Brain Teasers
2 < 1
Sue was disappointed after Bob showed her that 2 was not, in fact, equal to 1. She was not quite done, though. She told Bob that while 2 may not be equal to 1, she could show conclusively that 2 was in fact less than 1. Bob said it couldn't be done, of course, and Sue offered the following:
Proof:
1 < 2
Multiply the equation by log (0.5)
1 * log(0.5) < 2 * log(0.5)
Make the log factors exponents
log(0.5 ^ 1) < log(0.5 ^ 2)
Carry out the exponentiation
log(0.5) < log(0.25)
Raise 10 to the power of each side of the inequality
10^(log(0.5)) < 10^(log(0.25))
By definition of logarithms, 10^log(a) = a, so...
0.5 < 0.25
Then just multiply both sides by 4
2 < 1
Question:
Bob found the problem almost immediately. Can you?
Proof:
1 < 2
Multiply the equation by log (0.5)
1 * log(0.5) < 2 * log(0.5)
Make the log factors exponents
log(0.5 ^ 1) < log(0.5 ^ 2)
Carry out the exponentiation
log(0.5) < log(0.25)
Raise 10 to the power of each side of the inequality
10^(log(0.5)) < 10^(log(0.25))
By definition of logarithms, 10^log(a) = a, so...
0.5 < 0.25
Then just multiply both sides by 4
2 < 1
Question:
Bob found the problem almost immediately. Can you?
Hint
Remember that the logarithm of any number less than 1 is less than 0.Answer
When Sue multiplied the inequality by log(0.5), she was multiplying by a negative number, and multiplying an inequality by a negative number switches the direction. This first step should result in:1 * log(0.5) > 2 * log(0.5), which leads to 2 > 1, as expected.
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Comments
eh, im in algebra 2 and havent learned that log rule yet. not too bad if you knew that, but i didnt, so w/e. pretty good.
what the hell?
i did not get the question at all.
i did not get the question at all.
Apr 21, 2005
wat the heck was that!?!
nyahahahahahahahaha! I knew it hahahahahahahah!
What intrigues me is that this problem and others like it have been around for more than 50 years all over the world. It depends on your experience whether you find this interesting/boring/hard/easy.
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