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Which Method do you Use?

  1. Jan 17, 2009 #1
    Which method of cancellation do you do personally

    and which is better?

    I am worried that I am the only one who does (1)

    I am in 2nd year pure math right now and I am really worried I am

    doing this weird (even though I always get the correct answer)


    [​IMG]


    There is the eg ...When you have something multiplying the fraction,

    do you usually multiply it all up on top of the numerator and then canel

    or do you like to just cancel from the start like in (2)



    I think I use (1) because if I had something where I had to cancel out exponents too
    it would seem awkard to do it (2) to me eg



    [​IMG]


    also with something almost trivial like:

    [​IMG]


    I am not sure which MOST people do in their heads, it seems frustrating and trivial to do it like (1) ...but if I dont, and do it like (2) then how am I being consistent ...



    thanks for your help
    I did not explain myself last time, and its been frustrating me all week.
     
    Last edited: Jan 17, 2009
  2. jcsd
  3. Jan 17, 2009 #2

    symbolipoint

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    What is confusing you? Your methods "1" and "2" are equivalent. Can you FORMALLY express what property of numbers you are applying? You learned these properties in Algebra 1.

    For a number k not equal to zero and another number h not equal to zero (or should I specify they are Real numbers?), (k*h)/h = k
    Also, you understand that h/h = 1.
     
  4. Jan 17, 2009 #3
    Symbol Point, I get the rules, but

    (k*h)/h

    isnt EXACTLY the same was

    what's written above...

    the notation is still different even though I know they mean the same thing


    I just want to know what one most people apply








    whats confusing me is ...with every other rule, there seems to be one universal rule to go with it that can be put into symbols

    but for this, people just do it 2 ways or more ,---


    technically Im apply a^m/a^n=a^(m-n) ...but im not brining everything in this form EXACTLY ...like the other rules,
    the notation is still different
     
  5. Jan 17, 2009 #4
    i basically just want to know what most people do

    usually
     
  6. Jan 17, 2009 #5

    cristo

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    I would imagine most people do it the second way. It doesn't matter if you put in extra steps, though, since the methods are exactly the same.
     
  7. Jan 17, 2009 #6

    Mentallic

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    Memorising the formal steps of - lets say for example - constructing an induction proof would be better use of your time. The exact method you take in manipulating and simplifying your algebraic expressions is hardly important.
    There are numerous ways to express the same thing in mathematics, so why not take advantage of this and apply whichever method works best in each specific situation?

    There is no 'better', but there is 'more time efficient' and 'less prone to error'. All you need to do is weigh up these two against each other. While you waste time writing an extra step which most others (including myself) would skip, is it worth it to make the expression more clear for yourself?
     
  8. Jan 17, 2009 #7

    symbolipoint

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    PhysicsHelp12, what I stated in post #2 is a couple of generalities for Real numbers. The form really is the same (the first generality). Maybe part of what confuses you is that one of the factors, "e^(x)", is itself a variable expression, which you could just as well call by any variable you like, such as "h", so that h=e^(x).
     
  9. Jan 17, 2009 #8

    I don't see how it's an extra step at all...

    seeing as how you either multiply first and then cross out

    or cross out first and then multiply -both have 2 steps

    and you have to use the (1) one if you want to actually do the exponent rule a^M/a^N

    like I gave an eg of ...if you dont and just skip it, then youre not actually doing the rule

    youre just seeing the answer...
     
  10. Jan 17, 2009 #9

    cristo

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    It's an extra thing to write down. In your first example, you would not write the term with the 1 on the denominator: you would simple jump from your first expression to your last.
     
  11. Jan 17, 2009 #10

    I know that...I was just showing the full sequence of steps

    really there is only 2 in each case though

    for (1) multiply first, cross out

    (2) cross out first, then multiply

    both are done in your head, theres no writing anything down
     
  12. Jan 17, 2009 #11
    I dont get how people say (2) is better though,

    when it really only works for crossing out common factors (not if you have exponents too to cancel off)

    (1) works for all of them ...

    and (2) isnt really a rule ...its just a shortcut

    (1) is actually following the rules


    seems like people learned it in high school and never broke the habbit I still think (1) is better but w/e
     
  13. Jan 17, 2009 #12

    cristo

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    I don't get what you mean. Once you learn that a term h, say, is on the numerator unless otherwise stated, then it really doesn't matter whether you draw a line under it or not. Your two methods really are not different: I'm not entirely sure why you're so bothered by this. Do what you feel most comfortable doing!
     
  14. Jan 17, 2009 #13
    How do you do the second example without multiplying it first?

    Why would you do this...

    is it not easier to just multiply first for the second eg?
     
  15. Jan 17, 2009 #14

    Mentallic

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    Amen!
     
  16. Jan 18, 2009 #15
    Yea, but

    how do you do the second with method (2)

    ...dont most people use (1) ..

    because if its a mix like that im fine with it ...but if its not then im a bit worried

    i dont want to be the only one doing it like this ...even if its only because im confortable
     
  17. Jan 18, 2009 #16

    Mentallic

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    I've lost you there.

    Seriously though, I'm puzzled about what you actually want us to say/do for you. Maybe you want to make a poll on which method each voter prefers? I doubt this will be of any help though because it has already been clearly set out that the shorter, second choice is the more favourable method.

    If you are afraid that your subconcious reasoning will backfire one day, then by all means teach yourself to get into the habit of using the more appropriate method (whichever you feel that may be). I'm not saying to stress more about 'learning' something new; just use that method whenever these problems are encountered.

    Going to university is an opportunity to make the best out of your transition from a teenager to a young adult. With this transition comes more responsibility and can be your ticket to freedom. You must learn to make decisions for yourself sometimes!
     
  18. Jan 18, 2009 #17
    Ok, I actually like it SOMETIMES

    but I only see how to use it for crossing off *common factors*



    now when im doing exponents like a^m/a^n (as the second picture I posted)

    am I the only one who multiples it up first when theres things like that?
     
  19. Jan 18, 2009 #18

    Mentallic

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    Yes... err... no! um.... what was the question again?

    Do you mean the second picture in your very first post?

    [tex]x^2\frac{3x-1}{x}=\frac{x^2(3x-1)}{x}=x(3x-1)[/tex] ?

    You are not multiplying up, down, or even inside out! The first expression is equivalent to the second expression without any manipulation. In other words, all you did was reorganize (some might call it decorating). If this is seriously eating you up inside, do what any rational being would do. Weigh up the pros and cons of using each method, and come to a personal decision which satisfies YOU!
     
  20. Jan 18, 2009 #19
    Ok, do you 'decorate' it that way ...

    or what do you do in your head?
     
  21. Jan 18, 2009 #20

    Mentallic

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    No I don't take that extra step, seeing as both expressions are equivalent and I realize they are the same. Why change it around if I know I wont be getting anywhere?

    Simplify:

    Q1) [tex]a^2\frac{b}{a}[/tex]

    Q2) [tex]\frac{c^2d}{c}[/tex]

    Solution:

    A1) [tex]a^2\frac{b}{a}=\frac{a^2b}{a}=ab[/tex]

    A2) [tex]\frac{c^2d}{c}=c^2\frac{d}{c}=cd[/tex]

    Do you see how pointless the in-between step is?
     
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