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  1. #1 Don't laugh but......lol :-) 
    Forum Freshman uma_and_bill's Avatar
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    Right, I have encountered this a lot in my study and have used numbers like 0.0001 or even 0.000001 . I've managed fine using them but honestly have always wanted to ask this question but never got around to clarifying what I suspect. Right I think that the decimal in the first number I've mentioned is 1 thousandth and that the second number I've mentioned is ten thousandth (I think )

    Before I started studying I'd only ever seen number after the decimal range from 0 to 99 and then if it reached more than 99 then the number before that decimal point increase by one.


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  3. #2  
    Brassica oleracea Strange's Avatar
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    The first one is actually one ten thousandth. The trick is count how many digits there after the decimal point until you get to the first non-zero digit. That wasn't very clear, was it. What I mean is:

    0.1 = 1 tenth
    0.01 = 1 100th
    0.001 = 1 thousandth
    etc.


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  4. #3  
    Forum Freshman uma_and_bill's Avatar
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    so in the first example I've used, the most the number after the decimal point can possibly be before the 0 at the beginning becomes a one is 9,999 (I know people don't use commas anymore in number but it makes it easier for me :-) ) is that right?
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  5. #4  
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    I say you are right .9999 fills all the slots( a bit like in a fruit machine) and one more means they all drop to zero and the "0" to their left becomes a "1"

    So 0.00001 +0.9999 =1.0000 (or 1)

    Of course if the number was ,say 3.9999 and you added 0.0001 it would go up to 4.0000 (or 4)


    -but I don't know what you mean about not using commas.I thought you were still supposed (or allowed ) to do that.

    On the other hand I have never used the comma after the decimal point. Maybe that is what you meant?
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  6. #5  
    Forum Freshman uma_and_bill's Avatar
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    Thank you a ton for clarifying that, I suspected but hadn't quite committed. The fruit machine model was very easy to visualize. I once had a maths teacher that told me that conventionally we don't use the commas anymore. I suspect she was wrong. I've always used them any way. If its a long number with many of the same digits and no commas they all tend to merge and blur so I always use commas.
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  7. #6  
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    Quote Originally Posted by geordief View Post
    I say you are right .9999 fills all the slots( a bit like in a fruit machine) and one more means they all drop to zero and the "0" to their left becomes a "1"

    What's wrong with 0.99999, 0.999999, 0.9999999, ...?

    There are infinitely many slots.
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  8. #7  
    Forum Radioactive Isotope MagiMaster's Avatar
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    If you're going to use very large or very small numbers, it's much easier to switch to scientific notation. Instead of 0.000000000124, you can write 1.24 * 10^-10. Now you can almost forget about all those 0s and focus on the important digits. It also makes multiplication and division a lot easier to keep up with. (Addition and subtraction requires a bit of shifting things around, but not too much.)

    BTW, I've never seen commas (or other digit grouping marks) used after the decimal point. They are still used before it though.
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    Quote Originally Posted by MagiMaster View Post
    If you're going to use very large or very small numbers, it's much easier to switch to scientific notation. Instead of 0.000000000124, you can write 1.24 * 10^-10. Now you can almost forget about all those 0s and focus on the important digits. It also makes multiplication and division a lot easier to keep up with. (Addition and subtraction requires a bit of shifting things around, but not too much.)
    I wasn't sure if the OP was asking about how many digits are on the odometer. You might be right about there being conventions on when you switch to scientific notation.

    Quote Originally Posted by MagiMaster View Post
    BTW, I've never seen commas (or other digit grouping marks) used after the decimal point. They are still used before it though.

    I found this interesting discussion ...

    "In Spanish, Italian, and some other languages we use a point for thousands, and a comma for decimals, that is: 1.234.567,89"

    Now that's confusing to look at :-)

    Punto o coma decimal: Number 11,37 / 11.37 (Decimal Point / Comma) - WordReference Forums

    [It doesn't say anything about extra grouping commas to the right. I don't think they do that]
    Last edited by someguy1; June 12th, 2014 at 10:40 PM.
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  10. #9  
    Forum Radioactive Isotope MagiMaster's Avatar
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    I don't think there are any conventions on when to switch, but the more zeroes you have, the easier it is once you switch.

    I've seen some different grouping conventions, and yeah, it gets confusing quickly. (Some places even group in sets of four instead of three, IIRC.)
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  11. #10  
    Forum Freshman uma_and_bill's Avatar
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    Yes, I love scientific notation, I was terrified of some of the numbers I had seen before I worked out that I could just do that and save so much time, Its the reason my favourite number is ten. Before that my favorite number was 42
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  12. #11  
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    Quote Originally Posted by MagiMaster View Post
    (Some places even group in sets of four instead of three, IIRC.)
    Japanese does that in terms of the names for numbers; i.e. there is a word, man, for 10,000 so numbers go ten (ju), hundred (hyaku), thousand (sen), tenthousand (man), ten tenthousand (ju man) ... So instead of a word for million (which would be hyaku man) there is a word for 100,000,000.

    But you still put the commas every three digits, which can be confusing.
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  13. #12  
    ***** Participant Write4U's Avatar
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    [QUOTE=someguy1;573924]
    Quote Originally Posted by geordief View Post
    I say you are right .9999 fills all the slots( a bit like in a fruit machine) and one more means they all drop to zero and the "0" to their left becomes a "1"
    What's wrong with 0.99999, 0.999999, 0.9999999, ...? There are infinitely many slots.
    Accuracy? At cosmic (planck) scales, events which result in these type of numbers are abundant.

    the number 10^-26 is a little too small for a pocket calculator.
    "Art is the creation of that which evokes an emotional response, leading to thoughts of the noblest kind" (W4U)
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  14. #13  
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    [QUOTE=Write4U;574852]
    Quote Originally Posted by someguy1 View Post
    Quote Originally Posted by geordief View Post
    I say you are right .9999 fills all the slots( a bit like in a fruit machine) and one more means they all drop to zero and the "0" to their left becomes a "1"
    What's wrong with 0.99999, 0.999999, 0.9999999, ...? There are infinitely many slots.
    Accuracy? At cosmic (planck) scales, events which result in these type of numbers are abundant.

    the number 10^-26 is a little too small for a pocket calculator.
    You don't believe in the real numbers? Or am I misunderstanding you? Are you saying that you only believe in decimal numbers with 5 or 6 decimal places but no more?
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  15. #14  
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    Quote Originally Posted by someguy1 View Post
    quote]You don't believe in the real numbers? Or am I misunderstanding you? Are you saying that you only believe in decimal numbers with 5 or 6 decimal places but no more?
    As an old bookkeeper, I can assure you that four decimals is quite sufficient for everyday use.

    Try to write out 10^-36 (approx Planck length). Try your calculator, You'll get an "error" message.

    Just for comparison: The Planck length is about 10^-20 times the diameter of a proton, and thus is exceedingly small.

    btw. to raise a number by a negative number makes it smaller. 10^-4 = 10/10,000 = .0010 Thus a Planck length is about 10^-20 or .000000000000000000010 th "SMALLER" than the size of a proton, or something like it. Enough decimals for you?

    As I understand it, at Planck scale, matter loses its signature properties and becomes an "form of energy" (the uncertainty effect).
    Last edited by Write4U; June 18th, 2014 at 05:23 PM.
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    Quote Originally Posted by Write4U View Post
    Try to write out 10^-36 (approx Planck length)

    .000000000000000000000000000000000001 if I counted my 0's right. Should be 35 of them. What is the problem with this that you're having?


    Quote Originally Posted by Write4U View Post
    Just for comparison: The Planck length is about 10^-20 times the diameter of a proton, and thus is exceedingly small.
    I agree that the Planck length is small, but what does this have to do with the real numbers? I asked you if you believe in the real numbers. It's ok not to, I'm just trying to figure out where you're coming from here.

    Quote Originally Posted by Write4U View Post
    btw. to raise a number by a negative number makes it smaller. 10^-4 = 10/10,000 = .0010 Thus a Planck length is about 10^-20 or .000000000000000000010 the size of a proton, or something like it. Enough decimals for you?
    What do you mean enough? In the real number system you can have as many decimal positions as you like. What's wrong with 10^-100 or 10^-1000000?

    Quote Originally Posted by Write4U View Post
    As I understand it, at Planck scale, matter loses its signature properties and becomes a "form of energy" (the uncertainty effect).
    So do you believe in the real numbers or not? Do you even know what they are? I'm not understanding your response at all. You think there's a smallest positive number? What happens when you divide it by 2?
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  17. #16  
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    BTW, the Windows desktop calculator handles numbers like 10^-1000 easily. Also, any computer that implements IEEE single precision floating point numbers (most of them nowadays) can handle numbers down to around 10^-126. Most of those also implement double precision which works down to 10^-1022. (And I have done computations that stress the limits of a double precision floating point number, so there are uses for them.)

    Edit: Windows calculator seems to be using a decimal version of floating point numbers as it gets an underflow error just after 10^-9999.
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  18. #17  
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    Quote Originally Posted by MagiMaster View Post
    BTW, the Windows desktop calculator handles numbers like 10^-1000 easily. Also, any computer that implements IEEE single precision floating point numbers (most of them nowadays) can handle numbers down to around 10^-126. Most of those also implement double precision which works down to 10^-1022. (And I have done computations that stress the limits of a double precision floating point number, so there are uses for them.)

    Edit: Windows calculator seems to be using a decimal version of floating point numbers as it gets an underflow error just after 10^-9999.
    My point was convenience. I said I was an old bookkeeper, do I need to assert that real numbers (including decimals) were meaningful to me?

    My statement was that for "everyday life" the use of three or (rarely) four digits is more than sufficient. More sophisticated equipment (larger in size and computing power) is required to deal with decimals in the area of Physics and Cosmology. Was I wrong?

    Being that computers were invented to "deal with numbers" it would be expected that the greatest advances in computing power would lie in the area of computing........
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  19. #18  
    Forum Radioactive Isotope MagiMaster's Avatar
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    Your posts make your point less than clear. I do agree that most calculations don't need all that much precision, but there's a major difference between precision and number of digits. The number 0.000000012 only has two significant digits even though it has 10 digits total. (Imagine you measure something in meters, then convert that to centimeters. The number of digits changed, but the number of significant digits didn't. You don't know the length any more or less precisely just by writing the number down in different units.)

    Either way, big (or small) numbers don't require any sophistication to handle. Even most pocket calculators can handle at least 10^-8 and that's the most basic of basic.

    Also, your last sentence doesn't make sense to me. Computing power can be applied pretty much equally to any field.
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  20. #19  
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    Quote Originally Posted by MagiMaster View Post
    Your posts make your point less than clear. I do agree that most calculations don't need all that much precision, but there's a major difference between precision and number of digits. The number 0.000000012 only has two significant digits even though it has 10 digits total. (Imagine you measure something in meters, then convert that to centimeters. The number of digits changed, but the number of significant digits didn't. You don't know the length any n different units.)

    Either way, big (or small) numbers don't require any sophistication to handle. Even most pocket calculators can handle at least 10^-8 and that's the most basic of basic.

    Also, your last sentence doesn't make sense to me. Computing power can be applied pretty much equally to any field.
    I agree, universal constants are based in mathematics. But I thought we were discussing the need for certain etiquettes in mathematics.
    The number 0.000000012 only has two significant digits even though it has 10 digits total. (Imagine you measure something in meters, then convert that to centimeters. The number of digits changed, but the number of significant digits didn't. You don't know the length any n different units.)
    But we do know the length of meters and centimeters and that presents a true equivalency and a mathematical equation. But it "required" a translator, such as "CM" or "M".
    Thus the question, which is a more accurate measurement .0000012 (M) or .000000012 (CM)? the answer is the measurement in CM (it being the smaller increment).

    Now try the same numbers converting from yards to inches and see what obstacles those two significant numbers create (if we did not have computers to translate).

    You may also try to create a complex fractal. It'll fill your HD in a hurry. They are "Mathematical Monsters"
    btw, due to the ability of vanishing reiteration fractals are perfect for measurement of irregular surfaces at extremely small scales as illustrated in clip.
    Note: The variation in result of the measurement when smaller and smaller increments are used.
    https://www.youtube.com/watch?v=lmxJ1KDR_s0

    To think this process might occur at Planck scale (causal dynamic triangulation).
    Last edited by Write4U; June 19th, 2014 at 12:37 AM.
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    No, fractals will not fill your hard drive unless you're intent on decompressing them. "All the points in the complex plane, c, where the series z(n) = z(n-1)^2 + c does not diverge" is a complete description of the Mandelbrot fractal. Other fractals have similar concise descriptions. That's kind of the point of studying fractals. They stuff an impressive amount of apparent detail in to very simple descriptions. You can write rather small programs that will generate any section of the Mandelbrot set at any degree of resolution on demand. Decompressing it into an image file so an image displaying program can then do exactly the same thing is not a meaningful process.

    And no, cm is not more accurate than m. If your measurements only have two significant figures, they only have two significant figures. (Also, you wrote it backwards. 0.0012 m is equal to 0.12 cm.) And yes, you can mess yourself up by carelessly changing units, but generally changing units does not change the number of significant figures in your data.

    As an example, let's say you carefully measured something to be 1.23 feet. You're sure about those three digits, but not the next one, which (roughly) means the object you measured is somewhere between 1.225 and 1.235 feet. If you naively convert 1.23 feet to inches, you get 14.76 inches. Four significant figures all of a sudden, right? No. You should say the object is 14.7, not 14.76. You're no more certain about the 4th digit than you were before. If you convert the range to inches you get that the object is between 14.7 and 14.82 inches. Clearly, you have no idea what the fourth digit really ought to be.

    Significant digits are really just a short hand, simplified version of other error metrics. It'd probably be better to say the object is 1.23 plus or minus 0.005 feet instead. Then you can just convert both those numbers to inches and say the object is 14.76 plus or minus 0.06 inches. Both are just as accurate.

    Also, we're not discussing the need for anything, just why scientific notation (and now significant digits and other error metrics) are useful. Useful things tend to get used after all.
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  22. #21  
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    Quote Originally Posted by MagiMaster View Post
    No, fractals will not fill your hard drive unless you're intent on decompressing them. "All the points in the complex plane, c, where the series z(n) = z(n-1)^2 + c does not diverge" is a complete description of the Mandelbrot fractal. Other fractals have similar concise descriptions. That's kind of the point of studying fractals. They stuff an impressive amount of apparent detail in to very simple descriptions. You can write rather small programs that will generate any section of the Mandelbrot set at any degree of resolution on demand. Decompressing it into an image file so an image displaying program can then do exactly the same thing is not a meaningful process.

    And no, cm is not more accurate than m. If your measurements only have two significant figures, they only have two significant figures. (Also, you wrote it backwards. 0.0012 m is equal to 0.12 cm.) And yes, you can mess yourself up by carelessly changing units, but generally changing units does not change the number of significant figures in your data.

    As an example, let's say you carefully measured something to be 1.23 feet. You're sure about those three digits, but not the next one, which (roughly) means the object you measured is somewhere between 1.225 and 1.235 feet. If you naively convert 1.23 feet to inches, you get 14.76 inches. Four significant figures all of a sudden, right? No. You should say the object is 14.7, not 14.76. You're no more certain about the 4th digit than you were before. If you convert the range to inches you get that the object is between 14.7 and 14.82 inches. Clearly, you have no idea what the fourth digit really ought to be.

    Significant digits are really just a short hand, simplified version of other error metrics. It'd probably be better to say the object is 1.23 plus or minus 0.005 feet instead. Then you can just convert both those numbers to inches and say the object is 14.76 plus or minus 0.06 inches. Both are just as accurate.

    Also, we're not discussing the need for anything, just why scientific notation (and now significant digits and other error metrics) are useful. Useful things tend to get used after all.
    Thanks for your response. I made several careless errors, trying to make the point that ever smaller increments of measurements of irregular surfaces are more accurate and yield greater detail, than a gross approximation based on larger measurements.
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    That's a completely different point than the one I was making. That's called the coastline paradox and only applies to fractals, like coastlines. Finer and finer measurements of smooth objects don't have that problem.
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    Quote Originally Posted by MagiMaster View Post
    That's a completely different point than the one I was making. That's called the coastline paradox and only applies to fractals, like coastlines. Finer and finer measurements of smooth objects don't have that problem.
    Other than man made, are there any smooth objects in nature other than spheres?
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  25. #24  
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    The edges of most leaves are smooth beyond a certain scale. Plus, the coastline paradox only applies to edges. The surfaces of all leaves are smooth enough. Your skin is smooth enough. You hair is straight enough (even if it's curly). The lenses in your eyes are very smooth. Most objects in nature are not fractal in a way that brings up the coastline paradox. Your blood vessels are fractal, but in a different way. Either way, they carry a definite volume of blood and are composed of a definite volume of tissue.
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