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Super discussion about negative numbers on the BBC
Super radio discussion about the history of negative numbers:
http://www.bbc.co.uk/radio/aod/netwo...dio4/inourtime Andre Jute |
Super discussion about negative numbers on the BBC
In article .com,
Andre Jute wrote: Super radio discussion about the history of negative numbers: http://www.bbc.co.uk/radio/aod/netwo...dio4/inourtime It was good, wasn't it? Progs like this make R4 worth the entire TV licence. ;-) -- *I don't suffer from insanity, I enjoy every minute of it * Dave Plowman London SW To e-mail, change noise into sound. |
Super discussion about negative numbers on the BBC
When it gets into " i ", things get really interesting.
I was once greatly surprised running into a lecture on i once, from none-other than a purchasing agent for a box-maker... pre-computer. He showed me his equations for getting the maximum use of a given width and length of corrugated cardboard (roll) sheeting... each one included i. This guy was a high-school drop-out, but he used 'imaginary numbers' every day. Peter Wieck Wyncote, PA |
Super discussion about negative numbers on the BBC
On 10 Mar 2006 07:23:35 -0800, " wrote:
When it gets into " i ", things get really interesting. I was once greatly surprised running into a lecture on i once, from none-other than a purchasing agent for a box-maker... pre-computer. He showed me his equations for getting the maximum use of a given width and length of corrugated cardboard (roll) sheeting... each one included i. This guy was a high-school drop-out, but he used 'imaginary numbers' every day. Peter Wieck Wyncote, PA What is this "i" thing? I know all about "j". d Pearce Consulting http://www.pearce.uk.com |
Super discussion about negative numbers on the BBC
" i " is the square root of -1.
Peter Wieck Wyncote, PA |
Super discussion about negative numbers on the BBC
On 10 Mar 2006 07:32:55 -0800, " wrote:
" i " is the square root of -1. Peter Wieck Wyncote, PA No, "j" is the square root of -1. "i" must be something else entirely... d Pearce Consulting http://www.pearce.uk.com |
Super discussion about negative numbers on the BBC
http://mathforum.org/library/drmath/view/58251.html
Wrong side of the pond thing, you Brits? Or is it that you need to be ahead of us Colonials... and go all the way to j? Peter Wieck Wyncote, PA |
Super discussion about negative numbers on the BBC
"Don Pearce" wrote in message ... On 10 Mar 2006 07:32:55 -0800, " wrote: " i " is the square root of -1. Peter Wieck Wyncote, PA No, "j" is the square root of -1. "i" must be something else entirely... d Pearce Consulting http://www.pearce.uk.com i *is* the square root of -1, so is j and so is k. S. |
Super discussion about negative numbers on the BBC
Hi,
"Don Pearce" wrote in message ... On 10 Mar 2006 07:23:35 -0800, " wrote: What is this "i" thing? I know all about "j". My exam board (way back when there were 'O' levels) had some problems with this - people who took 'A' levels in both maths (where it was 'i') and physics (where it was 'j') often got them mixed up, and lost marks. There was a minor rebellion, and in the end the mathematicians lost. The square root of -1 was termed "j" from then on. It's probably a moot point now - the last Physics A level paper I looked at required no knowledge of either. a.c. electricity is now the domain of Universities, it would seem. Regards, Glenn. |
Super discussion about negative numbers on the BBC
Hi,
"Don Pearce" wrote in message ... On 10 Mar 2006 07:32:55 -0800, " wrote: " i " is the square root of -1. Peter Wieck Wyncote, PA No, "j" is the square root of -1. "i" must be something else entirely... Indeed. "i" is measured in amperes, and the square root of -1 isn't. Regards, Glenn. |
Super discussion about negative numbers on the BBC
On Fri, 10 Mar 2006 16:01:05 -0000, "Glenn Booth"
wrote: Hi, "Don Pearce" wrote in message ... On 10 Mar 2006 07:23:35 -0800, " wrote: What is this "i" thing? I know all about "j". My exam board (way back when there were 'O' levels) had some problems with this - people who took 'A' levels in both maths (where it was 'i') and physics (where it was 'j') often got them mixed up, and lost marks. There was a minor rebellion, and in the end the mathematicians lost. The square root of -1 was termed "j" from then on. It's probably a moot point now - the last Physics A level paper I looked at required no knowledge of either. a.c. electricity is now the domain of Universities, it would seem. Regards, Glenn. Addition an subtraction are for A level. Multiplication and division are now optional post grad modules. d Pearce Consulting http://www.pearce.uk.com |
Super discussion about negative numbers on the BBC
Hi,
wrote in message oups.com... http://mathforum.org/library/drmath/view/58251.html Wrong side of the pond thing, you Brits? Or is it that you need to be ahead of us Colonials... and go all the way to j? No, but it is considered sensible to differentiate between electrical current and the square root of -1. It's safer. People such as NASA might really screw up if the units are not well defined. ;-) Incidentally, the 6th Edition of the ASACOS Guidelines for preparation of ANSI standards suggests the use of "j" is preferable to the use of "i" to represent sqrt(-1). Regards, Glenn. |
Super discussion about negative numbers on the BBC
Hi,
"Don Pearce" wrote in message ... Addition an subtraction are for A level. Multiplication and division are now optional post grad modules. With a calculator, naturally. Regards, Glenn. |
Super discussion about negative numbers on the BBC
On Fri, 10 Mar 2006 16:37:41 -0000, "Glenn Booth"
wrote: Hi, "Don Pearce" wrote in message ... Addition an subtraction are for A level. Multiplication and division are now optional post grad modules. With a calculator, naturally. No other way is currently known - post doc research is underway. d Pearce Consulting http://www.pearce.uk.com |
Super discussion about negative numbers on the BBC
In article . com,
" wrote: When it gets into " i ", things get really interesting. I was once greatly surprised running into a lecture on i once, from none-other than a purchasing agent for a box-maker... pre-computer. He showed me his equations for getting the maximum use of a given width and length of corrugated cardboard (roll) sheeting... each one included i. This guy was a high-school drop-out, but he used 'imaginary numbers' every day. That seems far fetched, why would "equations for getting the maximum use of a given width and length of corrugated cardboard (roll) sheeting..." need to involve "i"? Regards, John Byrns Surf my web pages at, http://users.rcn.com/jbyrns/ |
Super discussion about negative numbers on the BBC
In article .com,
" wrote: http://mathforum.org/library/drmath/view/58251.html Wrong side of the pond thing, you Brits? Or is it that you need to be ahead of us Colonials... and go all the way to j? Math people and accountants call the square root of -1 "i", engineers and other such riff raff call it "j". Regards, John Byrns Surf my web pages at, http://users.rcn.com/jbyrns/ |
Super discussion about negative numbers on the BBC
"Don Pearce" wrote in message
... On Fri, 10 Mar 2006 16:37:41 -0000, "Glenn Booth" wrote: Hi, "Don Pearce" wrote in message ... Addition an subtraction are for A level. Multiplication and division are now optional post grad modules. With a calculator, naturally. No other way is currently known - post doc research is underway. Meanwhile, the term "mental arithmetic", having fallen into disuse, has been stolen by the medical industry and refers to a head count in a psychiatric hospital. What would today's 18 year olds make of a slide rule? Glenn. |
Super discussion about negative numbers on the BBC
Hi,
"John Byrns" wrote in message ... In article .com, " wrote: http://mathforum.org/library/drmath/view/58251.html Wrong side of the pond thing, you Brits? Or is it that you need to be ahead of us Colonials... and go all the way to j? Math people and accountants call the square root of -1 "i", engineers and other such riff raff call it "j". Accountants have no concept of "i" or "j" whatever. The have an entirely different definition of "imaginary numbers". They put them on your bill, labelled "hourly rate". Regards, Glenn. |
Super discussion about negative numbers on the BBC
why would "equations for getting the maximum use of a given width and length of corrugated cardboard (roll) sheeting..." need to involve "i"?
Well, one is subtracting from a given area, and the numbers subtracted are therefore 'negative'. As the numbers are _areas_, then we have a negative number that often needs to have its square root taken. Mathematically, this does not work. as -1 x -1 = 1. So. " i " is introduced to make the calculations work. This was a lecture mind... so here is the issue: feedstock is expensive, and there are thousands of sizes of boxes. These were corrugate boxes, with varying dimensions. This plant did not 'stock', but made to order, to size. The knives were capable of cutting in either horizontally or vertically and as finenly placed as necessary, as well as partial cuts. Set-up and roll changing were the costliest operations (as both involved machine shut-down), waste being the third largest cost. Several sizes of boxes could be run at once. He calculated how to run the orders to avoid waste and slivering, and to minimize the number of different widths of feedstock required. He chose to use the 'imaginary number i' in his process. It seemed to work as their scrap-pile was rather tiny to his great pride and joy. I pointed to Drexel U, as that is where I took my engineering courses. In the process of those courses, we were given some practical field applications of otherwise very dry concepts. This was one of them. Peter Wieck Wyncote, PA |
Super discussion about negative numbers on the BBC
John Byrns wrote: In article .com, " wrote: http://mathforum.org/library/drmath/view/58251.html Wrong side of the pond thing, you Brits? Or is it that you need to be ahead of us Colonials... and go all the way to j? Math people and accountants call the square root of -1 "i", engineers and other such riff raff call it "j". Regards, John Byrns Surf my web pages at, http://users.rcn.com/jbyrns/ I prefer to think of engineers as useful fellows; I know many both in electronics and automobiles who are the salt of the earth and have elegant minds besides. It's their inferiors, the "engineers" (and their hangers-on) who are clearly trailer park trash, know it, resent it loudly, and take out their frustration for being talentless on us. There is a useful American engineering use of "j" in the higher math of tubes to define all the relationships of phase shift. See for instance the work of Julian L Bernstein, the last associate dean of the RCA Institutes; he appears to have been in charge of the Day Schools, which probably beside the US Navy gave the finest tube education ever available. I have a complete mathematical transfer model in an Excel spreadsheet, based on Mr Bernstein's work, that I will discuss when the boards are calmer. Such use of "j" as at right angles (plus or minus something in our case) is particularly in line with the history of thought experiments of the square root of minus one (and other imaginary numbers) being at right angles to the continuum negative-positive. I recommend this super radio discussion about the history of negative numbers for the larger perspective; be quick though as it the BBC only keeps it up for a week: http://www.bbc.co.uk/radio/aod/netwo...l?radio4/inour... I would have liked to have heard more about how modern electronics is impossible without the square root of minus one, but I suppose in an introductory programme all one is entitled to is a glancing reference. Leaving the listener hungry for more is what defines a really good programme! Andre Jute |
Super discussion about negative numbers on the BBC
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Super discussion about negative numbers on the BBC
On 10 Mar 2006 07:54:43 -0800, " wrote:
http://mathforum.org/library/drmath/view/58251.html Wrong side of the pond thing, you Brits? Or is it that you need to be ahead of us Colonials... and go all the way to j? Nope, it depends on whether you're a mathematician or an EE. -- Stewart Pinkerton | Music is Art - Audio is Engineering |
Super discussion about negative numbers on the BBC
On Fri, 10 Mar 2006 16:58:53 -0000, "Glenn Booth"
wrote: "Don Pearce" wrote in message ... On Fri, 10 Mar 2006 16:37:41 -0000, "Glenn Booth" wrote: Hi, "Don Pearce" wrote in message .. . Addition an subtraction are for A level. Multiplication and division are now optional post grad modules. With a calculator, naturally. No other way is currently known - post doc research is underway. Meanwhile, the term "mental arithmetic", having fallen into disuse, has been stolen by the medical industry and refers to a head count in a psychiatric hospital. What would today's 18 year olds make of a slide rule? My Faber Castell Novo-Duplex would probably make a nice table ornament, but my Otis King would be a decent match for those extensible batons the piggies now use! :-) -- Stewart Pinkerton | Music is Art - Audio is Engineering |
Super discussion about negative numbers on the BBC
OK... how would you come up with the square root of -4? Practical
application, you are starting with so many square feet of feedstock, you are making 22 boxes each requiring two 4 square foot faces, two feet on a side and other sides may vary within certain parameters, and 12 boxes each requiring two 1 square foot faces. But the dimensions of the first box must be calculated to have the correct volume as a function of dimensions and not preclude the similar values for the second box. So, you are SUBTRACTING dimensions as square roots of total areas required for square cuts. As sq.rt. -4 does not calculate, but sq.rt. 4 x i does... that is how it comes in. Keep in mind that one *could* reverse the signs in one's head the reality is that all the areas calculated are *real*, but as there are many sign-changes in the calculation apart from negative number roots, the chance of error increases greatly. The elegant part of all this is that the " i " drops out at the end of the calculations, but it allows the rule of 8 (8 basic axioms of 'real' numbers) to apply during. As others have suggested, we have computers do this these days. The need for practical math has been relegated mostly to calculating tips in a restaurant. And few do even this, it seems. The history of Negative Numbers, remember? Peter Wieck Wyncote, PA |
Super discussion about negative numbers on the BBC
What would today's 18 year olds make of a slide rule?
They are clueless. I have a 14" K&E, double-sided. Hot-Sh*t in its day... Once upon a time, I could even use it. For calculus even... The kids (well over 18) only know what it is from me. Their kids? Not at all. Peter Wieck Wyncote, PA |
Super discussion about negative numbers on the BBC
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Super discussion about negative numbers on the BBC
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Super discussion about negative numbers on the BBC
"Hey, what's that.... crap snipped"
Do you ever stop pretending? Peter Wieck Wyncote, PA |
Super discussion about negative numbers on the BBC
The real number -1 = -1+0i = (1,180°) has angle 180 degrees (mod 360
degrees) and length 1. The purely imaginary number [0,1] = 0+i1 = (1,90°) has angle 90 degrees and length 1. Multiplying this point or number by itself, that is, squaring it, gives the point with length 1 ×1 = 1 and angle 90°+90° = 180°. So the product equals -1+0i = -1. We call i, the principal square root of -1. A second square root of -1 is obtained as follows. The imaginary number (0,-1) = 0+i(-1) = [1,-90°] has angle -90 degrees and length 1. Multiplying this point or number by itself, that is squaring it, gives the point with length 1 times 1 =1 and angle (-90°)+(-90°) = -180° = 180° (mod 360°). So this product equals -1+0i = -1 as well. This provides two square roots of -1 as both (1,+90°)2 = (1,+180°) = -1 and (1,-90°)2 = (1,-180°) = -1. I cannot reproduce a diagram. But what you get is four points on a graph. Set your knives to those points, and you can cut a straight cut on moving stock. Add other calculations, and you can graph other cuts to minimize waste. Remember, this was pre-desk-top-computer... well over 30 years ago and I was NOT the one making the calculations. The above are unashamedly cribbed from a website that also cribbed from another website... but also is dedicated to topology. Items like the Klein Bottle and the Mobius strip can be described mathematically. As I remember, both also use ' i ' as there are 'imaginary' conditions to be described as points in space or points on a plain. Peter Wieck Wyncote, PA |
Super discussion about negative numbers on the BBC
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Super discussion about negative numbers on the BBC
In article , Stewart Pinkerton
wrote: On 10 Mar 2006 07:54:43 -0800, " wrote: http://mathforum.org/library/drmath/view/58251.html Wrong side of the pond thing, you Brits? Or is it that you need to be ahead of us Colonials... and go all the way to j? Nope, it depends on whether you're a mathematician or an EE. Just to make things more confusing, you can also end up using 'j' for current density as well as 'i' for current... :-) The problem is that we don't have enough letters... Slainte, Jim -- Electronics http://www.st-and.ac.uk/~www_pa/Scot...o/electron.htm Audio Misc http://www.st-and.demon.co.uk/AudioMisc/index.html Armstrong Audio http://www.st-and.demon.co.uk/Audio/armstrong.html Barbirolli Soc. http://www.st-and.demon.co.uk/JBSoc/JBSoc.html |
Super discussion about negative numbers on the BBC
"flipper" wrote in message ... Which, btw, is exactly the same situation as those who don't think i exists. It doesn't. But then, "numbers" don't exist in any meaningful sense. You can have, say, five apples, but you can't just have five. It's just an idea. |
Super discussion about negative numbers on the BBC
"flipper" wrote in message ... On Mon, 13 Mar 2006 00:12:55 GMT, "Rich Wilson" wrote: "flipper" wrote in message . .. Which, btw, is exactly the same situation as those who don't think i exists. It doesn't. But then, "numbers" don't exist in any meaningful sense. If you go back and look at the context that's been removed you'll see that I said it's as 'real' as the 'real' numbers are. The point is it's 'reality' is conceptually no different regardless of what common conceptualization, or lack thereof, you now wish to obscure the matter with. I wasn't disputing your argument! Yes, i is as real as any other number. You can have, say, five apples, but you can't just have five. It's just an idea. Ideas 'exist'. They must since you apparently have one ;) I have an idea of, say, a machine that turns lard into banknotes. That doesn't mean it exists. Equally, I have an idea of the number "five". The idea exists, the number doesn't necessarily exist. |
Super discussion about negative numbers on the BBC
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Super discussion about negative numbers on the BBC
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Super discussion about negative numbers on the BBC
"Jon Yaeger" wrote in message ... in article , Rich Wilson at wrote on 3/12/06 9:14 PM: "flipper" wrote in message ... On Mon, 13 Mar 2006 00:12:55 GMT, "Rich Wilson" wrote: "flipper" wrote in message ... Which, btw, is exactly the same situation as those who don't think i exists. It doesn't. But then, "numbers" don't exist in any meaningful sense. If you go back and look at the context that's been removed you'll see that I said it's as 'real' as the 'real' numbers are. The point is it's 'reality' is conceptually no different regardless of what common conceptualization, or lack thereof, you now wish to obscure the matter with. I wasn't disputing your argument! Yes, i is as real as any other number. You can have, say, five apples, but you can't just have five. It's just an idea. Ideas 'exist'. They must since you apparently have one ;) I have an idea of, say, a machine that turns lard into banknotes. That doesn't mean it exists. Equally, I have an idea of the number "five". The idea exists, the number doesn't necessarily exist. Yes, but the idea of numbers is a priori; the lard converter is not. I'm not familiar with the latin phrase and the Wikipedia definition ain't helping... care to explain your argument in English? |
Super discussion about negative numbers on the BBC
in article , Rich Wilson at
wrote on 3/12/06 9:23 PM: "Jon Yaeger" wrote in message ... in article , Rich Wilson at wrote on 3/12/06 9:14 PM: "flipper" wrote in message ... On Mon, 13 Mar 2006 00:12:55 GMT, "Rich Wilson" wrote: "flipper" wrote in message ... Which, btw, is exactly the same situation as those who don't think i exists. It doesn't. But then, "numbers" don't exist in any meaningful sense. If you go back and look at the context that's been removed you'll see that I said it's as 'real' as the 'real' numbers are. The point is it's 'reality' is conceptually no different regardless of what common conceptualization, or lack thereof, you now wish to obscure the matter with. I wasn't disputing your argument! Yes, i is as real as any other number. You can have, say, five apples, but you can't just have five. It's just an idea. Ideas 'exist'. They must since you apparently have one ;) I have an idea of, say, a machine that turns lard into banknotes. That doesn't mean it exists. Equally, I have an idea of the number "five". The idea exists, the number doesn't necessarily exist. Yes, but the idea of numbers is a priori; the lard converter is not. I'm not familiar with the latin phrase and the Wikipedia definition ain't helping... care to explain your argument in English? This link should do it: http://www.importanceofphilosophy.co...l_APriori.html Jon |
Super discussion about negative numbers on the BBC
"Jon Yaeger" wrote in message ... in article , Rich Wilson at wrote on 3/12/06 9:23 PM: "Jon Yaeger" wrote in message ... in article , Rich Wilson at wrote on 3/12/06 9:14 PM: "flipper" wrote in message ... On Mon, 13 Mar 2006 00:12:55 GMT, "Rich Wilson" wrote: "flipper" wrote in message ... Which, btw, is exactly the same situation as those who don't think i exists. It doesn't. But then, "numbers" don't exist in any meaningful sense. If you go back and look at the context that's been removed you'll see that I said it's as 'real' as the 'real' numbers are. The point is it's 'reality' is conceptually no different regardless of what common conceptualization, or lack thereof, you now wish to obscure the matter with. I wasn't disputing your argument! Yes, i is as real as any other number. You can have, say, five apples, but you can't just have five. It's just an idea. Ideas 'exist'. They must since you apparently have one ;) I have an idea of, say, a machine that turns lard into banknotes. That doesn't mean it exists. Equally, I have an idea of the number "five". The idea exists, the number doesn't necessarily exist. Yes, but the idea of numbers is a priori; the lard converter is not. I'm not familiar with the latin phrase and the Wikipedia definition ain't helping... care to explain your argument in English? This link should do it: http://www.importanceofphilosophy.co...l_APriori.html OK... so "a priori" knowledge is knowledge that has no reference to reality... i.e. knowledge about things that don't exist. |
Super discussion about negative numbers on the BBC
in article , Rich Wilson at
wrote on 3/12/06 9:50 PM: "Jon Yaeger" wrote in message ... in article , Rich Wilson at wrote on 3/12/06 9:23 PM: "Jon Yaeger" wrote in message ... in article , Rich Wilson at wrote on 3/12/06 9:14 PM: "flipper" wrote in message ... On Mon, 13 Mar 2006 00:12:55 GMT, "Rich Wilson" wrote: "flipper" wrote in message ... Which, btw, is exactly the same situation as those who don't think i exists. It doesn't. But then, "numbers" don't exist in any meaningful sense. If you go back and look at the context that's been removed you'll see that I said it's as 'real' as the 'real' numbers are. The point is it's 'reality' is conceptually no different regardless of what common conceptualization, or lack thereof, you now wish to obscure the matter with. I wasn't disputing your argument! Yes, i is as real as any other number. You can have, say, five apples, but you can't just have five. It's just an idea. Ideas 'exist'. They must since you apparently have one ;) I have an idea of, say, a machine that turns lard into banknotes. That doesn't mean it exists. Equally, I have an idea of the number "five". The idea exists, the number doesn't necessarily exist. Yes, but the idea of numbers is a priori; the lard converter is not. I'm not familiar with the latin phrase and the Wikipedia definition ain't helping... care to explain your argument in English? This link should do it: http://www.importanceofphilosophy.co...l_APriori.html OK... so "a priori" knowledge is knowledge that has no reference to reality... i.e. knowledge about things that don't exist. My understanding of a priori as it applies to math is that we have a concept of numbers that seem to be beyond argument, but on closer inspection, is actually based upon faith. We can't really "prove" the number 5, for example, but we operate on the assumption that "5" is true. As an article of faith, perhaps it is "more empirical" (if such a non-sequitur can be used to describe something that is based upon "faith") than, say, the imaginary concept of a lard converter. We don't really depend upon concept of a lard converter for understanding other concepts as we do numbers. It is a bit of an abstract and subtle distinction. Perhaps a philosopher or professional parser such as Mr. Byrnes can correct me or shed more light on the distinction, if indeed there is one? ;-) Jon |
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