Easy way to do the X^3 = 12^9:
X^3 = (12^3)^3
x = 12^3
Which is where I usually stop since I'm more of a pure math type of person and rarely bother with arithmetic involving more than 1 digit numbers.
(Calculations are for computers, the beauty is in the proof).
But alright, x = 12 * 12 * 12 = 144*12 = 1728.
As for the initial question, that's not really addition. What you're doing is taking the cardinality of set union, where repeated items only count once. It's a simple matter to show:
| A union B | <= |A| + |B|
In fact, this follows from the more general result:
|A| + |B| = | A union B| + | A intersect B|
f(x) = x^3 + 4x^2 - 5
f'(x) = 3x^2 + 8x
Thus, the average velocity where x = 5 is 115.
Just to be picky
, that's the instantaneous velocity. But yeah, wonderful stuff.
As for 0 = 1, all these proofs contain some subtle flaw that people aren't meant to notice. If you look hard enough, you can always find a false step, or a false assumption.
Consider:
(Step1) 1 = 1^(2*1/2)
(Step2) = (1^2)^1/2
(Step3) = ((1)^2)^1/2
(Step4) = ((-1)^2)^1/2, since (-1)^2 = 1 = (1)^2
(Step5) = (-1)^(2*1/2)
(Step6) = (-1)^1
(Step7) = -1
Which of course you can use to your hearts desire to "prove" other results, such as:
1 = -1
1 + 1 = -1 + 1
2 = 0
1 = 0 (divide both sides by 2)
So, anyone care to guess which of the above steps was false?
Anyways, these "proofs" often like to make use of square roots but there are other examples that misuse the Well-Ordering princiople, or a proof by smallest counter example.
As an example of the Well-Ordering Principle, (every nonempty set of natural numbers contains a least element) we can "prove" that every natural number is "interesting". (And no, this isn't exactly a false proof, just really silly).
Proof: Suppose for the sake of contradiction that not every natural number is interesting. Let X be the set of all the non-interesting natural numbers. By assumption, X is non-empty. By the Well-Ordering principle, let x be the smallest element of X.
Now, 0 is Certainly interesting, it's the identity element for addition, and anything *0 = 0. Similarily, 1 is the multiplicative identity, it's the only unit in the set of natural numbers, 2 is the only even prime. Certainly all interesting numbers.
So what is x? It's the first natural number that is not interesting, and that makes it very interesting. => <=
(Granted, the last line is pretty ridiculous, but hopefully you can appreciate the humor in it.)
As for Hacker's question, if we are allowed the assumption that two sides of a triangle and the inclosing angle completely define the triangle (which is certainly true).
Then suppose AD bisects BC and AD is perpendicular to BC, then we have the |BD| = |DC| (the lengths are equal). But then since AD is perpendicular to BC, we have angle BDA = 90 degrees = angle CDA. But then we have for triangle, BDA, sides BD and AD enclosnig a 90 degree and, while for triangle CDA, sides DC and AD enclose a 90 degree angle, but |DC|=|BD| and certainly |AD|=|AD|, so these two triangles are quivalent. Hence the third sides BA and CA have equal length. But that contradicts triangle ABC is scalene. Thus if AD bisects BC, it is not perpendicular.
Here's some funny puzzles for the calculus types:
Note: For all functions below, the domain must be defined on all real numbers.
1) Find a function which is discontinuous at infinitely many points
2) Find a function which is discontinuous at an uncountably infinite number of points.
3) Find a function which is continuous at only a countably infinte number of points.
3b) but whose integral is continuous everywhere.
4) Find a function which is continuous at exactly one point.
5) Fnid a function which is discontinuous everywhere.
6) Find a continuous function which if not differentiable anywhere.
