of natural numbers is defined as follows (and some stuff has to be proven as we go along - I'll just state the results). 0 = {} (zero is the empty set) 1 = {0} (the set containing zero) 2 = {0, 1} 3 = {0, 1, 2} 4 = {0, 1, 2, 3} et cetera. This set is defined by the "axiom of infinity" in the Zarmelo-Frankel set theory. Then you need to show that if m and n are two of the natural numbers above and there is a 1-1 mapping between them, they are identical. Then define m x n (multiplication) as follows: The ordered pair (a, b) is {{a}, {a, b}} If S and T are two sets, then: S X T = {(s, t) : s is in S and t is in T} (I used upper-case X for the cross product above.) To calculate m x n, construct the set m X n (this is just a set), and then find the unique natural number that can be mapped 1-1 onto this set. Thus, if m = 2 and n = 3 m = {0, 1}, n = {0, 1, 2} m X n = {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2)} Define f mapping m X n to 6 = {0, 1, 2, 3, 4, 5} as follows: f: (0, 0) -> 0 (0, 1) -> 1 (0, 2) -> 2 (1, 0) -> 3 (1, 1) -> 4 (1, 2) -> 5 Show f is 1-1 and onto and you've proved that 2x3 = 6. - Doctor Tom, The Math Forum http://mathforum.org/dr.math/ |
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