Consider this (gee, I wish writing math on webpage is prettier and more convenient, the square root is a pain):
What's the answer? Well it is i × i and of course the answer is -1.
But wait, √(xy) = √x√y, so subsitute both x and y as -1, we have: √(-1 × -1), which is √(1), which is...+1.
Oh no, I just proved 1 = -1! I show this to the bank and all the debt I owe becomes my money, woohoo.
What's wrong here? (to be continued...)
Do I have any reader out there wanting to comment?
1 comment:
Sadly I have no readers willing to comment, perhaps all my readers are robotic web crawlers.
Ok, √ (xy) CANNOT be broken to √x√y when both are negative... because that would upset the sign as seen. If both are negative, then come on, cancel them first so it is √(|x|)√(|y|). We sometimes just have to be careful in algebra operations... if you ever need to do them in real life.
√ is a special operation... We are only interested in the positive result. For example, √4 = 2. We don't care about -2 even if -2×-2 = 4. Special case is that imaginary number i, where √-1 = i.
Digression:
The informed students know that there are always n answers in n-th root problems. For example, what's the cube root of 8? 2 you say? yes that's one answer. What about the other 2? oh mine, they lie in that circle of equal distant in the real-imaginary plane.
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