Following on from yesterday's little puzzler.
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It seem that many of us are convinced that -∞ is larger than
0so I thought I'd try and explain why that isn't the case, even though it does seem to make sense. Let's look at what "greater than" actually means (in all cases I'll use integers but it's exactly the same for floating point numbers). 1 is greater than 0, 2 is greater than both 1 and 0, 3 is greater than 2, 1, and 0, and so on: the general case is "if you add a positive number* to a value, you get a value that is greater than the original":X + n > X where n is any positive number. Similarly, "less than" comes down to:X - n < X where n is any positive number. And it works:2 > 1 because 1 + 1 == 2; 3 > 1 because 1 + 2 == 3; ...1 < 2 because 2 - 1 == 1; 1 < 3 because 3 - 2 == 1; ...And we can use "greater than" and "Less than" for find maxima and minima for a set of numbers. We can find the smallest positive number by taking any positive number as a starting point and repeatedly subtracting 1 until we reach a non-positive value (which will be zero): 1 was the last, so it's the smallest positive number. Everyone here has agreed on that! But when we look for the largest negative number it seems that some people are mistaking the absolute magnitude of a value for the value itself, and saying that the largest negative number is -∞ But that's not the case: just as numbers get smaller as you approach 0 from the positive side, they don't start getting bigger again as you move away into the negative side:1 > 0; 1 > -1; 1 > -2Slide that sideways and it's clearer for negative numbers:0 > -1; 0 > -2; 0 > -3-1 > -2; -1 > -3; -1 > -4So to find the largest negative number, we start with any negative number as a starting point and repeatedly adding 1 until we reach a non-negative value (which will be zero): -1 was the last, so that's the largest negative number. Make sense? * Zero is neither positive nor negative because the definition of both those terms stems from the direction of X from 0."I have no idea what I did, but I'm taking full credit for it." - ThisOldTony "Common sense is so rare these days, it should be classified as a super power" - Random T-shirt
oh no this has prompted me to post for the first time ever... > the general case is "if you add a positive number to a value, you get a value that is greater than the original" That is your postulate, not a fact or proof. -1 is greater than -2 only if you assume this is true. I propose another: To divide a quantity or object in half is to produce two halves that are each smaller than the original whole. Divide a number in half, the result is the smaller number. > Let's look at what "greater than" actually means... We all know language is ambiguous. It could actually mean many different things. Of course no one is arguing that (-1 > -2) doesn't evaluate to true in your programming language of choice* :) That's just pragmatic. *except maybe c++ in some cases...
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Very interesting riddle, almost got a brain freeze :-D If you read carefully, the answer to get to 30 is quite obvious, unfortunately our brains are not wired that way.. :((
My first reaction, too. The first time I encountered this puzzle was an oral presentation. Made for some interesting notes, until one does the math correctly. The key is "where is the money", not "who spent what".
"A little time, a little trouble, your better day" Badfinger
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oh no this has prompted me to post for the first time ever... > the general case is "if you add a positive number to a value, you get a value that is greater than the original" That is your postulate, not a fact or proof. -1 is greater than -2 only if you assume this is true. I propose another: To divide a quantity or object in half is to produce two halves that are each smaller than the original whole. Divide a number in half, the result is the smaller number. > Let's look at what "greater than" actually means... We all know language is ambiguous. It could actually mean many different things. Of course no one is arguing that (-1 > -2) doesn't evaluate to true in your programming language of choice* :) That's just pragmatic. *except maybe c++ in some cases...
The formal mathematical proof that
1 + 1 = 2runs to 360 pages of arcane symbols, and I don't understand a single page of it. I'm not going to try and modify that to formally proveX + n > X where n is a positive valuebecause that proof would derive from1 + 1 = 2. Instead, I suggest you show any example which is consistent with1 + 1 = 2whereX + n <= X where n is a positive value. If you are right and I am wrong (which I'm fully prepared to believe) it should be simple for you :D"I have no idea what I did, but I'm taking full credit for it." - ThisOldTony "Common sense is so rare these days, it should be classified as a super power" - Random T-shirt AntiTwitter: @DalekDave is now a follower!
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The formal mathematical proof that
1 + 1 = 2runs to 360 pages of arcane symbols, and I don't understand a single page of it. I'm not going to try and modify that to formally proveX + n > X where n is a positive valuebecause that proof would derive from1 + 1 = 2. Instead, I suggest you show any example which is consistent with1 + 1 = 2whereX + n <= X where n is a positive value. If you are right and I am wrong (which I'm fully prepared to believe) it should be simple for you :D"I have no idea what I did, but I'm taking full credit for it." - ThisOldTony "Common sense is so rare these days, it should be classified as a super power" - Random T-shirt AntiTwitter: @DalekDave is now a follower!
That is simple: you have a hole in the ground and add a positive amount of soil to it. Is the hole now bigger or smaller? The hole is negative volume. Divide that hole in half, the half hole is smaller than the whole hole.
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It seem that many of us are convinced that -∞ is larger than
0so I thought I'd try and explain why that isn't the case, even though it does seem to make sense. Let's look at what "greater than" actually means (in all cases I'll use integers but it's exactly the same for floating point numbers). 1 is greater than 0, 2 is greater than both 1 and 0, 3 is greater than 2, 1, and 0, and so on: the general case is "if you add a positive number* to a value, you get a value that is greater than the original":X + n > X where n is any positive number. Similarly, "less than" comes down to:X - n < X where n is any positive number. And it works:2 > 1 because 1 + 1 == 2; 3 > 1 because 1 + 2 == 3; ...1 < 2 because 2 - 1 == 1; 1 < 3 because 3 - 2 == 1; ...And we can use "greater than" and "Less than" for find maxima and minima for a set of numbers. We can find the smallest positive number by taking any positive number as a starting point and repeatedly subtracting 1 until we reach a non-positive value (which will be zero): 1 was the last, so it's the smallest positive number. Everyone here has agreed on that! But when we look for the largest negative number it seems that some people are mistaking the absolute magnitude of a value for the value itself, and saying that the largest negative number is -∞ But that's not the case: just as numbers get smaller as you approach 0 from the positive side, they don't start getting bigger again as you move away into the negative side:1 > 0; 1 > -1; 1 > -2Slide that sideways and it's clearer for negative numbers:0 > -1; 0 > -2; 0 > -3-1 > -2; -1 > -3; -1 > -4So to find the largest negative number, we start with any negative number as a starting point and repeatedly adding 1 until we reach a non-negative value (which will be zero): -1 was the last, so that's the largest negative number. Make sense? * Zero is neither positive nor negative because the definition of both those terms stems from the direction of X from 0."I have no idea what I did, but I'm taking full credit for it." - ThisOldTony "Common sense is so rare these days, it should be classified as a super power" - Random T-shirt
tl;dr "larger" is ambiguous "larger" can mean "greater than" "larger" can mean "greater magnitude than" And that is the issue. English is fickle.
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tl;dr "larger" is ambiguous "larger" can mean "greater than" "larger" can mean "greater magnitude than" And that is the issue. English is fickle.
English developed from communication of daily experiences by common folk over two millennia. Hence the English system of units, which look arcane but were very practical. It is flexible and adaptable, but that can lead to confusion. Look at the knots tied in mangled lamguage by lawyers trying to pin down in writing agreements that to ordinary people seem obvious!
