impossible statements

[mechanicalbrain]

OK here's a new one:
Theorem: n=n+1

Proof:
(n+1)^2 = n^2 + 2*n + 1

Bring 2n+1 to the left:
(n+1)^2 - (2n+1) = n^2

Substract n(2n+1) from both sides and factoring, we have:
(n+1)^2 - (n+1)(2n+1) = n^2 - n(2n+1)

Adding 1/4(2n+1)^2 to both sides yields:
(n+1)^2 - (n+1)(2n+1) + 1/4(2n+1)^2 = n^2 - n(2n+1) + 1/4(2n+1)^2

This may be written:
[ (n+1) - 1/2(2n+1) ]^2 = [ n - 1/2(2n+1) ]^2

Taking the square roots of both sides:
(n+1) - 1/2(2n+1) = n - 1/2(2n+1)

Add 1/2(2n+1) to both sides:
n+1 = n

[Alan Anderson]

Originally Posted by mechanicalbrain

[ (n+1) - 1/2(2n+1) ]^2 = [ n - 1/2(2n+1) ]^2

Taking the square roots of both sides:

I hate it when square roots go bad.

Seriously, there are times when you have to remember that there is usually more than one root for a given expression. For example, the square roots of 9 are 3 and -3.

If you evaluate the expressions inside the brackets, you find that they can be reduced to 1/2 on the left side and -1/2 on the right. Naively taking the square root of their squares loses the minus sign, and breaks the equality. A proper result at this step requires an arbitrary choice of the negative root for one of the sides of the equation.

[Dan Zollman]

Here's my argument:

If we go back to the elementary school definition of division, we can say that in division, the answer is equal to the size of each part when something is divided into a certain number of parts. In other words, if we have the expressions 12/3 = x, x is 4 because when 12 is divided into 3 parts, there is four in each part. Also, x is the number of times that 3 goes into four.

In another definition, when we have the equations x/y = z, then z is the number that you can multipy by y to get x. In other words, if 12/3 = z, then z = 4 because 4 is the number you can multiply 3 by to get 12.

Having those two definitions to use, we can look at the problem of dividing by zero. When we ask, "What is 5/0?" we do not come up with an answer because there is no number that you can mulitiply by 0 to get 5. This is the main reason why the rules of algebra do not allow you to divide by zero.

In the 1 = 2 proof above, it is true that one step divides by zero. However, in that step, the numerator is also zero. Let's look at that situation:
0/0 = ?
What number can we multiply by zero to get zero? Any number!
And this is why I say that 0/0 = 0/0 can give you 1 = 2.

Even though this proof is logically valid, the idea that 0/0 is not practical in Algebra. This is why the rules of Algebra say that you cannot divide by zero, even when the numerator is zero.The fact alone that 1 does not equal 2 is enough reason that it will ultimately be said that this proof (and others like it) are useless.

I still think it's a fun idea to play with.

Now, what about the second proof?

[Alan Anderson]

Originally Posted by worldbringer

Now, what about the second proof?

It's another example of roots gone bad:

...take the 4th root of 1...

There are four 4th roots of 1:
1
-1
i
-i
Raise any of these to the 4th power and you get 1.

The main thing to remember from this is that taking roots is not something you can safely do to both sides of an equation unless you can figure out which root is required in order to satisfy the result.

[Greg Ross]

Proof by intimidation:

• Horses have an even number of legs.
• Behind they have two legs, and in front they have fore-legs.
• This makes six legs (which is certainly an odd number of legs for a horse.)
• The only number that is both even and odd is infinity.
• Therefore, horses have an infinite number of legs.

Q.E.D.

[Alan Anderson]

In the same spirit:

• If a > b and b > c, then a > c.
• Nothing is better than complete happiness.
• A peanut butter sandwich is certainly better than nothing.
• Therefore, a peanut butter sandwich is better than complete happiness.

Q.E.D.

[nehalita]

not related directly but....

Reading this reminded me of a theorem that I learned in a logics class, i'm sure takers of logic are familiar with this. I forgot what it's called but I personally call it the Law of Contradiction (or The "Anything" Tautology):
[S^~S] => C
This means if you take a statement that is true and its negation (the opposite) is also true, then you can imply anything from this contradiction.

For example, If man can be happy and unhappy at the same time, then the sky is falling. You can literally apply anything since "C" is merely a variable and the statements it can represent is endless.

If anyone wants the proof for this tautology, I can do it. By the way, since this is a tautology, it is always true.

So... I leave with this statement.
If the Ice Breakers liquid ice is ice and not ice, then the secret to hapiness is pi.

[billbo911]

Originally Posted by nehalita

So... I leave with this statement.
If the Ice Breakers liquid ice is ice and not ice, then the secret to hapiness is pi.

I think if you ask Dave Lavery, the secret of happiness is Donuts.

[phrontist]

Originally Posted by billbo911

I think if you ask Dave Lavery, the secret of happiness is Donuts.

Therefore...

doughnuts = Pi

Since we are operating in dave's universe that means that one of us is greater then Pi, because "if you look closely you will see, something between Pi you and me."

[Denman]

Originally Posted by Alan Anderson

It's another example of roots gone bad:

There are four 4th roots of 1:
1
-1
i
-i
Raise any of these to the 4th power and you get 1.

The main thing to remember from this is that taking roots is not something you can safely do to both sides of an equation unless you can figure out which root is required in order to satisfy the result.

Aka the fourth roots of unity. You can take roots from any imaginary number. If you think about it geometrically using argand diagrams, the differences between the angle of the roots is the argument of the complex number.
I'll post a better thing tomorrow or when i'm thinking lol

[Dan Zollman]

.333333 is not a perfect representation of 1/3. .9999999 is not a perfect representation of 1. However, this is the best way to express 1/3 and 1 (being 3 * 1/3), and this is what is generally accepted, so .999999 = 1 because that's the way we define such decimals even though it's not perfect.

[Alan Anderson]

Originally Posted by worldbringer

.333333 is not a perfect representation of 1/3. .9999999 is not a perfect representation of 1.

However, .333... is a perfect decimal representation of 1/3 ( which equals 3/9), and .999... is a perfect decimal representation of 1 (which equals 9/9). Those three trailing dots ("ellipses") are there for a reason, and that reason is why repeating decimals can be used to represent any rational number.