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
Quote:
1 was the last, so it's the smallest positive number. Everyone here has agreed on that!
Not the flat earthers. (they abound) :laugh:
>64 It’s weird being the same age as old people.
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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
And again I balk. "greater than" and "largest" are not synonymous for me, and "largest", when speaking of negative numbers, is non-sensical because negative things don't have "largeness." :rolleyes:
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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
I never doubted for a minute that you were correct, I just didn't think hard enough about it.
The difficult we do right away... ...the impossible takes slightly longer.
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Explain this: the sum of all positive integers on to infinity equals minus 1/12 The Great Debate Over Whether 1+2+3+4..+ ∞ = -1/12 | Smart News| Smithsonian Magazine[^]
Paul6124 wrote:
on to infinity equals minus 1/12
As it says "only equals -1/12 because the mathematicians redefined the equal sign." You can also prove other things by ignoring and/or redefining terms and assumptions in mathematics. For example it is generally accepted that you cannot prove in Euclidean geometry that parallel lines do not intersect. However you can prove that if you assume that a right triangle has a 90 degree angle. So trade one assumption for another. So in terms of the prior post one can redefine the problem by asserting that negatives can be bigger if the absolute value is bigger. Thus redefining what 'bigger' means in terms of the standard for Number Theory.
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OriginalGriff wrote:
* Zero is neither positive nor negative because the definition of both those terms stems from the direction of X from 0.
Well that's bullpucky. I've been looking around and I have seen only circular definitions/properties for "positive" -- first they assume that zero is not positive and then they define a property which may or may not be consistent with that definition. Zero is positive. And so am I.
PIEBALDconsult wrote:
Well that's bullpucky. I've been looking around and I have seen only circular definitions/properties for "positive"
Not sure I understand your point. There are many assumptions and term definitions in mathematics. Proofs are then based on both of those. If the terms/definitions are not accepted/understood then the proof becomes invalid (at least for one person.) I am rather certain that negative and positive are and always have been definitions. No one attempts to prove them. Not to mention of course that semantics of language makes this even more confusing. For example provide a definition for the word 'table' which includes all tables but excludes all other objects. Because of that people are always going to be limited in attempting to provide exact definitions. Including in mathematics.
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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
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And again I balk. "greater than" and "largest" are not synonymous for me, and "largest", when speaking of negative numbers, is non-sensical because negative things don't have "largeness." :rolleyes:
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"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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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
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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
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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
I'm not sure about your explanation, but I sense you are making the mistake I referred to yesterday, viz. basing your argument on the way number theory is implemented in the computer languages with which you are familiar, rather than on formal number theory itself. To answer the question we need the advice of pure mathemeticians on how magnitude of negative numbers is defined. How computers deal with it is clouded, as usual, by practicalities and compromise. However I doubt if there are any pure mathematicians lurking in this forum!
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Maybe - if you change "non-positive value" in your last sentence to "non-negative value".
Oops! Fixed :thumbsup:
"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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How about -π: it has lots of figures and a minus sign :laugh:
Mircea
e^iπ is the largest negative integer, I'd say.
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e^iπ is the largest negative integer, I'd say.
e_i_π FTFY.
Software Zen:
delete this; -
e_i_π FTFY.
Software Zen:
delete this;Eye thang ewe. ;-)
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Eye thang ewe. ;-)
You're welcome... I think. Kind of sounds like an indecent proposal to a sheep, but to quote the immortal Marty Feldman, "Suit yourself; I'm easy."
Software Zen:
delete this; -
Paul6124 wrote:
on to infinity equals minus 1/12
As it says "only equals -1/12 because the mathematicians redefined the equal sign." You can also prove other things by ignoring and/or redefining terms and assumptions in mathematics. For example it is generally accepted that you cannot prove in Euclidean geometry that parallel lines do not intersect. However you can prove that if you assume that a right triangle has a 90 degree angle. So trade one assumption for another. So in terms of the prior post one can redefine the problem by asserting that negatives can be bigger if the absolute value is bigger. Thus redefining what 'bigger' means in terms of the standard for Number Theory.
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Paul6124 wrote:
on to infinity equals minus 1/12
As it says "only equals -1/12 because the mathematicians redefined the equal sign." You can also prove other things by ignoring and/or redefining terms and assumptions in mathematics. For example it is generally accepted that you cannot prove in Euclidean geometry that parallel lines do not intersect. However you can prove that if you assume that a right triangle has a 90 degree angle. So trade one assumption for another. So in terms of the prior post one can redefine the problem by asserting that negatives can be bigger if the absolute value is bigger. Thus redefining what 'bigger' means in terms of the standard for Number Theory.
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In my case (non native english speaker) "large" is for me more associated with size, not value. That's why I would usually think first on the biggest module in negative, meaning -∞. But... as I have had a lot of such tricky questions, I tend to wait a second, put back the obvious answer and pay a lot of more attention to the wording while activating the paranoic mode. So at the end I found the right solution.
M.D.V. ;) If something has a solution... Why do we have to worry about?. If it has no solution... For what reason do we have to worry about? Help me to understand what I'm saying, and I'll explain it better to you Rating helpful answers is nice, but saying thanks can be even nicer.
I (native speaker) would agree there. To me, there's a difference between "greater" and "larger", and between "less" and "smaller". Greater/less include the sign whereas larger/smaller refer to the absolute magnitude.
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I can't explain that, it involves division and I can't do that. :-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!
The becomes Teh...
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Mount Everest isn't high at all; it's at ground level.
Perfect, it might have the highest/tallest peak, still at ground level... :-D
