[math-fun] keeping things in order
How do you write your polynomials, ax^2+bx+c or c+bx+ax^2? Traditionally it seems the first way is most common. Why? The second would seem more natural. Don't we generally like to have things (like exponents) increasing from left to right? By contrast, for power series we have a_0+a_1x+a_2x^2+ . . ., (never. . .+a_2x^2+a_1x+a_0). For that matter, why do we write our numerals so that ten is written 10 rather than 01? As with polynomials, the digits of a numeral represent coefficients of powers of ten in decreasing order from left to right. Aha, I get it! Of course. They're Arabic numerals and therefore were read by their creators from right to left. That doesn't explain the polynomials though, or did we get them from the Arabs too? Some further random related observations. 1. Reversing the digit order convention would make addition and multiplication feel more natural since in applying the algorithms one first learns the units digit, then the tens and so on. When we "carry" we would move digits "forward", when we borrow we'd move them backwards. 2. How would you feel about pi ~95414.3? It would certainly take getting used to. 3. Lets look at the "names" for the numerals. English: seven-teen, eight-teen,. ., so far so good, increasing exponent order, but then we do twenty one, twenty two, Arabic order. French: At least it's consistently Arabic. dix-sept, dix-huit, and vingt six, vingt sept. Spanish, Italian the same, but, 4.German: consistently anti-Arabic, achtzehn, neunzehn, zwanzig, ein und zwanzig, but after 100 we get, for example, 123 is ein hundert drei und zwazig so 123 gets permuted to 132, the middle digit comes last! 5. The above suggests that the different digit orderings is a Romance vs. Germanic language thing. However, going back to the Latin is no help since they didn't have the Arabic place system. For example 18 I just learned is "duodevigniti". How would a Roman say "You owe me XXXIV denarius" I wonder.
While this is fun speculation for written forms, it is actually a serious issue for algebraic software. The answer seems to come down to which operations are more common: comparisons v. additions. It would be interesting to take a poll of symbolic algebraic manipulation (SAM) systems to see how they represent polynomials & integers. At 06:37 PM 3/1/2005, David Gale wrote:
How do you write your polynomials, ax^2+bx+c or c+bx+ax^2?
Traditionally it seems the first way is most common. Why? The second would seem more natural. Don't we generally like to have things (like exponents) increasing from left to right? By contrast, for power series we have
a_0+a_1x+a_2x^2+ . . ., (never. . .+a_2x^2+a_1x+a_0).
For that matter, why do we write our numerals so that ten is written 10 rather than 01? As with polynomials, the digits of a numeral represent coefficients of powers of ten in decreasing order from left to right. Aha, I get it! Of course. They're Arabic numerals and therefore were read by their creators from right to left. That doesn't explain the polynomials though, or did we get them from the Arabs too? Some further random related observations.
1. Reversing the digit order convention would make addition and multiplication feel more natural since in applying the algorithms one first learns the units digit, then the tens and so on. When we "carry" we would move digits "forward", when we borrow we'd move them backwards.
2. How would you feel about pi ~95414.3? It would certainly take getting used to.
3. Lets look at the "names" for the numerals.
English: seven-teen, eight-teen,. ., so far so good, increasing exponent order, but then we do twenty one, twenty two, Arabic order.
French: At least it's consistently Arabic. dix-sept, dix-huit, and vingt six, vingt sept. Spanish, Italian the same, but,
4.German: consistently anti-Arabic, achtzehn, neunzehn, zwanzig, ein und zwanzig, but after 100 we get, for example, 123 is ein hundert drei und zwazig so 123 gets permuted to 132, the middle digit comes last!
5. The above suggests that the different digit orderings is a Romance vs. Germanic language thing. However, going back to the Latin is no help since they didn't have the Arabic place system. For example 18 I just learned is "duodevigniti". How would a Roman say "You owe me XXXIV denarius" I wonder.
----- Original Message ----- From: "David Gale" <gale@math.berkeley.edu> To: <math-fun@mailman.xmission.com>; <davis@math.toronto.edu>; <mrcsgardner@hotmail.com> Sent: Tuesday, March 01, 2005 9:37 PM Subject: [math-fun] keeping things in order
2. How would you feel about pi ~95414.3? It would certainly take getting used to.
...95141.3 =~ ip naht os erom neve, seY.
At 11:57 PM 3/1/2005 -0500, you wrote:
2. How would you feel about pi ~95414.3? It would certainly take getting used to.
...95141.3 =~ ip naht os erom neve, seY. \
OUCH!!! dg
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I think it's entirely due to the differences in writing. Arabic, Hebrew, Hindi from right to left. Most of the rest from left to right -- is this due to an inherent right- or left-handedness? The arabic system (and perhaps even more so the Iraqi one which goes back nearly 4000 years) was obviously superior for purposes of calculation, so was gratefully borrowed by the Western world. So we got used to the less logical way of putting the big numbers first. There is a practical advantage -- (provided you read from left to right) that you get to know the most significant digits first (tho not how significant until you get to the end). For polynomials, I guess that we just got used to putting the big things first, i.e., on the left. R. On Tue, 1 Mar 2005, David Gale wrote:
How do you write your polynomials, ax^2+bx+c or c+bx+ax^2?
Traditionally it seems the first way is most common. Why? The second would seem more natural. Don't we generally like to have things (like exponents) increasing from left to right? By contrast, for power series we have
a_0+a_1x+a_2x^2+ . . ., (never. . .+a_2x^2+a_1x+a_0).
For that matter, why do we write our numerals so that ten is written 10 rather than 01? As with polynomials, the digits of a numeral represent coefficients of powers of ten in decreasing order from left to right. Aha, I get it! Of course. They're Arabic numerals and therefore were read by their creators from right to left. That doesn't explain the polynomials though, or did we get them from the Arabs too? Some further random related observations.
1. Reversing the digit order convention would make addition and multiplication feel more natural since in applying the algorithms one first learns the units digit, then the tens and so on. When we "carry" we would move digits "forward", when we borrow we'd move them backwards.
2. How would you feel about pi ~95414.3? It would certainly take getting used to.
3. Lets look at the "names" for the numerals.
English: seven-teen, eight-teen,. ., so far so good, increasing exponent order, but then we do twenty one, twenty two, Arabic order.
French: At least it's consistently Arabic. dix-sept, dix-huit, and vingt six, vingt sept. Spanish, Italian the same, but,
4.German: consistently anti-Arabic, achtzehn, neunzehn, zwanzig, ein und zwanzig, but after 100 we get, for example, 123 is ein hundert drei und zwazig so 123 gets permuted to 132, the middle digit comes last!
5. The above suggests that the different digit orderings is a Romance vs. Germanic language thing. However, going back to the Latin is no help since they didn't have the Arabic place system. For example 18 I just learned is "duodevigniti". How would a Roman say "You owe me XXXIV denarius" I wonder.
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Hold on, this line of analysis doesn't seem to add up. Writing tends to originate as transcribed speech, and the natural order for magnitudes leads from the most-significant: "A fiery stream issued and came forth from before him; a thousand thousands ministered unto him, and ten thousand times ten thousand stood before him; the judgment was set, and the books were opened." --Daniel 7:10. In ancient times when these things arose I can't imagine anything else would've flown: King: "How many myriads of the enemy stand before us?" Sage: "Oh exalted one, their number leaves none when divided into tenths; after administering that decimation you will be pleased to discover the remainder is zero; moreover continuing in this way we again achieve nothing; next..." King: "Enough of this, off with his head!" Page: "Too late my lord, they've already breached our gates!" Nor does it seem natural for someone writing in a right-to-left script such as Arabic to reverse direction just for numbers, having to skip over adequate space, etc. So how did it arise? We've heard that the Arabic signs for individual digits were adopted, but are we just assuming that the original positional ordering was preserved? Surely there are scholars who know all about this (I'll bet Knuth does). Similar motivations probably applied to polynomials: that it's a cubic was likely more interesting at the outset than the value of the constant term. Of course it's probably too much to expect this sort of thing to make any kind of sense. Note that we enlightened moderns are most likely conducting this conversation using tools with parts labeled 1234567890 (followed by QWERTYUIOP...!<;-).
P'r'aps I didn't make myself clear. P'r'aps most readers read from left to right. The Babylonians wrote numbers 1 to 9; then if they wanted tens or sixties or whatever, they wrote them to the left of the units. They may have read them (i.e., looked at them) in the same way. But, when it was important to know (convey?) the rough size of the number, they may well have developed a convention of saying them (making actual noises) in some other way. But saying preceded writing. And saying numbers, except for the first very few, was presumably an estimate, using small numbers of appropriately chosen units (hands, scores -- all fingers & toes, and progressively larger and no doubt vaguer things). Except for the Babylonian mathematicians, who clearly knew what they were doing, I doubt if there was much call for statements as precise, say, as `I caught 47 fish' -- more likely `I caught dozens of fish' and, if there were angling competitions, then they would be carefully counted and `scored' and even records kept and broken. R. On Wed, 2 Mar 2005, Marc LeBrun wrote:
Hold on, this line of analysis doesn't seem to add up.
Writing tends to originate as transcribed speech, and the natural order for magnitudes leads from the most-significant:
"A fiery stream issued and came forth from before him; a thousand thousands ministered unto him, and ten thousand times ten thousand stood before him; the judgment was set, and the books were opened." --Daniel 7:10.
In ancient times when these things arose I can't imagine anything else would've flown:
King: "How many myriads of the enemy stand before us?"
Sage: "Oh exalted one, their number leaves none when divided into tenths; after administering that decimation you will be pleased to discover the remainder is zero; moreover continuing in this way we again achieve nothing; next..."
King: "Enough of this, off with his head!"
Page: "Too late my lord, they've already breached our gates!"
Nor does it seem natural for someone writing in a right-to-left script such as Arabic to reverse direction just for numbers, having to skip over adequate space, etc. So how did it arise? We've heard that the Arabic signs for individual digits were adopted, but are we just assuming that the original positional ordering was preserved?
Surely there are scholars who know all about this (I'll bet Knuth does).
Similar motivations probably applied to polynomials: that it's a cubic was likely more interesting at the outset than the value of the constant term.
Of course it's probably too much to expect this sort of thing to make any kind of sense. Note that we enlightened moderns are most likely conducting this conversation using tools with parts labeled 1234567890 (followed by QWERTYUIOP...!<;-).
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When we are beginning readers, we read each letter of each word left to right and sound it out. When we become experienced readers, we read the words as units. In fact, our brains autoprocess some of the fluff words (the, of, etc) to the point that we are not even conscious of them. There is an interesting demonstration of this, where one is supposed to quickly count the number of "f"s that occur in a paragraph. More often than not, people will tend to miss the "f"s in the word "of", because our brains shield us from perceiving the word. Even when you know the trick, you have to be really careful not to miss any of the "of"s. At any rate, I think some of this applies to numbers as well. Once we learn how to read numbers, we simply scan them as units, not concerned with the digits unless we need to be. When we read: State and local agencies had 708022 full-time sworn personnel and 311474 full-time civilian employees in 2000. Certainly, we read the words left to right, but we don't read the digits of the numbers left to right any more than we read the letters of the words left to right. The numbers and words are absorbed as units. We only become interested in the individual characters when we run into a difficulty (new word, misspelling) or have need of precise information. thereareevenscriptslikeancientgreekwheretherearenovisiblecuessuch aspunctuationwordbreaksorcasetodistinguishwordsfromcharacters eveninthismostadverseofconditionsyourbraincanpickoutwordsfrom theseaofcharactersasyouwillrealizewhenyouhavereadthisparagraph .left-to-right reading text Arabic in embedded often are right-to-left read numbers ,example foR .units as read are numbers and words the since ,difference of lot whole a make n'twould it ,text within words reading of direction the from different was numbers or words within characters reading of direction the if that think even I ----- Original Message ----- From: "Richard Guy" <rkg@cpsc.ucalgary.ca> To: <ham>; "math-fun" <math-fun@mailman.xmission.com> Cc: <davis@math.toronto.edu>; <mrcsgardner@hotmail.com> Sent: Wednesday, March 02, 2005 10:28 AM Subject: Re: [math-fun] keeping things in order
I think it's entirely due to the differences in writing. Arabic, Hebrew, Hindi from right to left. Most of the rest from left to right -- is this due to an inherent right- or left-handedness?
The arabic system (and perhaps even more so the Iraqi one which goes back nearly 4000 years) was obviously superior for purposes of calculation, so was gratefully borrowed by the Western world. So we got used to the less logical way of putting the big numbers first. There is a practical advantage -- (provided you read from left to right) that you get to know the most significant digits first (tho not how significant until you get to the end).
=David Wilson When we are beginning readers, we read each letter of each word left to right and sound it out. When we become experienced readers, we read the words as units.
Which perhaps explains in a nutshell why the "whole language" approach to reading was apparently such a pedagogical disaster.
thereareevenscriptslikeancientgreekwheretherearenovisiblecues...
Somewhere I heard of an interesting experiment contrasting "agglutination" versus "isolation" in languages: when correcting enunciation mistakes speakers start over on what they perceive as word boundaries, rather than phonemes--giving, eg, English more granular error-recovery than, say, German.
...Arabic...foR...n'twould...
Fun!nuF
Quoting Richard Guy <rkg@cpsc.ucalgary.ca>:
I think it's entirely due to the differences in writing. Arabic, Hebrew, Hindi from right to left. Most of the rest from left to right -- is this due to an inherent right- or left-handedness?
Yes. If you write with pen and ink, you can keep from smearing what you just wrote. Painting with a brush or incising clay tablets, it doesn't much matter. Who knows what stone cutters have to deal with (but paying them calls for an economy of verbiage)?
... There is a practical advantage -- (provided you read from left to right) that you get to know the most significant digits first (tho not how significant until you get to the end).
More recently, that was the difference between the Motorola 6800 and the Intel 8080.
For polynomials, I guess that we just got used to putting the big things first, i.e., on the left.
But power series work the other way around. No doubt to avoid that last term ... . - hvm ------------------------------------------------- www.correo.unam.mx UNAMonos Comunicándonos
dgale>How do you write your polynomials, ax^2+bx+c or c+bx+ax^2? Traditionally it seems the first way is most common. Why? The second would seem more natural. Don't we generally like to have things (like exponents) increasing from left to right? By contrast, for power series we have a_0+a_1x+a_2x^2+ . . ., (never. . .+a_2x^2+a_1x+a_0). Never say never. (c163) sum(a[k]*x^k,k,0,2) 2 (d163) a x + a x + a 2 1 0 (c164) taylor(%,x,0,2) = taylor(%,x,inf,0) 2 2 (d164)/T/ a + a x + a x + . . . = a x + a x + a + . . . 0 1 2 2 1 0 henry>It would be interesting to take a poll of symbolic algebraic manipulation (SAM) systems to see how they represent polynomials In some systems, however you want: (c165) (powerdisp:true,d163) 2 (d165) a + a x + a x 0 1 2 (c166) horner(%,x) (d166) a + x (a + a x) 0 1 2 (c167) (powerdisp:false,%) (d167) x (a x + a ) + a 2 1 0 thane>The trouble of learning emacs and LaTeX is repaid many times over in mathematical writing. Especially if your CAS(SAM) will do some TeXing for you: (c168) tex(d163 = %) % A[2]*X^2+A[1]*X+A[0] = X*(A[2]*X+A[1])+A[0] $$ a_{2}\,x^{2}+a_{1}\,x+a_{0}=x\,\left(a_{2}\,x+a_{1}\right)+a_{0} $$ --rwg PS That diagonal Cal\Nevada border--is it a great circle, rhumb line, or what? If the former, then its compass bearing varies from end to end. If the latter, it isn't straight.
RWG wrote:
PS That diagonal Cal\Nevada border--is it a great circle, rhumb line, or what? If the former, then its compass bearing varies from end to end. If the latter, it isn't straight.
It's whatever the United States Coast and Geodetic Survey produce when you ask them for a "straight" line between two points. See the 1980 U.S. Supreme Court case "California v. Nevada", 447 U.S. 125, at http://laws.findlaw.com/us/447/125.html --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.
participants (8)
-
David Gale -
David Wilson -
Henry Baker -
Marc LeBrun -
mcintosh＠servidor.unam.mx -
Michael Kleber -
R. William Gosper -
Richard Guy