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Discussion Groups / English Usage / January 2010A basic algebraic operation
Hello to all, while discussing the "n times bigger" subject I failed to find the proper English term for a basic algebraic operation which we call Aequivalenzumformung (literally: equivalence transformation) in German. In fact, there is an article in German wikipedia <http://de.wikipedia.org/wiki/%C3%84quivalenzumformung> but no version in English or any other language. The term describes a method to treat linear equations: Put an operand to both sides which doesn't touch the equivalence: 103 + x = 107 => 103 - 103 + x = 107 - 103 => x = 4 My question: How do you call this in English? Is there a term like "equivalent transformation" or the like? Thanks & Ciao, Paul Paul Schmitz-Josten skrev: > Aequivalenzumformung (literally: equivalence transformation) in German. Aequivalen-zum-formung ? (Nein, ich weiss) > The term describes a method to treat linear equations: > Put an operand to both sides which doesn't touch the equivalence: > 103 + x = 107 => 103 - 103 + x = 107 - 103 => x = 4 In Danish we do not have a specific name for it. We just say "subtract on both sides" or "move to the other side" which also works for multiplication. It is implied that the operator must change + <-> - * <-> /  SignatureBertel, Denmark Bertel Lund Hansen set the following eddies spiralling through the space-time continuum: > Paul Schmitz-Josten skrev: > [quoted text clipped - 15 lines] > + <-> - > * <-> / Sometimes in English we will say "cancel the t's" where t represents something that appears more than once in the equation, and dropping them from the equation does not affect it ("touch the equivalence" as you say). The equation must first be written in a form where t appears twice, and the cancellation is represented by drawing a line through each appearance. It is more commonly applied to factors appearing in the numerator and denominator of an expression, but it can be used either side of an = sign. To apply to your equation: 103 + x = 107 Split the 107: 103 + x = 103 + 4 Cancel the 103's (unfortunately I can't represent strikeout here) x = 4 Deliberately cancelling a disguised 0 from the numerator and denominator of a fraction, is the basis of much mathematical mumbo-jumbo. Signatureξ:) Proud to be curly Interchange the alphabetic letter groups to reply Wed, 13 Jan 2010 10:34:05 +0000 from Prai Jei <pvstownsend.zyx.abc@ntlworld.com>: > Sometimes in English we will say "cancel the t's" where t represents > something that appears more than once in the equation, and dropping them > from the equation does not affect it ("touch the equivalence" as you say). And every teacher will try top get you not to say that, because then students try to "cancel the t's" in t + 8 ----- = t + 3 t + 7 SignatureStan Brown, Oak Road Systems, Tompkins County, New York, USA http://OakRoadSystems.com Shikata ga nai... > while discussing the "n times bigger" subject I failed to find the > proper English term for a basic algebraic operation which we call [quoted text clipped - 10 lines] > My question: How do you call this in English? > Is there a term like "equivalent transformation" or the like? When I taught algebra back in the 1970s, I called this "doing the same thing to both sides" as the general principle involved in "solving equations." I don't remember any general name like you suggest. Not that it helps you, but I remember that talking about balance scales worked best to convey this somewhat nonintuitive procedure. If you start with two trays in balance (the equation), what changes will destroy the balance and which will preserve it? SignatureBest -- Donna Richoux >> while discussing the "n times bigger" subject I failed to find the >> proper English term for a basic algebraic operation which we call [quoted text clipped - 14 lines] > thing to both sides" as the general principle involved in "solving > equations." I was taught to "move something from one side of the equation to the other and change the sign" and it always seemed to be a rather arbitrary rule until I realized that it was equivalent to your much better way of putting it. When teaching I always used wording very similar to yours. > I don't remember any general name like you suggest. Nor do I. > Not that it helps you, but I remember that talking about balance scales > worked best to convey this somewhat nonintuitive procedure. If you start > with two trays in balance (the equation), what changes will destroy the > balance and which will preserve it? Signatureathel >>> while discussing the "n times bigger" subject I failed to find the >>> proper English term for a basic algebraic operation which we call [quoted text clipped - 24 lines] > >Nor do I. I recall two terms from years ago: "simplification" of an equation and "cancellation". Simplification can be done on one or both sides of an equation. Cancellation is one method of simplification in which the same thing is done to both sides. This explains simplification: http://www.mathleague.com/help/algebra/algebra.htm .... To find a solution for an equation, we can use the basic rules of simplifying equations. These are as follows: 1) You may evaluate any parentheses, exponents, multiplications, divisions, additions, and subtractions in the usual order of operations. When evaluating expressions, be careful to use the associative and distributive properties properly. 2) You may combine like terms. This means adding or subtracting variables of the same kind. The expression 2x + 4x simplifies to 6x. The expression 13 - 7 + 3 simplifies to 9. 3) You may add any value to both sides of the equation. 4) You may subtract any value from both sides of the equation. This is best done by adding a negative value to each side of the equation. 5) You may multiply both sides of the equation by any number except 0. 6) You may divide both sides of the equation by any number except 0. .... I think that 3 to 6 are methods of cancellation. SignaturePeter Duncanson, UK (in alt.usage.english) Peter Duncanson (BrE) in <cnfrk59614b0g4uetf15dju83r1prrp588@4ax.com>: >I recall two terms from years ago: "simplification" of an equation and >"cancellation". [quoted text clipped - 9 lines] > To find a solution for an equation, we can use the basic rules of > simplifying equations. These are as follows: <snip> "Rules of simplification" sounds nice to me. Thanks a lot, Paul > Peter Duncanson (BrE) in <cnfrk59614b0g4uetf15dju83r1prrp588@4ax.com>: > [quoted text clipped - 16 lines] > > Thanks a lot, I can't agree, though. Simplifying is what you do to an expression, such as by combining similar terms. To speak of "simplifying equations" I think is one person's invention. He might as well have written "rules for the solution of equations." It certainly does not *stand* for the "doing the same thing to both sides" concept. The only term I can think of that hasn't been mentioned yet is "isolate" -- to "isolate the variable" is the goal in all the manipulation, to get "x" by itself on one side of the equation and any other values on the other. It's not the answer to your question about doing the same thing to both sides, but it is a widely used term for the overall purpose. SignatureBest -- Donna Richoux > > Peter Duncanson (BrE) in <cnfrk59614b0g4uetf15dju83r1prrp588@4ax.com>: > > [quoted text clipped - 28 lines] > other. It's not the answer to your question about doing the same thing > to both sides, but it is a widely used term for the overall purpose. Good common sense, but not what the OP asked for. The concept of an equivalence transformation clearly implies a group of such transformations. Mentioning any one transformation in particular just doesn't do it. I still think the only answer is no, English speaking teachers have not gone to this level of pedantry, Jan Athel Cornish-Bowden skrev: > I was taught to "move something from one side of the equation to the > other and change the sign" and it always seemed to be a rather > arbitrary rule until I realized that it was equivalent to your much > better way of putting it. When teaching I always used wording very > similar to yours. I was given the good explanation when I learned this at school, and I gave the same careful explanation as a teacher. But afterwards we just used the short phrase when solving (un)equations. SignatureBertel, Denmark > Hello to all, > [quoted text clipped - 12 lines] > My question: How do you call this in English? > Is there a term like "equivalent transformation" or the like? Neither is there in Dutch. It would seem that German teachers are a bit more pedantic than those in other parts of the world, Jan >> Hello to all, >> [quoted text clipped - 18 lines] > >Jan In my limited mathematical experience, I have not come across a term that is used for these redundant operations. Strictly speaking, what you have is not a transformation: that make the German term highly idiomatic. > >> Hello to all, > >> [quoted text clipped - 23 lines] > have is not a transformation: that make the German term highly > idiomatic. If you want to be pedantic you can say that we are dealing here with the equivalence group of all forms of the equation under addition. Subtracting 103 is an equivalence transformation in the group. Why use three words when thirty will do? Jan >> >> Hello to all, >> >> [quoted text clipped - 28 lines] >of all forms of the equation under addition. >Subtracting 103 is an equivalence transformation in the group. That'll work, and nicely done. I'm sure every freshman in a high school algebra class will appreciate such ad hoc reasoning, too. >Why use three words when thirty will do? > >Jan It does fill in class time nicely, though. It is awkward to find that you've finished your lecture in twenty minutes, when the class expects an hour's worth of time for the money they spent on being there. >> >> Hello to all, >> >> [quoted text clipped - 29 lines] > > Why use three words when thirty will do? Don't forget about multiplying both sides by the same (non-zero) quantity. SignatureRoland Hutchinson He calls himself "the Garden State's leading violist da gamba," ... comparable to being ruler of an exceptionally small duchy. --Newark (NJ) Star Ledger ( http://tinyurl.com/RolandIsNJ ) > >> >> Hello to all, > >> >> [quoted text clipped - 31 lines] > > Don't forget about multiplying both sides by the same (non-zero) quantity. Sure, the multipliction groep too. And more, if you insist, Jan > Don't forget about multiplying both sides by the same (non-zero) quantity. Oh, you're allowed to multiply both sides of an equation by zero, but it wouldn't do you any good, because you'd only wind up with the statement that 0 = 0. It is, of course, dividing by zero that is forbidden. SignatureBest -- Donna Richoux Donna Richoux in <1jccyzp.1520vizmrd4lrN%trio@euronet.nl>: (Roland Hutchinson:) >> Don't forget about multiplying both sides by the same (non-zero) quantity. > >Oh, you're allowed to multiply both sides of an equation by zero, but it >wouldn't do you any good, because you'd only wind up with the statement >that 0 = 0. ...and that is why Roland's statement seems correct to me in the context of solving, simplifying or transforming equations.+0, -0, *0 "do no good" to this task ;-> >It is, of course, dividing by zero that is forbidden. IANAM. I don't think so - it only fails to return an unequivocal result. Ciao, Paul > Donna Richoux in <1jccyzp.1520vizmrd4lrN%trio@euronet.nl>: > [quoted text clipped - 8 lines] > solving, simplifying or transforming equations.+0, -0, *0 "do no good" to > this task ;-> It's true that I can't think offhand of a reason why you would want to add zero to both sides of an equation. But doing so would leave the equation unchanged, not leave you with the trivial statement that zero equals zero. > >It is, of course, dividing by zero that is forbidden. > > IANAM. I don't think so - it only fails to return an unequivocal result. It fails to return a useful or valid result, in the field of secondary school algebra, so students are told not to do it. Here's the reasoning as to why you can't divide by zero. Suppose N is a non-zero number, and you divide it by zero. Whatever N/0 is, let's call that X. So what do we know about X? From the relationship of division and multiplication, we know that X times 0 must equal N. But wait a minute, anything times 0 equals 0, and we started off saying that N was not zero. So there isn't any value of X that will make sense of the statement that N/0 = X. N/0 is considered to be an undefined, meaningless statement. Then we can look at the special case of 0/0 = X, but you can work out the consequences of that one. SignatureBest -- Donna Richoux Donna wrote on Sat, 16 Jan 2010 01:29:37 +0100: >> Donna Richoux in <1jccyzp.1520vizmrd4lrN%trio@euronet.nl>: >> [quoted text clipped - 9 lines] >> the context of solving, simplifying or transforming >> equations.+0, -0, *0 "do no good" to this task ;-> > It's true that I can't think offhand of a reason why you would > want to add zero to both sides of an equation. But doing so [quoted text clipped - 5 lines] >> IANAM. I don't think so - it only fails to return an >> unequivocal result. > It fails to return a useful or valid result, in the field of > secondary school algebra, so students are told not to do it. > Here's the reasoning as to why you can't divide by zero. > Suppose N is a non-zero number, and you divide it by zero. > Whatever N/0 is, let's call that X. So what do we know about [quoted text clipped - 4 lines] > the statement that N/0 = X. N/0 is considered to be an > undefined, meaningless statement. > Then we can look at the special case of 0/0 = X, but you can > work out the consequences of that one. The normal mathematical statement is that division by zero is undefined. SignatureJames Silverton Potomac, Maryland Email, with obvious alterations: not.jim.silverton.at.verizon.not > Donna wrote on Sat, 16 Jan 2010 01:29:37 +0100: > [quoted text clipped - 3 lines] > The normal mathematical statement is that division by zero is > undefined. And then someone brings up l'Hôpital's rule. SignatureEvan Kirshenbaum +------------------------------------ HP Laboratories |If to "man" a phone implies handing 1501 Page Mill Road, 1U, MS 1141 |it over to a person of the male Palo Alto, CA 94304 |gender, then to "monitor" it |suggests handing it over to a kirshenbaum@hpl.hp.com |lizard. (650)857-7572 | Rohan Oberoi http://www.kirshenbaum.net/ Evan wrote on Fri, 15 Jan 2010 17:19:01 -0800: >> Donna wrote on Sat, 16 Jan 2010 01:29:37 +0100: >> [quoted text clipped - 3 lines] >> The normal mathematical statement is that division by zero is >> undefined. > And then someone brings up l'Hôpital's rule. I'm not going to get into that but a good discussion is given at http://mathworld.wolfram.com/LHospitalsRule.html SignatureJames Silverton Potomac, Maryland Email, with obvious alterations: not.jim.silverton.at.verizon.not >> The normal mathematical statement is that division by zero is >> undefined. > > And then someone brings up l'Hôpital's rule. Ayaah--you can't fool me: that's not normal; it's tangents, innit. SignatureRoland Hutchinson He calls himself "the Garden State's leading violist da gamba," ... comparable to being ruler of an exceptionally small duchy. --Newark (NJ) Star Ledger ( http://tinyurl.com/RolandIsNJ ) > > Donna wrote on Sat, 16 Jan 2010 01:29:37 +0100: > > [quoted text clipped - 5 lines] > > And then someone brings up l'Hôpital's rule. A mistake, since that is not part of arithmetic, Jan >> Donna wrote on Sat, 16 Jan 2010 01:29:37 +0100: >> [quoted text clipped - 5 lines] > >And then someone brings up l'Hôpital's rule. And poles. SignatureRegards, Chuck Riggs, An American who lives near Dublin, Ireland and usually spells in BrE >> Donna Richoux in <1jccyzp.1520vizmrd4lrN%trio@euronet.nl>: >> [quoted text clipped - 13 lines] > equation unchanged, not leave you with the trivial statement that zero > equals zero. Typically, you'd do it if you didn't realize that that's what you're doing. Take "4-x = 4". The two ways to solve this are to subtract four from both sides and then multiply both sides by -1 or to add x to both sides and then subtract four from both sides. If you choose the second way, you find out when you're done that by adding x you added zero. Which is fine and helps you find the answer. >> >It is, of course, dividing by zero that is forbidden. >> [quoted text clipped - 15 lines] > Then we can look at the special case of 0/0 = X, but you can work > out the consequences of that one. That's the real problem, because it can cause you to lose solutions. If the thing you're solving is "x^2 = 2x" and you divide by x, you only get one of the two solutions, since the limit as x approaches zero of x^2/x is 2, not x. SignatureEvan Kirshenbaum +------------------------------------ HP Laboratories |On a scale of one to ten... 1501 Page Mill Road, 1U, MS 1141 |it sucked. Palo Alto, CA 94304 kirshenbaum@hpl.hp.com (650)857-7572 http://www.kirshenbaum.net/ >>> Donna Richoux in <1jccyzp.1520vizmrd4lrN%trio@euronet.nl>: >>> [quoted text clipped - 45 lines] >only get one of the two solutions, since the limit as x approaches >zero of x^2/x is 2, not x. This is eighth or ninth-grade stuff, FCS. SignatureRegards, Chuck Riggs, An American who lives near Dublin, Ireland and usually spells in BrE >>That's the real problem, because it can cause you to lose solutions. >>If the thing you're solving is "x^2 = 2x" and you divide by x, you >>only get one of the two solutions, since the limit as x approaches >>zero of x^2/x is 2, not x. > > This is eighth or ninth-grade stuff, FCS. Possibly now. We didn't get into limits until tenth or eleventh and derivatives and l'Hôpital's rule until twelfth. Earlier, we were just told "It's undefined". SignatureEvan Kirshenbaum +------------------------------------ HP Laboratories |It is error alone which needs the 1501 Page Mill Road, 1U, MS 1141 |support of government. Truth can Palo Alto, CA 94304 |stand by itself. | Thomas Jefferson kirshenbaum@hpl.hp.com (650)857-7572 http://www.kirshenbaum.net/ >>>That's the real problem, because it can cause you to lose solutions. >>>If the thing you're solving is "x^2 = 2x" and you divide by x, you [quoted text clipped - 6 lines] >derivatives and l'Hôpital's rule until twelfth. Earlier, we were just >told "It's undefined". Forget my nasty response, yesterday, Evan. In my haste, I misread your equation. SignatureRegards, Chuck Riggs, An American who lives near Dublin, Ireland and usually spells in BrE Donna Richoux in <1jcdzx6.1wfjn04rd0youN%trio@euronet.nl>: >> (Roland Hutchinson:) >> >> Don't forget about multiplying both sides by the same (non-zero) quantity. [quoted text clipped - 11 lines] >equation unchanged, not leave you with the trivial statement that zero >equals zero. only if you omit *0 from my statment. (dividing by zero) >> IANAM. I don't think so - it only fails to return an unequivocal result. > >It fails to return a useful or valid result, in the field of secondary >school algebra, so students are told not to do it. Here's the >reasoning as to why you can't divide by zero. <snip> As I said: I am not a mathematician (sp?) - IANAM. Therefore I'm not going to follow you into that OT maze. ;-> Ciao, Paul > > Donna Richoux in <1jccyzp.1520vizmrd4lrN%trio@euronet.nl>: > > [quoted text clipped - 13 lines] > equation unchanged, not leave you with the trivial statement that zero > equals zero. You would want the addition group generated by the equation to have a unit element. > > >It is, of course, dividing by zero that is forbidden. > > [quoted text clipped - 11 lines] > the statement that N/0 = X. N/0 is considered to be an undefined, > meaningless statement. Equivalently, division is repeated subtraction. Repeated subtraction of zero will never reduce N to something smaller than N, so the result cannot be defined. Jan Wed, 13 Jan 2010 09:20:35 +0100 from Paul Schmitz-Josten <alossola@web.de>: > Hello to all, > [quoted text clipped - 12 lines] > My question: How do you call this in English? > Is there a term like "equivalent transformation" or the like? "Subtracting the same number from both sides". Or, in upper-lavel math classes, "adding negative 103 to both sides". There is a term "equivalence transformation" but I don't think it would apply here. I don't know of a formal term for "doing the same thing to both sides of an equation". SignatureStan Brown, Oak Road Systems, Tompkins County, New York, USA http://OakRoadSystems.com Shikata ga nai... Paul Schmitz-Josten: >> The term describes a method to treat linear equations: >> Put an operand to both sides which doesn't touch the equivalence: [quoted text clipped - 3 lines] >> My question: How do you call this in English? >> Is there a term like "equivalent transformation" or the like? Stan Brown: > "Subtracting the same number from both sides". Or, in upper-lavel > math classes, "adding negative 103 to both sides". Agreed. There is no general phrase for this; we just say "do the same thing to both sides", or more likely, we speak of the specific change made to both sides. For example: "add X to both sides", "subtract X from both sides", "multiply both sides by X", "divide both sides by X", "negate both sides", "square both sides", "take the square root of both sides", "raise both sides to the Xth power", "take the cosine of both sides", whatever. In all cases X might be an expression. Incidentally, note how in English I converted the expression X to an ordinal Xth by adding "th". In German would you write "X.", or is that use of "." reserved for numerals? In other branches of the threads, some people spoke of "simplifying" and "canceling". Those things are what you are doing in the *second* transformation, when you replace "103 - 103 + x = 107 - 103" with "x = 4". "Simplifying" is the general expression; "canceling" means simplifying by removing terms that add to 0 or factors that multiply to 1. (Note that "terms" are added or subtracted while "factors" are multiplied.) In this case you are canceling when you delete the "103 - 103" part. Of course, in practice when you do one of these transformations and then simplify, you don't usually write out the intermediate step. You go directly from "103 + x = 107" to "x = 4" and assume people can follow what's going on. In that case you are also likely to describe the operation as "simplifying". In the specific case of an equation like "xt = 4t", you solve it -- if t is not 0 -- by first dividing both sides by t to get "xt/t = 4t/t" and then canceling the t's on each side to get "x = 4". Since the middle step is typically omitted, this may also be described as canceling the t's in the original equation. This reminds me: 64 / 16 = 4 / 1 = 4. See, 6 is not 0, so you can just cancel the 6's, right? :-) SignatureMark Brader First, the next time you buy a house, get one that msb@vex.net costs exactly $100,000. It makes the math easier. Toronto -- John Gilmer My text in this article is in the public domain. Mark Brader filted: >Agreed. There is no general phrase for this; we just say "do the same >thing to both sides", or more likely, we speak of the specific change [quoted text clipped - 3 lines] >sides", "raise both sides to the Xth power", "take the cosine of both >sides", whatever. In all cases X might be an expression. Up to the first two examples, that's from Euclid's "common notions"....r SignatureA pessimist sees the glass as half empty. An optometrist asks whether you see the glass more full like this?...or like this? Mark Brader: >> Agreed. There is no general phrase for this; we just say "do the same >> thing to both sides", or more likely, we speak of the specific change >> made to both sides. For example: "add X to both sides", "subtract X >> from both sides"... R.H. Draney: > Up to the first two examples, that's from Euclid's "common notions". Smart fellow, that Euclid. SignatureMark Brader, Toronto | "But going repeatedly back and forth in time is msb@vex.net | cheating. Anybody can do that!" --Paul Kriha Mark Brader filted: >Mark Brader: >>> Agreed. There is no general phrase for this; we just say "do the same [quoted text clipped - 6 lines] > >Smart fellow, that Euclid. That's why they named a street in Cleveland for him....r SignatureA pessimist sees the glass as half empty. An optometrist asks whether you see the glass more full like this?...or like this? Thu, 14 Jan 2010 11:48:58 -0600 from Mark Brader <msb@vex.net>: > This reminds me: 64 / 16 = 4 / 1 = 4. See, 6 is not 0, so you can just > cancel the 6's, right? :-) The one I learned is 95/19. :-) SignatureStan Brown, Oak Road Systems, Tompkins County, New York, USA http://OakRoadSystems.com Shikata ga nai... |
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