Multiplication is defined as repeated addition. 3x5 = 5+5+5 How do we define 10/2? Do we start out with 10 and subtract 2 X number of times?
Division is defined as the inverse operation of multiplication (in the same way as subtraction is defined as the inverse of addition). "the result of the division of p by q is n" p / q = n where n is such that q * n = p For subtraction: a - b = c where c is such that a = b + c Bye! Crisp
You have posted this on a number of different forums and every response tells you "No, your first assertion is wrong. Multiplication is NOT defined as 'repeated addition'." What does that tell you?
At first glance, I don't see a problem with defining multiplication as multiple additions. Why is this 'wrong'? I was first taught multiplication as being the way to figure out how many cells on a checkboard.... but this seems to be equivalent to adding up the number of cells on each row a certain number of times.
Multiplication comes from tables. It is however repeated addition. A = B + B ; is 2*B P = A + B ; is 3*B To divide (an integer) subtract until you reach 0 while counting the subtractions. The count is the division result. D = B/A Count while Try = B - A until Try = 0 Let B = 4 and A = 2 Count 1 4-2 = 2 Count 2 2-2 = 0 These are very simple computer programs.
Because that particular definition falls apart for negative numbers and irrational numbers. For example, there is no simple way to compute √2 · √3 or -4 · -3 using the multiple addition methodology. The reason that multiplication is initially taught as repeated addition is to connect it with a concept children already are versed in, namely addition. However, when more Mathematical maturity is gained, one learns the definition of multiplication that is not constrained to rational numbers, and realizes that the old "definition" was simply a special case of the general definition.
Multiplication is most certainly NOT repeated addition: 2m x 6m = 12 m^2 =/= 6m + 6m The dimension changes with multiplication but not with addition.
Good points. I didn't think about the units part. This is something we I never really thought ofPlease Register or Log in to view the hidden image! How would you actually define multiplication.
If multiplication is not repteated addition there must be some other way to prove that p * q = n. I guess it's defined geometrically with a rectangle with length p and width q?
p*q = n is such because we define it as such. We could just as easily say that p*q = 0 for all p =/= q and p*p = 1, and that would satisfy all the definitions for a field in the algebraic sense. As far as I know there isn't really a "proof" that p*q=n, but (i) I don't know a lot about number theory and (ii) if it didn't we'd have problems. Also keep in mind that this only works for a certain class of mathematical objects. Matrices aren't fields, so obviously p*q is different from q*p, if it's defined at all, etc.
Yeah, but matrixes have strict definitions for multiplication. They are based on a process. Multiplacation has to have some specific definition as to how we get a result.
This is true, but that algorithm is based upon the multiplication of scalars, which, if they obey different multiplication rules, are going to behave differently. The definition of matrix multiplication is such more for what it does to basis vectors under linear transformation than for anything else.
Does anybody know how computers perform multiplication? Do they have some kind of table or do they repeat addition or what is it? The product can't simply pop up from p * n = q magically.
I don't know much about computers, but from what I could make of an article I just read, it goes through the standard multiplication procedure you are taught from the beginning, only in binary (which is a much smaller order in terms of possible results of multiplication). So, to multiply the numbers 101 and 110 the computer takes 0 * 101 + 1 * 101 [adding a zero so this is 1010] + 1 * 101 [adding yet another zero so this is 10100] and it comes out with 11110, and, sure enough, 5*6 = 30
Yea, that is how I figured out binary math like 110101 * 01110 or 1111011 + 1010001... it is the same thing as in decimal. You can even prove it.. just convert the binary to decimal, add. Convert this back to binary. See if this result agrees with the binary addition. I cannot seem to get other people to see this though. It is hard for me to teach it. Well, it is hard for me to explain anything. Mainly because I see things in pictures and I cannot think verbally (not self-talk). But, this is the wrong thread and fourm to be dicussing this.
It's probably not trial and error cause the computer doesn't know what answer is right to begin with. (Well actually it must one way or another, just maybe not in that strict sense.) It would be interresting to know, if nobody answer here I guess I'll have togo googling. My guess is that there is somekind of "computerized" abacus pre-programmed.
