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
"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 -
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 don't think anyone doubted that -1 is the greatest negative integer. It is just that the concept of "large negative" can be easily interpreted by "a number with a lot of figures and minus sign in front of it".
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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
A simple method I use is: If number A is to the right of another number B on the usual number line, then A is larger of the two. Otherwise B is larger. Consequently the largest of a set of numbers is the rightmost on the number line. (Of course, two numbers can both be equal, in which case this question doesn't arise).
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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
First of all, thanks for this brain teaser I think our brains have to struggle a lot to answer the questions a.) What is the largest negative integer number b.) What is the smallest negative integer number What surprises me is when I am asked the questions (which I asked myself after reading your puzzle): a.) Is -1 greater than -2? vs b.) Is -2 less than -1? Myself can answer question b.) much more easily :confused:
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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
OriginalGriff wrote:
It seem that many of us are convinced that -∞ is larger than 0
That has me worried about the state of our profession! :omg: Disclaimer: I didn't read all of the >50 replies to your original message, considering the solution you posted in the ≈5th message as self-evident and not a reason for debate. After that point I just shook my head in disbelief. :sigh:
Mircea
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I don't think anyone doubted that -1 is the greatest negative integer. It is just that the concept of "large negative" can be easily interpreted by "a number with a lot of figures and minus sign in front of it".
How about -π: it has lots of figures and a minus sign :laugh:
Mircea
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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
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[^]
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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[^]
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!
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How about -π: it has lots of figures and a minus sign :laugh:
Mircea
You are evil... :laugh: :laugh: :laugh:
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.
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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
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.
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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, as I stated in JavaScript:
-1 > Number.NEGATIVE_INFINITY; // true
If you only believe C# then:
Console.WriteLine(-1 > Double.NegativeInfinity ); // true
Since programming languages do model mathematics I think this should help to understand this. However, I am no mathematician and defer to anyone with a Math degree on this. :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
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.
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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.
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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
Yes. The Mariana Trench is deeper than Mount Everest is high. But that doesn't mean the trench is "bigger".
"Before entering on an understanding, I have meditated for a long time, and have foreseen what might happen. It is not genius which reveals to me suddenly, secretly, what I have to say or to do in a circumstance unexpected by other people; it is reflection, it is meditation." - Napoleon I
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Yes. The Mariana Trench is deeper than Mount Everest is high. But that doesn't mean the trench is "bigger".
"Before entering on an understanding, I have meditated for a long time, and have foreseen what might happen. It is not genius which reveals to me suddenly, secretly, what I have to say or to do in a circumstance unexpected by other people; it is reflection, it is meditation." - Napoleon I
Mount Everest isn't high at all; it's at ground level.
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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
it makes sense. the problem is the term "largest negative integer". Does the "largest non-negative integer" = infinity ? if so, then the reverse would be "largest negative integer" which by symmetry would be -infinity. The problem is mixing language and mathematics. 3rd grade math revisited.
"A little time, a little trouble, your better day" Badfinger
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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.
