Decimals

Discussion in 'The Cesspool' started by Hertz, Apr 16, 2012.

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  1. Hertz Hz Registered Senior Member

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    A fraction expressed as a decimal should be the number to the left of the decimal point, plus the number to the left divided by the number to the right of the decimal point (should it not?) Isn't the fraction a fraction of the integer (whole number)

    For example:

    10.1=10+10/.1=10+100=110

    Is this correct?
     
  2. rpenner Fully Wired Valued Senior Member

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    No. A finite decimal expression \(A = a_n a_{\tiny n-1} a_{\tiny n-2} \dots a_{\tiny 2} a_{\tiny 1} a_{\tiny 0} . a_{\tiny -1} a_{\tiny -2} a_{\tiny -3} \dots a_{\tiny 2-m} a_{\tiny 1-m} a_{\tiny -m} \; = \; \sum_{k=-m}^{n} a_{\tiny k} 10^{k}\). m here is the number of digits to the right of the decimal point.

    Multiplying by \(10^j\) moves the decimal point j positions to the right.

    So if \(b_{\tiny k + m} = a_{\tiny k}\) then it follow that \(B = b_{\tiny n+m} b_{\tiny n+m-1} b_{\tiny n+m-2} \dots b_{\tiny 2} b_{\tiny 1} b_{\tiny 0} \; = \; \sum_{k=0}^{n+m} b_{\tiny k} 10^{k} \; = \; \sum_{k=-m}^{n} a_{\tiny k} 10^{k+m} \; = \; 10^{\tiny m} \sum_{k=-m}^{n} a_{\tiny k} 10^{k} = 10^{m} A\) is an integer.

    Thus 10.1 has one digit of the right of decimal point so (10)(10.1) = 101 or 10.1 = 101/10.
     
  3. Hertz Hz Registered Senior Member

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    So 10.1=10+(10/1)=11?
     
  4. rpenner Fully Wired Valued Senior Member

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    No. 10.1 is between 10 and 11, so ten times 10.1 must be between ten times 10 and ten times 11.

    \(\begin{eqnarray} 10 & = & 10.0 & = & \frac{100}{10} \\ & & 10.1 & = & \frac{101}{10} \\ 11 & = & 11.0 & = & \frac{110}{10} \end{eqnarray}\)

    Similarly, 12.34 = 1234/100
     
  5. Aqueous Id flat Earth skeptic Valued Senior Member

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    No. 10.1 = 10 + 1/10

    Or

    10.1 = 1 x 10[sup]1[/sup] + 1 x 10[sup]-1[/sup]

    whereas

    11 = 1 x 10[sup]1[/sup] + 1 x 10[sup]0[/sup]

    so

    11 - 10.1 = 1 x 10[sup]0[/sup] - 1 x 10[sup]-1[/sup]
     
  6. Aqueous Id flat Earth skeptic Valued Senior Member

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    No. Shifting a digit right is equivalent to dividing by 10. "0.1" means "1/10".
     
  7. Emil Valued Senior Member

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    10.1 = 1 x 2[sup]1[/sup] + 0 x 2[sup]0[/sup]+ 1 x 2[sup]-1[/sup] =10+1/2 :D
     
  8. Aqueous Id flat Earth skeptic Valued Senior Member

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    Note: 2[sup]1[/sup] ≠ 10[sub]10[/sub]
     
  9. Emil Valued Senior Member

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    2[sup]1[/sup][sub]10[/sub] ≠ 10[sub]10[/sub]
    2[sup]1[/sup][sub]10[/sub] = 10[sub]2[/sub]
     
  10. rpenner Fully Wired Valued Senior Member

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    Emil - the thread name "Decimals" restricts the topic to the base-ten system. In addition, you made at least one mistake in your post #7, even though that post is one line long. If you assume you are writing binary digits on the left and right sides of that equality, you cannot use a 2 in your fraction and be consistent. More consistent: \(10.1_{\tiny 2} = \frac{101_{\tiny 2}}{10_{\tiny 2}} = \frac{5}{2}\)
     
  11. Emil Valued Senior Member

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    Whoops ... you're right.
    edit,
    10.1[sub]2[/sub] = 1 x 2[sup]1[/sup] + 0 x 2[sup]0[/sup]+ 1 x 2[sup]-1[/sup] =10[sub]2[/sub]+1[sub]2[/sub]/10[sub]2[/sub]
     
  12. Hertz Hz Registered Senior Member

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    If .1 is one tenth then 10/1=1 so 10.1=10+10/1=11.
     
  13. prometheus viva voce! Moderator

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    No, 10/1 does not equal 1.

    Feel free to have another go. :)
     
  14. Hertz Hz Registered Senior Member

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    I meant 10/1=10. Sorry.

    10.1=10+(10/1)=20
     
  15. rpenner Fully Wired Valued Senior Member

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    As has been said in many ways before, what you have written is incorrect. Wikipedia explains:
    http://en.wikipedia.org/wiki/Tenth
    The American dictionary folks at Merriam-Webster write:
    http://www.merriam-webster.com/table/dict/number.htm
    So tenth = a tenth = one tenth = 1/10 = 0.1.

    1/2 + 1/2 = 1
    1/3 + 1/3 + 1/3 = 1
    1/4 + 1/4 + 1/4 + 1/4 = 1
    1/5 + 1/5 + 1/5 + 1/5 + 1/5 = 1
    1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 = 1

    0.5 + 0.5 = 1
    0.2 + 0.2 + 0.2 + 0.2 + 0.2 = 1
    0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 = 1

    .1 is one tenth.
    1/10 is also one tenth.
    So .1 = 1/10.

    10 = 10/1
    10 = 100/10

    10.1 = 10 + .1 = 100/10 + 1/10 = 101/10
    10.1 = 10 + .1 = 10 + 1/10

    The equals sign, "=", is very important -- it is an assertion of the truth that what is on the left side is always equal to the right side. If you assert things in public that turn out to be untrue, then you damage your reputation. In other posts you have written, you have also abused the equals sign and other math notation.

    A general expectation is by age 11 (Elementary School Grade 5), all students should "recognize and generate equivalent forms of commonly used fractions, decimals, and percents;" and "understand the effects of multiplying and dividing whole numbers;" http://www.nctm.org/standards/content.aspx?id=7564
    Sadly, this goal is not always achieved, nor is material always retained, which is the premise of the Jeff Foxworthy show: "Are You Smarter Than a 5th Grader?"
     
  16. funkstar ratsknuf Valued Senior Member

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    This smells like trolling. In any case, regardless of rpenner's valiant efforts, I think this site is not served well by having to explain decimals.
     
  17. Fraggle Rocker Staff Member

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    We're here to serve. If there are some really young people here who don't quite understand the concept, there's no harm in explaining it. If they like the place and stick around, five years from now we can explain relativity to them. ;)
     
  18. Captain Kremmen All aboard, me Hearties! Valued Senior Member

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    Exactly right.
    Every time I give you 10 dollars and 10 cents, you give me 11 dollars.
     
  19. funkstar ratsknuf Valued Senior Member

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    Point taken.
     
  20. James R Just this guy, you know? Staff Member

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    Surely the opening post is either an example of breathtaking inanity or trollbait.
     
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