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#1 2009-08-12 11:44:20

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,535

Define "Simplification"

I am trying to make a master page on Simplification, and want to give clear and easy instructions to learners.

But I have stumbled at the very start, how to define "Simplest form"

Is it "The least number of terms" or "No Parentheses"?

But I would prefer this: (x+1)(x+2)
To this: x[sup]2[/sup] + 2x + 2

And which of these is better:

This: 2x + 2/(x-3)
Or this: (2x[sup]2[/sup] - 6x + 2)/(x-3)

Is this: 1/√2
Better than this: √2/2 (rationalized denominator) ... ?

I can give general advice (combine like terms, extract common factors), but an overall definition eludes me.

Help me make it "simple" for students smile


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#2 2009-08-13 00:54:40

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,237

Re: Define "Simplification"

Hi MathsisFun;
I don't know of an overall definition, I don't think there is one.

MathsisFun wrote:

But I would prefer this: (x+1)(x+2)
To this: x^2 + 2x + 2

I would prefer the factored form also. It gives more info and uses less operations.

This: 2x + 2/(x-3)
Or this: (2x^2 - 6x + 2)/(x-3)

This one really looks like the first is simpler.

Is this: 1/√2
Better than this: √2/2 (rationalized denominator) ... ?

Like the √2/2 better and it's a bit more numerically stable.

Last edited by bobbym (2009-08-13 01:00:26)


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#3 2009-08-14 12:05:23

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,535

Re: Define "Simplification"

Thanks bobby. Good reply.

Anyone else have opinions on this?


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#4 2009-08-14 18:36:52

Identity
Member
Registered: 2007-04-18
Posts: 934

Re: Define "Simplification"

hmm it's kinda hard to define...
Every form is useful for some purpose... and I guess that would be my definition: The simplest form is the form which is most useful for a specific purpose, so the notion is relative.

For example, you could argue that

is more simplified than
, but it would be easier to integrate the second rather than the first.

And take for example

If you wanted the asymptotes you would go for

but if you wanted to integrate you would go for

Even with

, while most people would say that
is the most simplified form, that's only because it's easier to evaluate. If we aren't worried about evaluating, but simply math theory, then there's there's no reason why we couldn't say
is more simplified.

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#5 2009-08-16 13:58:45

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,535

Re: Define "Simplification"

Good points.

How about:

Simpler: easier to use


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#6 2009-08-16 14:11:45

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,237

Re: Define "Simplification"

Hi MathsisFun;

Same problem as simpler. How do you define easier to use? Less operations, less terms, shorter in length, smaller numbers? The best simplification depends on what you are trying to do.

Look at this:


or

Now, supposing you wanted to evaluate them numerically so you would know how much something costs.
The first one certainly is shorter and simpler but the second one which is a truncated Taylor series of e^x allows you to calculate values like e^(1/2) in your head. While the first one is useless for computation. Which is better 3.14159265... or the symbol π? Depends on what you require. I think you can only suggest some general guidelines.

Last edited by bobbym (2009-08-16 14:22:17)


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#7 2009-08-16 16:38:48

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,535

Re: Define "Simplification"

Well ... I have put some of these ideas down in concrete form, so we have something to discuss.

What do you think of: Simplifying in Algebra


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#8 2009-08-16 18:55:55

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,237

Re: Define "Simplification"

Hi MathsisFun;

Looked at the page, looks good to me. Maybe somebody else has an opinion about it.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Offline

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