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Re: [Phys-L] Conceptual Physics Course



I believe the inequality (1/x + 1/y) =/= x+y was a typo. Earlier in the thread the question related to "the reciprocal of (1/x + 1/y)".

Multiplying both sides of (1/x + 1/y) =?= x+y (which I don't think was the question) by xy gives (y+x) =?= (x+y)xy, which can only be true if x+y=0 or xy=1, i.e., if y is either the additive or multiplicative inverse of x. That makes sense, when you look at the equation. If y = -x, both sides are automatically 0. If y = 1/y, then each side is equal to x + 1/x.

Back to the real question, how to easily convince students that 1/(1/x + 1/y) =/= x+y: What about simply multiplying both sides by whatever it takes to clear out the fractions? That's not a trick, that's a reliable algebraic technique students should think of using. We proceed:

Start with 1/(1/x + 1/y) =?= x+y

Multiply both sides by (1/x + 1/y): 1 =?= (x+y)(1/x + 1/y)

Expand: 1 =?= 1 + x/y + y/x + 1

Simplify: x/y + y/x + 1 =?= 0

Multiply both sides by xy: x^2 + y^2 + 1 =?= 0

Now for any real numbers x and y, x^2 >=0 and y^2 >=0, so x^2 + y^2 >=0 and x^2 + y^2 + 1 >= 1.

So the last equation _can't_ be 0 -- unless we allow imaginary or complex numbers.



Another idea:

Not plugging in numbers, but still trying a special case: Let y = x:

Is 1/(1/x + 1/x) =?= x + x

Simplify: 1/(2/x) =?= 2x

Simplify: x/2 =?= 2x

Not unless x = 0, and then the left-hand side has divide-by-zero problems.

This should remind us of combinations of resistors or capacitors in series or parallel. Combining resistors in series add the resistances, combining them in parallel adds their inverses and inverts the answer. Using two 1000 ohm resistors, we can either make 2000 ohms or 500 ohms. It's clear what's happening: in the series case the current must pass through double the resistance, in the parallel case we have provided two paths, effectively halving the resistance. When thought about in this way, the two procedures are as different as ... doubling vs. halving! No way should we expect the results to be the same.

:-)

Testing your formula by trying it out with special cases is a useful strategy from time to time. It's not that much different from looking at the results of your button-pushing on the calculator and thinking whether the answer could possibly make sense or not. (It's frightening how often students turn the crank, write down and answer that a moment's thought would determine is unreasonable.)

Ken Caviness


-----Original Message-----
From: phys-l-bounces@mail.phys-l.org [mailto:phys-l-bounces@mail.phys-l.org] On Behalf Of LaMontagne, Bob
Sent: Thursday, 17 May 2012 10:12 PM
To: Phys-L@Phys-L.org
Subject: Re: [Phys-L] Conceptual Physics Course

Multiply both sides by xy. The result is obviously nonsense.

Bob at PC

________________________________________
From: phys-l-bounces@mail.phys-l.org [phys-l-bounces@mail.phys-l.org] on behalf of Paul Lulai [plulai@stanthony.k12.mn.us]
Sent: Thursday, May 17, 2012 8:43 PM
To: Phys-L@Phys-L.org
Subject: Re: [Phys-L] Conceptual Physics Course

I (perhaps mistakenly) assumed the question was asking one thing and implying another.
There was a conversation about students being uncomfortable using variables and preferring numbers. I read some of the posts, but not all. I apologize if I missed something. From what I read, I got the impression that a theme of something similar to: students need to practice using variables so they can see general principles, understand the nature of what is going on etc...
This question asked how to prove (1/x + 1/y) =/= x+y Is there a clever (or not clever), simple way to show a kid with weak algebra skills the above? We all default to plug numbers in. The kids are probably a bit weaker than we are at algebra. They default to plugging numbers in much more quickly to a wider range of problems. If we go to plugging numbers in to prove something does or does not work, why shouldn't we expect them to do the same?

Have a good one.
Paul.

________________________________________
From: phys-l-bounces@mail.phys-l.org [phys-l-bounces@mail.phys-l.org] on behalf of John Denker [jsd@av8n.com]
Sent: Thursday, May 17, 2012 3:09 PM
To: Phys-L@Phys-L.org
Subject: Re: [Phys-L] Conceptual Physics Course

On 05/17/2012 08:22 AM, Jeffrey Schnick wrote:
On a related note, what do you do to convince a person in a lasting
manner that the reciprocal of (1/x + 1/y) is not, in general, x+y?

Well, when I was in kindergarten, the emphasized the idea of "Check Your Work".

In this case, even the simplest check would tell the tale.
I'm pretty sure the reciprocal of (1/x + 1/y) is not *ever* equal to x+y, not for any real-valued x and y, except in the trivial case where they are both zero.

Apparently the authors of certain widely-used introductory physics texts have never heard of "Check Your Work". I say that because of the astonishing amount of bogus physics in those books ... stuff that would not withstand even a moment's scrutiny.

Still, we can hold students to a higher standard. We can remind them to "Check Your Work".


=================

Here's another suggestion, more focused on this particular
case: Tell them to choose y=1 or some other positive constant, and then plot z = 1/(1/x + 1/y) as a function of x. Also show z = x and z = x+y on the same plot.

It should be clear that x+y is always larger than x, while 1/(1/x + 1/y) is always smaller (for positive real-valued x and y). This is something well worth knowing, in the context of resistors, which presumably have positive real-valued resistances.

====

As a related theoretical exercise: Define the mathematical operator "‖" (pronounced "parallel") according to

1
x‖y := ------------
1/x + 1/y

Then ask students: prove or disprove:
a) is ‖ commutative?
b) is ‖ associative?
c) does multiplication distribute over ‖?
d) does ‖ distribute over addition?
e) does addition distribute over ‖?
f) based on the above, would you say that ‖ has the same
operator structure as addition, or multiplication, or
subtraction, or what?

In real-world electrical engineering, the concept of x‖y is well known and quite useful.

Also this is an example of the spiral approach to learning.
We are spiraling back to the axioms of arithmetic, reinforcing them and extending them ... making *connections* between old ideas and new.
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