I know how to do synthetic division, but I do not understand why it works.
122334
Synthethic division is commonly taught, but I have never actually had a proof/explanation shown to me. Why does it work?  Work So Far  I related the "$x$" to powers to 10, and then proceeded to relate synthetic division to non-polynomial division, but couldn't seem to find the correlation.  Research So Far  My teacher doesn't seem to have a valid explanation for why it works. A   doesn't provide any good results either. All I seem to get is a Yahoo answers link with a badly formatted proof that makes it hard to understand and a physics forum link that links synthetic division to "normal division" by relating the "x" to 10, a conclusion I have already arrived at.
108892
I am having trouble understanding why the Euclidean algorithm for finding the GCD of two numbers always works?   I found some resources here (), and here().  But I am a little confused about how they approach it here. I understand that if we have two numbers, a and b, then the greatest common divisor of a and b has to be less than a, and if a divides b, then a will have to be the GCD.   But I am confused about what happens when:  b=a*q+r So now, we are saying that we take a/r, correct? Why should we do this at all?
11832
I suspect this has something to do with thermodynamics and the  and I'm interested in a mathematical derivation of this result.
75776
This is a very silly question since nobody will actually do this because it makes very little sense in the real world but I just want to know how would it actually work if possible.  For example let us take an amount of 2 dollars and 2 cents and multiply that buy 2 dollars and 2 cents. Would the result be 4 dollars and 4 cents? The only way to make sense out of this for me is to treat them as vectors and multiply components.
52719
Possible Duplicate:          My calculator (as well as WolframAlpha) gives me the approximation:  $$0.2078795763507619085469...$$  But I don't understand how exponentiating two purely imaginary constructs yields a real (albeit irrational) number. When I do $i^{i+1}$ it gives me an imaginary number as well as $(i+1)^i$. So why does $i^i$ fall into that precise point where it is real and no longer imaginary? What is happening? I understand that exponentiation is not repeated multiplication, and it wouldn't make sense to multiply $i$ by itself $i$ times (because it would only yield $i$, $-i$, $1$, or $-1$). So what are we doing behind the scenes to get such a number?
98367
We all know that multiplication is the inverse of division, and therefore $x\div{x}\cdot{x}=x$ But what if $x=0$? $0\div0$ is undefined so $0\div0\cdot0$ should be too, but whatever happens when we divide that first $0$ by $0$ should be reversed when we multiply it by $0$ again, so what is the right answer? 0, undefined, or something else entirely?
42801
I noticed that , given an operation like $\frac{n}{m},\;n,m \in N$ that result in a periodic decimal number, computes really fast the length of the period.  E.g. $\frac{3923}{6173}$ has a period of 3086: .  I was wondering how this computation is done: is there some method to do this (except the trivial one of executing the division and looking for a sequence repetition) ?
73205
Why are the only (associative) division algebras over the real numbers the real numbers, the complex numbers, and the quaternions?  Here a division algebra is an associative algebra where every nonzero number is invertible (like a field, but without assuming commutativity of multiplication).  This is an old result proved by Frobenius, but I can't remember how the argument goes.  Anyone have a quick proof?
29000
What's the correct way to do division using a backslash? If I write $a/bc + d$, will that be equal to $(a/(bc))+d$?   Basically, if I place a slash, will I divide by what's directly behind the slash $(b)$, the term that's behind the slash $(bc)$, or everything after the slash $(bc+d)$?
68946
Unlike with addition, subtraction and multiplication where the order of operations will lead to an unambiguous result when the rules of BIDMAS (or PEMDAS) are applied, this does not hold true for division.  For example:  5 + 3 x 5 x 7 - 2 = 5 x 7 x 3 + 5 - 2 = -2 + 5 + 7 x 5 x 3  Of course, BIDMAS states we must group our multiplications together but within that grouping the order of multiplications makes no difference to the result.  Furthermore, following the rules of BIDMAS and performing the multiplications first, the subsequent order of the arithmetic also makes no difference.  Right?    In the above example, for instance, 3 x 5 x 7 = 105.  So: 5 + 105 - 2 = -2 + 105 + 5, etc...  However, operation order DOES affect the result when dividing!  For example:  (1/4)/27 does NOT equal 1/(4/27).    We know this, right!  I always took it for granted that the order of operations when presented with multiple divisions must be performed in the order it is presented; ie from left to right.  That is what we were taught at kindergarten, right?  So:  a/b/c/d = ((a/b)/c)/d  Right?    Well that's what I believed all my life and took this for granted.  But now my sanity is being challenged by Casio's most recent scientific calculators which do not obey this rule!!!!  If you were to type in a/b/c/d into a modern Casio scientific calculator using its fraction key for '/' rather than the division button, the calculator yields a/(b/(c/d))!!!!!  Using the division button instead of the fraction key gives a conflicting (but what I believe to be the correct) answer; that is:  a divided by b divided by c divided by d = ((a divided by b) divided by c) divided by d  I feel sure that the older Casio scientifics (pre 1991) gave the correct result following the rules as I have explained above.  But I cannot get hold of one to check my theory.  So my question is: What is the correct order of operation with multiple consecutive divisions?  That is, what is the order of division for:  a/b/c/d   as written without any parenthesis?  Are the new Casio calculators giving the wrong answer or have I been living a lie?  PS:  I am aware that the calculators give what I believe is the correct answer if  the right cursor key is pressed after the keying in of each denominator.  Bu that is a fudge.  By using the right cursor, you are applying implicit parenthesis.  As inputted, either the calculators are giving the wrong answers or I have been living a lie.  Can someone please confirm the order of division should indeed be from left to right!  PS:  I have searched everywhere I could on the internet and in many maths text books but have not been able to find anything that addresses the ambiguity in the order of multiple consecutive division operations.
121569
From this paper:     Shouldn't $du$ be $dt$? And also how do you get from that line to the final result if $du$ is not $dt$?
33895
A method to finding the LCM of two numbers is to use prime factoring. I know how to do this, but logically why does this work in finding the lcm
89905
For example, when I divided $1$ by $98$, I got an amazing result of $0.0102040816326530612244897...$ where so many numbers get multiplied by two every time in the right pattern with some carrying.  Also, when I divided $1$ by $998$, I got an amazing result of $0.0010020040080160320641282...$.  I have a longer result from my cool memory, which is $0.001002004008016032064128256513026052104208517034068136272545090180360721442885...$What explains why this doubling thing is happening?  I love it when I see it!  Also I don't know if it repeats (I'm talking about 1/998).  I think it does because I looked it up and it said 498 numbers repeat every time, so it's rational.  I mean seriously, how does this doubling happen?  I hope I can receive good information from you!
35643
I have read the explanation for this in several textbooks, but I am struggling to understand it via Archimedes' principle. If someone can clarify with a diagram or something so I can understand or a clear equation explanation that would be great.
48304
I've tried both calculations on Wolfram Alpha and it returns different results, but I can't get a grasp of why it is like that. From my point of view, both calculations should be the same, as $2.5=25/10,$ and $(-2)^{2.5}$ is equal to $(-2)^{25/10},$ relying on a general rule $(a^m)^n=a^{mn}$.  Links to sources:
54549
Hello,  in the picture there is an equation I can not understand. What I do not understand is how to get to the first step. I have been thinking and thinking but can not find an explanation. Could somebody please explain? Thanks.
29219
Suppose you have two vectors a and b that you want to take the dot product of, now this is done quite simply by taking each corresponding coordinate of each vector, multiplying them and then adding the result together. At the end of performing our operation we are left with a constant number.  My question therefore is what can we do with this number,why do we calculate it so to speak? I mean it seems almost useless to me compared with the cross product of two vectors (where you end up with an actual vector).