Enough seriousness for the moment. This one is just for fun.
Okay, there is an error in here (obviously), but see if you can find it. (Note: The "^" symbol means "to the power of", so that "a^2" means "a to the power of 2" or "a squared".)
Let a=b
Thus: a^2=ab
Add a^2 to both sides: a^2 + a^2 = a^2 + ab
Or: 2a^2 = a^2 + ab
Subtract 2ab from both sides: 2a^2 – 2ab = a^2 + ab – 2ab
Or: 2a^2 – 2ab = a^2 – ab
This can be factored to: 2(a^2 – ab) = 1(a^2 – ab)
If you divide both sides by a^2 – ab, you get: 2 = 1
5 comments:
“If you divide both sides by a^2 – ab…”
Then you are dividing by zero. Forbidden.
I had an excellent teacher for analysis and for calculus in high school. I remember him doing this on the chalkboard as a warning to us to be careful in proving things.
One of his favorite expressions was, “Don’t put Descartes before de horse.” In 2009 I heard from a former classmate that the teacher is still alive, at around age 85.
You would want to solve the parenthetical problems before dividing at the end I would think. I'm not very good at this. But 2 * 0 = 1* 0.
My contention has always been that the opening equation is false since A cannot equal B because both A and B are different variables within the same equation. In different equations A might equal B, ie equation 1 A=1 B=2, and equation 2 A=2 B=3, then A=B, but it doesn't work that way. But then, my math skills have never been all that great.
Sorry, David, but a "variable" in math is defined as a "variable". It can be the same numeric value. "Variable" simply means "currently unknown".
Danny was close on that, but Anonymous had the right answer. If a=b, then a^2 - ab = 0. End of the effort.
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