4.11.2005
Math disproven!
Hey guys:
Sorry I haven't posted for a while; I've been really busy. More about that later.
Kendra White, President of the CHC Student Council, and I were having an ultra-logical discussion. It's fun, but it sometimes gets too weird. . . .
Like this:
In formal logic, it is sometimes possible to reason from an incorrect assumption to a correct conclusion using a valid argument form.
However, in math, this is impossible. All = signs are absolutes, so you cannot go from an incorrect equation to a correct equation.
Until now.
Observe:
{beginning (incorrect)} (n) = (-n)
{square entire equation} (n) x (n) = (-n) x (-n)
{finish} n2 = n2
{square root} n = n
There you have it. From "n = -n" (incorrect), we get "n = n" (correct).
Have we detected a problem with algebra?
For a short time, comments will be enabled so that you can tell me what you think.
Have fun!
In Him,
David S. MacMillan III
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23 comments:
I am taking the absolute square root in the example.
Have I read C.S. Lewis??? And how!
That's true lamewing, but still the possible outcomes are:
n = -n
-n = -n
n = n
-n = n
So it really doesn't make a difference. It still results in at least 2 correct outcomes.
but you can't square pi because pies are round, cornbread is square. duh....
Anyway, you have to take an absolute value anyway because (in this world, anyway), we are governed by absolute standards - be they spiritual (sin, righteousness), temporal (law, lawlessness), or social (harm, beneficial action).
Isn't that a major equivocation on the word "absolute"? :-)
So I wonder if this "rule" may be made to apply in issues other than math? I mean, still maintaining logical properties, but with different components... Have you come up with any other examples?
Wait a minute, wait a minute...
You can't square both sides of the equation n= -n to reach a valid conclusion because it is a rule in mathematics that whether you add, subtract, divide, or multiply, you always have to do the same thing on both sides of the equation.
Here's your argument:
1. false premise: n= -n
2. procedure: n x n=-n x -n
3. conclusion: n=n
Here's the correct way to do it:
1. false premise: n= -n
2. procedure: n x n=-n x n
3. conclusion: n= -n
See the difference in the procedure?
Wow, Pete! I guess math is reliable after all.
Gabriel, LOL! :)
Yeah, it's impossible to get truth from falsity while following strict logical procedures and not making a mistake... Like David did. :)
Algebraically, it is perfectly viable to square both sides of an equation. We do it all the time, trust me. Ask your algebra teacher "Can I square both sides of an equation?" He'll say yes.
"Algebraically, it is perfectly viable to square both sides of an equation. We do it all the time, trust me. Ask your algebra teacher "Can I square both sides of an equation?" He'll say yes."
Hi David,
My dad is a school teacher and math major in college, and I verified it with him that it is a mathematical flaw to square both sides of an equation unless you are squaring the same amount on both sides.
I repeat, it is impossible to get truth from non-truth.
Did you read my whole explanation?
Algebraically, it is perfectly viable to square both sides of an equation.
Let me see if you're right:
1/2(4)=2
1/4(16)=4
And:
2(2)=4
4(4)=16
I guess you are right after all. But still, square roots can yield both negative and positive results, so the square root of n^2 could better be represented as ±n.
My dad is a school teacher and math major in college, and I verified it with him that it is a mathematical flaw to square both sides of an equation unless you are squaring the same amount on both sides.
Weird . . . I always thought that it was perfectly legit, 'cause they do it all the time with variables in the equation in order to figure out the answer.
I left a message at 1-800-ALGEBRA and a message with Dr. Jay L. Wile. Hopefully I can get some feedback that will clear this up :-).
Gabriel, your example carries no pertinence with this idea of truth resulting from non-truth. Why? Your example contained two real numbers one of which was not negated.
You said:
1/2(4)=2
1/4(16)=4
And:
2(2)=4
4(4)=16
Now, squaring both sides of an equation works, but only if the amounts on both sides are the same. If x and n are two different real numbers, You cannot multiply one number by n, and the number on the other side of the equal sign by x without breaking mathematical law.
Thus, to get a valid equation by squaring both sides, it must be something like this:
{premise (true)}: 4=4
{procedure (valid)}: 4^2=4^2
{conclusion (also true)}: 16=16
Now, here is the nonsense and fallaciousness of what is being said as true:
{premise (false)}: 5=3
{procedure (valid)}: 5^2=3^2
{conclusion (utter nonsense}: 25=9
I'm sure we all agree this is completely inane.
The only other way it can be done would be by inserting absolute value. We do so:
{premise (true)}: /-7/=/7/
{procedure (valid)}: /-7/^2=/7/^2
{conclusion (valid)}: 49=49
However, this process has been done for years, and is not the same as saying:
(n)=(-n)
(n)^2=(-n)^2
n^2=n^2
This is definitely not the same thing as I did previously. This distinction may be what you are missing.
According to both Dr. Jay L. Wile of Apologia Science (HighSchoolScience.com) and the math experts with at 1-800-ALGEBRA (VideoText Interactive), it is perfectly feasible to square both sides of any equation, but that it can result in nonsense if certain conditions are not met . . . kinda like dividing by zero but not illegal under algebraic rules.
{premise (false)}: 5=3
{procedure (valid)}: 5^2=3^2
{conclusion (utter nonsense}: 25=9
Of course this example starts with an inaccurate premise and ends in an inaccurate conclusion. However, in my example we start with an inaccurate premise and end in an accurate conclusion.
David, I never said it isn't feasable to square both sides of an equation. It is perfectly acceptable, but only if both sides of the equation are equal.
Did you show your particular example to those two enterprised you mentioned? Did they say that you can get truth from non-truth? Or did you just ask them if it's okay to square both sides of an equation?
But David, the right conclusion to your example would be ±n=±n, not n=n.
And besides, when you square an equation, you are multiplying each side of the equation by the same number. So even if you can square an equation and get an accurate answer, it doesn't mean that you can square a false equation and have it remain false.
"And besides, when you square an equation, you are multiplying each side of the equation by the same number."
No, you're not. You're multiplying each number on either side of the equation by itself, and unless both the numbers on both sides of the equation are equal, you are not multiplying both of them by the same amount.
Pete,
I was assuming the equation was a true equation, not a false one.
Did you show your particular example to those two enterprised you mentioned? Did they say that you can get truth from non-truth?
I did show them the example; they agreed that you can do it, but it results in an algebraic inconsistency. The guy at 1-800-ALGEBRA told me that whenever you square both sides of an equation that is not true, you can get a correct result, but if you use variables like I did, any rational number will work in the final equation but only zero works in the original equation. Squaring both sides of an equation is a little like dividing by zero . . . but you are always allowed to do it. It just messes things up occasionally.
Hi David, sorry I haven't had time to get back in awhile...
Well, that's what I've been saying all along. You CAN do it, but most of the time, it won't work with real numbers. ;)
But the problem was that while you obeyed one rule, you disobeyed another. You squared the equation, but you didn't multiply it by the same amount on both sides.
When you have variables, for instance:
7n3-3n2=33/n2+15
you don't know if you are multiplying both sides by the same number.
That's why we can do it.
Starting with an incorrect equation messes up the system, but they have to do it all the time in calculus and computer mathematics. It's really wierd, but it sorta works kinda maybe. :-)
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