16% of threads on the first page are silly threads about percents

[UnderstandForgive]

This is an old one, but I still like it:

a = b
a^2 = ab
a^2 + a^2 = ab + a^2
2a^2 = a^2 + ab
2a^2 - 2ab = a^2 + ab - 2ab
2a^2 - 2ab = a^2 - ab
2(a^2 - ab) = 1(a^2 - ab)
2 = 1

Find the error.

The error was when you wrote the last part when you wrote "2 = 1"
2a^2 - 2ab = a^2 - ab is like saying 0=0
2(a^2 - ab) = 1(a^2 - ab) "2*0 = 1*0" then divide the zeros. Oh wait you can't do that, it doesn't tell you anything.

When you divide both terms you better make sure that you are not dividing by 0.

Also you can make many numbers equal one another by this method
42(a^2 - ab) = 1(a^2 - ab)
42 = 1

 

[Resha Caner]

I don't think it's an error, so much as I think it's an optical illusion.

No. There is an actual error.

 

[AV1611VET]

No. There is an actual error.

Is it: multiplying by zero?

 

[Resha Caner]

When you divide both terms you better make sure that you are not dividing by 0.

Yep, you got it. There is a division by 0, which is invalid. What makes it hard to see is that the division is implicit, not explicit. It is easy for people to trivialize the first step (a = b), but it is not trivial. One needs to fully understand the implications of making such statements.

This is a thing I harp on all the time - especially in this forum: know your initial conditions, know your assumptions. And I mean know them.

So what about Gabriel's trumpet? Is that intuitive? As my calculus teacher put it: Does it make sense that if you filled the trumpet with paint, you wouldn't have enough to paint the trumpet?

 

[AV1611VET]

There is a division by 0, which is invalid.

If 7/7 =1.

If 20%/20% = 1.

If marshmallows/marshmallows = 1.

Does 0/0 = 1?

 

[leftrightleftrightleft]

This is an old one, but I still like it:

a = b
a^2 = ab
a^2 + a^2 = ab + a^2
2a^2 = a^2 + ab
2a^2 - 2ab = a^2 + ab - 2ab
2a^2 - 2ab = a^2 - ab
2(a^2 - ab) = 1(a^2 - ab)
2 = 1

Find the error.

a^2 - ab = 0

Can't divide by zero.

 

[leftrightleftrightleft]

If 7/7 =1.

If 20%/20% = 1.

If marshmallows/marshmallows = 1.

Does 0/0 = 1?

No it's undefined. It's a non-number.

If 0/0 = 1, then you can make any number equal any other number, which makes no sense.

 

[Resha Caner]

If 7/7 =1.

If 20%/20% = 1.

If marshmallows/marshmallows = 1.

Does 0/0 = 1?

No. It's possible you were a victim of "process" teaching - something I despise (though I have to admit it has its place). In process teaching, students are taught to memorize steps for solving certain types of problems rather than gaining an understanding of the axioms and theorems. Such a method leads to the type of question you ask: This is the process. Why isn't it always true?

It's pretty rare for a process to work 100% of the time.

So, not all numbers were created equal (though I believe certain American liberals have begun a campaign to stop discriminating against numbers. You know, irrational numbers should be allowed to marry each other and also join the integer club. Oh, and they don't believe in imaginary or transcendent numbers). Ha! That was fun! Sorry, back to the topic ...

All numbers are not created equal. The 5 most important numbers in mathematics are 1, 0, pi, e, and i. Those numbers do some things that other numbers don't do.

So, think about what division is, not the process for division. Division breaks things into parts. How can you take a thing and break it up so that is has no parts? In a sense, you're trying to turn something into nothing. That is what you are trying to do when you divide by zero ... or at least that's the best layman's explanation I can give. The issue is dealt with more rigorously in calculus (series, limits, L'Hopital's rule, etc.).

- - -

Rigor is another thing people don't seem to learn from process teaching. Rigor is very important to mathematics. As an example, your marshmallow example lacks rigor. Not only are marshmallows not numbers, but there are different classes of marshmallows. So, if I were to equate numbers to marshmallows based on mass (rather than the unit assignment I assume you intended), then the answer is different.

Suppose 1 regular marshmallow equals the mass of 4 miniature marshmallows. Then, the division problem you suggested could yield an answer of 4 (among others). Rigor (in part) means exhaustive definition

 

[Resha Caner]

Is no one going to bite on Gabriel's trumpet, or is the topic just too boring?

 

[lesliedellow]

Lies, damn lies and statistics.

 

[lesliedellow]

Another one I like is Gabriel's trumpet:

Gabriel's Horn - Wikipedia, the free encyclopedia

At least it can be described in terms of a formula, and visualised - sort of, which is more than can be said for the snow flake curve.

 

[Resha Caner]

At least it can be described in terms of a formula, and visualised - sort of, which is more than can be said for the snow flake curve.

Do you mean the fractal curve or something else?

 

[lesliedellow]

Do you mean the fractal curve or something else?

I mean the fractal curve.

I remember many years ago expressing doubts to a university lecturer about whether the thing actually existed. You couldn't write down a formula for it, you couldn't draw it, and it was extremely difficult to see how it could be handled mathematically.... in exactly what sense did it exist?

Usually in mathematics, even if an abstraction can't be visualised, you can still give it a formulaic expression.

 

[Resha Caner]

I mean the fractal curve.

I remember many years ago expressing doubts to a university lecturer about whether the thing actually existed. You couldn't write down a formula for it, you couldn't draw it, and it was extremely difficult to see how it could be handled mathematically.... in exactly what sense did it exist?

Usually in mathematics, even if an abstraction can't be visualised, you can still give it a formulaic expression.

Hmm. Depending on how many years you're talking about and who the professor was, I would have expected more from him. The formulas and graphs do exist:

Koch snowflake - Wikipedia, the free encyclopedia

 

[lesliedellow]

Hmm. Depending on how many years you're talking about and who the professor was, I would have expected more from him. The formulas and graphs do exist:

Koch snowflake - Wikipedia, the free encyclopedia

I have seen the formulas, but they are not on quite the same plane as differential geometry (for instance), and the "graphs" are only pretty pictures of polygons with finite length.

I graduated in 1981, so the conversation would probably have been in 1978-79.

 

[Resha Caner]

I graduated in 1981, so the conversation would probably have been in 1978-79.

What was your degree? It's possible your knowledge outstrips mine.

I have seen the formulas, but they are not on quite the same plane as differential geometry (for instance), and the "graphs" are only pretty pictures of polygons with finite length.

As I said, maybe your depth in mathematics exceeds mine, but I've never thought of fractals as just "pretty pictures." The formulas exist to calculate the length of the curve and the area enclosed. The whole idea of finite segments progressing through self-similarity toward some kind of continuous figure - the link through fractal dimensions of things like a line and a plane - is fascinating to me.

I guess ideas of infinity just always intrigue me.

 

[lesliedellow]

What was your degree? It's possible your knowledge outstrips mine.

Mathematics.

 

[Resha Caner]

Mathematics.

Ah, well, then you are beyond me. My father and brother-in-law both have degrees in mathematics though, so I'm at least a mathematician by relation.

So what do you do? What's your favorite area of mathematics.

 

[lesliedellow]

So what do you do?

I am a freelancer who earns a living partly through computer programming and partly through writing.

What's your favorite area of mathematics.

Functional analysis I suppose. Potential Theory.

 

[Resha Caner]

I am a freelancer who earns a living partly through computer programming and partly through writing.

Ha! Are you my long lost twin?

Though my dad's degree is mathematics, he taught both that and computer science. He was one of the first to teach it at the high school level in the Midwest, and at one time was touring around helping get other computer science programs started. Computer science was a big hobby of mine in high school, and my dad said he was surprised when I chose a career in mechanical engineering instead. I've always lived in the border lands, so my grad thesis was on robotic controls - kind of a combination of engineering and computer science.

Another high school passion was writing (actually, I have too many hobbies). But, I didn't like English (grammar) and didn't like the idea of freelancing, so that was never a serious consideration for a career. Still, I'm proud of my list of amateur publications. I think my name still comes up in web searches for fiction writing, but I'm not doing much of it anymore.

So, what do you write?

Functional analysis I suppose. Potential Theory.

It gets better and better. In my work on engines and transmissions, we do a lot of frequency analysis - Fourier transforms. For diesel engines specifically, there is a lot of nonlinear behavior, and I became rather disenchanted with what frequency analysis could do for me. I played with wavelets for a time, and then started looking for other transforms that were better suited to the physics of the problem. I found a few things I really liked, but could never get the support I needed.

It basically ended when one of my mentors told me I'd have to go back to school, finish a PhD, and try to build a program around my idea. That's not a mountain I wanted to climb.