- #36
disregardthat
Science Advisor
- 1,866
- 34
Landau said:Easy: pi/2+pi/2=pi.
That's like saying 2*3 is log(2)+log(3).
Landau said:Easy: pi/2+pi/2=pi.
How can a true statement be like a false statement?Jarle said:That's like saying 2*3 is log(2)+log(3).
Landau said:How can a true statement be like a false statement?
Besides, I don't think my answer was so bad. It recognises the fundamental relation between multiplication of complex numbers and scaling rotation, i.e. the algebra representation
[tex]\mathbb{c}\to M_2(\mathbb{R})[/tex]
[tex]z=x+yi\mapsto \left(\begin{array}{cc}x & -y\\ y & x \end{array}\right).[/tex]
So if we stick to the unit circle, there is not scaling and only rotation. And the composition ('product') of two rotations (in 2d) amounts to adding the angles.
Hurkyl said:Multiplication is bilinear. That means (a+b)c = ac + bc and a(b+c) = ab + ac. In other words, the distributive property holds.
In the very special case that "a" can be written as repeated addition of a multiplicative unit:
a = 1 + 1 + 1 + ... + 1then "ab" can be written as repeated addition of b:
ab = (1 + 1 + ... + 1)b = 1b + 1b + ... + 1b = b + b + ... + b
It's hard to say, since the first year engineering student never responded.brydustin said:And I'm sure that this was very helpful to the first year engineering student...
JyN said:So, if multiplication was indeed repeated addition, there would only be two elementary operations. Addition, and subtraction. And since subtraction is inverse addition, that would mean that division is repeated subtraction, and it certainly isn't.
Hurkyl said:If you think of division only in terms of how it relates to multiplication, then naturally division will seem like it's working backwards. :tongue:
Then could you explain what you mean by starting at the destination and working backwards to figure out how to get there?apeiron said:But I wasn't. So if you care to offer a more constructive reply...
Hurkyl said:Then could you explain what you mean by starting at the destination and working backwards to figure out how to get there?
If I gave you any randomly chosen real number and asked you to split it into x equal portions, and you had no access to multiplication tables or other forms of prior knowledge, how in terms of a mathematical operation would you proceed?
I'm thoroughly confused with this. The way real numbers are normally specified, there is a straightforward division algorithm. Just follow the steps until you're done (or have enough precision, as the case may be).apeiron said:If I gave you an additive, subtractive or multiplicative question, you could say hang on and I'll use this operation to crank out the answer. The size of each step, and the total number of steps, is specified. So no problems.
But with division, even if the number of steps has been specified in the question, the size of them isn't. It is precisely what you have to discover somehow.
I just want to add that there are even ways of representing numbers that are especially convenient for division. e.g. representing any positive real number x by the decimal expansion of log(x). (I'm not being frivolous with this -- I have really seen this used) Also division is rather simple in the the prime factorization representation of rational numbers.Hurkyl said:I'm thoroughly confused with this. The way real numbers are normally specified, there is a straightforward division algorithm. Just follow the steps until you're done (or have enough precision, as the case may be).
Studiot said:You will find the answer in Euclid, dear soul.
It is a very simple and elementary construction that used to be taught to 11 year olds.
Hurkyl said:I'm thoroughly confused with this. The way real numbers are normally specified, there is a straightforward division algorithm. Just follow the steps until you're done (or have enough precision, as the case may be).
Hurkyl said:I just want to add that there are even ways of representing numbers that are especially convenient for division. e.g. representing any positive real number x by the decimal expansion of log(x). (I'm not being frivolous with this -- I have really seen this used) Also division is rather simple in the the prime factorization representation of rational numbers.
Please try to be far more specific than you have been. I have to guess at the fine details of what you mean.apeiron said::sigh: If only the answer were so simple as long division.
As should be clear, the issue is the precision. The answer for simple constructive operations is always going to be exact. But for division, answers are only going to be effective. You have to introduce a cut-off on the number of decimal places as a further pragmatic choice.
Hurkyl said:Please try to be far more specific than you have been. I have to guess at the fine details of what you mean.
Long division is an exact operation on decimal numerals. Every computable operation on real numbers will have to deal with precision issues of some sort. Even addition.
When applied to decimals that represent rational numbers (because at some point a sequence of digits repeats forever), a slight modification allows long division to terminate after a finite number of steps.
Incidentally, when applied to rational numbers represented as a quotient of integers, the division algorithm and multiplication algorithms are pretty much identical.
Eh? There were two and a half pages addressing the original question. You brought up a new claim -- that division is somehow working backwards from a destination to a construction, but yet you are not thinking in terms of division being an inverse of multiplication, and that this is somehow a fundamental difference between division and other arithmetic operations, rather than just being one of many ways to view division.apeiron said:In fact forget the whole question because you clearly are not interested in actually addressing it, just talking around it forever.
In fact forget the whole question because you clearly are not interested in actually addressing it, just talking around it forever.
Studiot said:I gave you an answer that is absolutely precise, but you chose to do exactly what you are accusing others of - you ignored it.
Studiot said:You will find the answer in Euclid, dear soul.
It is a very simple and elementary construction that used to be taught to 11 year olds.
Hurkyl said:Eh? There were two and a half pages addressing the original question. You brought up a new claim -- that division is somehow working backwards from a destination to a construction, but yet you are not thinking in terms of division being an inverse of multiplication, and that this is somehow a fundamental difference between division and other arithmetic operations, rather than just being one of many ways to view division.
If "addressing" your point means unquestioningly buying into your assertion, then yes, I am uninterested in "addressing" it.
And since subtraction is inverse addition, that would mean that division is repeated subtraction, and it certainly isn't. As a side note: I actually remember seeing it like this when i was very young and first learning about arithmetic. And, because i saw multiplication as repeated addition, it seemed to me that division was really not like the others.
I don't know about precise, but that's the feyest attempt at an insult I've seen in a long time.
Studiot said:However the fact remains that up to the late 1960s boys in their first year in an English grammar school would be taught the construction, from Euclid, that I was referring to.
In those days such a construction was used by engineers and draughtsmen and a version appeared on many boxwood scales of that time.
Studiot said:You have now divided the original line perfectly into n equal parts.
But are you saying it is a geometric representation of either inverse multiplication or repeated subtraction as arithmetic operations?
Studiot said:I said before that I answered a specific question you made, asking for a proceedure to "divide a given random number into x equal parts" and thought you also implied that you did not think this could be done.
I am rather suprised you have not heard of it since in another thread you claimed a classical education within the British/Irish system.
If I gave you any randomly chosen real number and asked you to split it into x equal portions, and you had no access to multiplication tables or other forms of prior knowledge, how in terms of a mathematical operation would you proceed?
Studiot said:I can only understand what is written.
Conforms exactly to the question I answered and later paraphrased.
It was about the OP point that "division seems different" and so about the precise nature of that difference in number theory.
You can't answer the larger question that was posed.
apeiron said::sigh: If only the answer were so simple as long division.
As should be clear, the issue is the precision. The answer for simple constructive operations is always going to be exact. But for division, answers are only going to be effective. You have to introduce a cut-off on the number of decimal places as a further pragmatic choice.
apeiron said:It seems like the prime number factorisation problem. You have to guess repeatedly to crack the answer. There is no simple iterative operation to employ.
Jarle said:This is purely due to the choice of representation by decimals. Division is a wholly constructive operation, but it is simply the case that not all rational and real numbers can be written as a finite decimal expansion in base 10. There isn't anything imprecise about the result of an operation that iteratively gives you the base 10 digits of a given real or rational number, it gives you exactly what you want.
You talked about the necessity to introduce limits when it comes to real numbers - as a sort of flaw, but it is in this sense real numbers are defined (or can be defined) - as limits.
Actually, there are simple constructive methods to give you the full prime factorization of any given integer. It is not necessary to guess at any point. Even the most naive (ineffective) ones will not involve any guesswork.
I can't guess what you mean from this description.apeiron said:Yes, division can be a wholly constructive operation (namely, repeated subtraction)
I can't guess what you mean from this description.but only because a further "natural" step has been taken in breaking the symmetry of the number line by choosing a base 10 numbering system.
I can't guess what you mean from this description.Whereas the numberline is a linear additive concept, we are now laying over the top of it a geometric expansion which gives us "counting in orders of magnitude and decimal scale".
I can't guess what you mean from this description.The challenge was to connect something that is essentially discrete (a string of points) with what also had to be essentially continuous (a line) and breaking the scale of counting in this way, using a base as a further constraint, seems like the way it has been done.
I can't guess how you draw that conclusion from this description.So anyway, the answer for me now goes clearly beyond the original question about the nature of division and is clearly part of all the conversations about irrational numbers and infiities.
I can't guess what you mean from this description.The numberline is founded on the notion of "one-ness". And that is a symmetric or single-scale concept. But as soon as you introduce an asymmetry, a symmetry-breaking constraint - such as any base system starting even from base 2 - then there is something new. A connection is forged between the original point-like discreteness and the continuity implied by a numberline. Scale is broken geometrically over all scales. Allowing then measurement down to the "finest grain".
Hurkyl said:Try using math instead of prose.