The Mathematical Proof Thread

[Whateverist]

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That seems like more a matter of definitions and making conventions compatible.

If you think of x^3: as 1•x•x•x

and x^2 as: 1•x•x

and x^1 as: 1•x

then x^0 should be simply: 1

Note that 3 isn't a factor in x^3 any more than 2 is a factor in x^2.  So there is no reason 0 should be a factor in x^0, the usual worry.  Note that every factorization includes 1 trivially.  x^1 does too.  x^0 = 1 because there are no factors of x in it at all.  But every factorization includes 1, trivially.

There are better justifications for this, but I found this way the most satisfying to students.

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heya, hoya, hoo, I'm not reading this until I give it some thought myself but thanks, I'll read this when I'm done crying

My apologies.  I would have used hide tags but I didn't think of this as a proof.

Suggestion:  Lets use hide tags after the statement of the proposition if we post any more proofs so that others can have a go at it first.

[deleteduser12345]

heya, hoya, hoo, I'm not reading this until I give it some thought myself but thanks, I'll read this when I'm done crying

My apologies.  I would have used hide tags but I didn't think of this as a proof.

Suggestion:  Lets use hide tags after the statement of the proposition if we post any more proofs so that others can have a go at it first.

It's cool :p

[Kernel Sohcahtoa]

My kid is studying some stuff in her 8th grade advanced honors algebra class. I'll have to run this by her to see if she understands. Once you start throwing letters into a math equation, I get lost. Letters don't belong with numbers for us dumb folk.

In high school and in college, I was actually not interested in math at all.  However, two years ago, I became interested in it and have taught myself (no classes or formal education; I'm simply an independent learner, nothing more) high school algebra, pre-calc, trig, calc I,II,III (I absolutely loved the u substitution), differential equations (odes w/ a brief intro to pdes), elementary linear algebra, and discrete math (I'm currently learning this). My point in making this recollection is that with the exception of basic linear algebra and discrete math, I was definitely more focused on the numerical and computational aspects of subjects, rather than gaining a true appreciation for the underlying theory. Hence, I was too grounded in computational thinking, and I can tell you that it is entirely normal to be thrown off by letters, as they represent a shift in thinking (from the specific to the general) which takes time to properly cultivate.
 
As a result, when studying proofs (at least in the beginning) it may be helpful to work out a few numerical examples just to see the concept being concretely illustrated. For example 2 is even because, 2=2*1 (one is an integer).  Likewise, 4 and 6 are even because they can be written as 4=2*2 and 6=2*3, 2 and 3 are integers.  Hence, this process can be extended to a more general understanding of even numbers: an integer n is even if n=2a for some integer a (the definition of an even number). Hence, all the letters do is acknowledge what we already know to be true in individual cases: it extends those known facts and connects them to a broader general theory.  Once you get more acclimated with proofs, then you can often do what we just did above in reverse: prove a theorem in more general terms via definitions, lemmas, or other theorems and then reinforce that general understanding with specific examples and cases in order to gain a complete understanding of the mathematical concepts.

[robvalue]

Another way of looking at why x^0 = 1

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From the rules of powers, x^(m-n) = x^m / x^n

for example 8 = 2^3 = 2^(5-2) = 2^5 / 2^2 = 32/4 = 8

x^0 = x^(1-1) = x^1 / x^1 = 1

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[deleteduser12345]

My kid is studying some stuff in her 8th grade advanced honors algebra class. I'll have to run this by her to see if she understands. Once you start throwing letters into a math equation, I get lost. Letters don't belong with numbers for us dumb folk.

In high school and in college, I was actually not interested in math at all.  However, two years ago, I became interested in it and have taught myself (no classes or formal education; I'm simply an independent learner, nothing more) high school algebra, pre-calc, trig, calc I,II,III (I absolutely loved the u substitution), differential equations (odes w/ a brief intro to pdes), elementary linear algebra, and discrete math (I'm currently learning this). My point in making this recollection is that with the exception of basic linear algebra and discrete math, I was definitely more focused on the numerical and computational aspects of subjects, rather than gaining a true appreciation for the underlying theory. Hence, I was too grounded in computational thinking, and I can tell you that it is entirely normal to be thrown off by letters, as they represent a shift in thinking (from the specific to the general) which takes time to properly cultivate.
 
As a result, when studying proofs (at least in the beginning) it may be helpful to work out a few numerical examples just to see the concept being concretely illustrated. For example 2 is even because, 2=2*1 (one is an integer).  Likewise, 4 and 6 are even because they can be written as 4=2*2 and 6=2*3, 2 and 3 are integers.  Hence, this process can be extended to a more general understanding of even numbers: an integer n is even if n=2a for some integer a (the definition of an even number). Hence, all the letters do is acknowledge what we already know to be true in individual cases: it extends those known facts and connects them to a broader general theory.  Once you get more acclimated with proofs, then you can often do what we just did above in reverse: prove a theorem in more general terms via definitions, lemmas, or other theorems and then reinforce that general understanding with specific examples and cases in order to gain a complete understanding of the mathematical concepts.

You taught yourself so much in two years? That's quite awesome, damn.

[A Handmaid]

In high school and in college, I was actually not interested in math at all.  However, two years ago, I became interested in it and have taught myself (no classes or formal education; I'm simply an independent learner, nothing more) high school algebra, pre-calc, trig, calc I,II,III (I absolutely loved the u substitution), differential equations (odes w/ a brief intro to pdes), elementary linear algebra, and discrete math (I'm currently learning this). My point in making this recollection is that with the exception of basic linear algebra and discrete math, I was definitely more focused on the numerical and computational aspects of subjects, rather than gaining a true appreciation for the underlying theory. Hence, I was too grounded in computational thinking, and I can tell you that it is entirely normal to be thrown off by letters, as they represent a shift in thinking (from the specific to the general) which takes time to properly cultivate.
 

Could you tell me what material you used to learn Calc II and III and differential equations?

[A Handmaid]

Also, the proof that an irrational number to an irrational number can equal a rational number is one of my favorites.

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Proof:
Consider A=sqrt(2)^sqrt(2). 
If A is rational, we are done. 
If A is irrational, 
A^sqrt(2)=(sqrt(2)^sqrt(2))^sqrt(2)                      (x^y)^z=x^(y*z)
A^sqrt(2)=sqrt(2)^(sqrt(2)*sqrt(2))
A^sqrt(2)=sqrt(2)^2
A^sqrt(2)=2
And we are done.

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[Kernel Sohcahtoa]

In high school and in college, I was actually not interested in math at all.  However, two years ago, I became interested in it and have taught myself (no classes or formal education; I'm simply an independent learner, nothing more) high school algebra, pre-calc, trig, calc I,II,III (I absolutely loved the u substitution), differential equations (odes w/ a brief intro to pdes), elementary linear algebra, and discrete math (I'm currently learning this). My point in making this recollection is that with the exception of basic linear algebra and discrete math, I was definitely more focused on the numerical and computational aspects of subjects, rather than gaining a true appreciation for the underlying theory. Hence, I was too grounded in computational thinking, and I can tell you that it is entirely normal to be thrown off by letters, as they represent a shift in thinking (from the specific to the general) which takes time to properly cultivate.
 

Could you tell me what material you used to learn Calc II and III and differential equations?

Yes, sir.  For the entire calculus sequence, I used Calculus 10th edition by Ron Larson and Bruce Edwards.  This book also has a supplemental website called calcchat.com, which elaborates on all of the odd exercises (this was very useful).  This text book covers differential calculus through and including multi-variable and vector calculus.  However, a lot of the credit goes to the website Paul's Online notes.  Paul's notes provided an outstanding instructional template for learning calculus, as he goes into the details and thoroughly works the problems along with providing many practice problems (he explains these very well too).  Paul's notes was invaluable for Calc II.  The series and sequences chapter in the Larson text was a bit sparse, but Paul's notes did a great job of explaining index shifts and clearly explained why all the various tests of convergence and divergence worked.  In addition, the following resources were also useful:

mathispower4u

Professorrobbob

Krista King

In addition, I also used Paul's notes to learn differential equations . However, Paul only has the example problems in his notes (which were still amazing and extremely useful) and no additional practice problems like he had for calculus.  As a supplemental aid for working more problems, I purchased a Schaum's differential equations outline 4th edition by Richard Bronson and Gabriel B. Costa.  Hence, Paul's notes made the desire to learn differential equations a reality (I loved Laplace transforms).  I hope these resources may be of use to you Handmaid. Thanks for your inquiry. Live long and prosper

[Kernel Sohcahtoa]

Handmaid, for differential equations, I found that this video explained the concept of a phase portrait very well.  IMO, she explained this more clearly than Paul did in his notes. Good luck with your intellectual endeavors, sir, and thank you for posting your beautiful proof

[bennyboy]

Cleary, the odds are 1:1.  I assert this, can't prove it.

Boru

Well, it's easily proven. It could be true, or not, so it's clearly fifty fifty.

. . . how X-tians see the BOP.