Which Method of Cancellation is Most Effective in Mathematical Problem Solving?

[PhysicsHelp12]

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)


http://img183.imageshack.us/img183/6484/mathegcd6.jpg


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



http://img103.imageshack.us/img103/1507/matheg2ti5.jpg


also with something almost trivial like:

http://img183.imageshack.us/img183/7785/matheg3pt0.jpg


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.

 

[symbolipoint]

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.

 

[PhysicsHelp12]

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 I am apply a^m/a^n=a^(m-n) ...but I am not brining everything in this form EXACTLY ...like the other rules,
the notation is still different

 

[PhysicsHelp12]

i basically just want to know what most people do

usually

 

[cristo]

i basically just want to know what most people do

usually

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.

 

[Mentallic]

Memorising the formal steps of - let's 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?

Which method of cancellation do you do personally

and which is better?

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?

 

[symbolipoint]

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).

 

[PhysicsHelp12]

Memorising the formal steps of - let's 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?

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 don't and just skip it, then youre not actually doing the rule

youre just seeing the answer...

 

[cristo]

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

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.

 

[PhysicsHelp12]

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.

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, there's no writing anything down

 

[PhysicsHelp12]

I don't 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) isn't 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

 

[cristo]

I don't 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)

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!

 

[PhysicsHelp12]

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?

 

[Mentallic]

I'm not entirely sure why you're so bothered by this. Do what you feel most comfortable doing!

Amen!

 

[PhysicsHelp12]

Yea, but

how do you do the second with method (2)

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

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

i don't want to be the only one doing it like this ...even if its only because I am confortable

 

[Mentallic]

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 subconscious 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!

 

[PhysicsHelp12]

Ok, I actually like it SOMETIMES

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



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

am I the only one who multiples it up first when there's things like that?

 

[Mentallic]

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

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

$$x^2\frac{3x-1}{x}=\frac{x^2(3x-1)}{x}=x(3x-1)$$ ?

am I the only one who multiples it up first when there's things like that?

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!

 

[PhysicsHelp12]

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

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

$$x^2\frac{3x-1}{x}=\frac{x^2(3x-1)}{x}=x(3x-1)$$ ?


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!

Ok, do you 'decorate' it that way ...

or what do you do in your head?

 

[Mentallic]

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 won't be getting anywhere?

Simplify:

Q1) $$a^2\frac{b}{a}$$

Q2) $$\frac{c^2d}{c}$$

Solution:

A1) $$a^2\frac{b}{a}=\frac{a^2b}{a}=ab$$

A2) $$\frac{c^2d}{c}=c^2\frac{d}{c}=cd$$

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

 

[PhysicsHelp12]

Ok, yea

so it's better to just 'see it'

and if I can't then do that middle step?

is that normal then

 

[PhysicsHelp12]

I think I do just see it ...

so then that's normal?

I only do the intermediate step becaue I thought it was proper to do

 

[Mentallic]

Ok, yea

so it's better to just 'see it'

and if I can't then do that middle step?

is that normal then

Yes that's what I've been saying the whole time! Glad to see it's sinking in though

I think I do just see it ...

so then that's normal?

I only do the intermediate step becaue I thought it was proper to do

No you can skip doing these pointless 'steps' which aren't really steps since the actual definition of a step is to advance in the desired direction. The decoration is unnecessary.

 

[PhysicsHelp12]

Ok thank you

 

[HallsofIvy]

It is remarkable to me that this went on for 24 posts.

 

[Redbelly98]

Which method of cancellation do you do personally

and which is better?

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

Since it's easier for you to think in terms of method (1), and it's a valid method, then it's perfectly okay to continue using method (1).

 

[Mentallic]

It is remarkable to me that this went on for 24 posts.

A strong and persuasive debate usually stretches to around 30 doesn't it?