Decimals

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

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  1. Hertz

    Hertz Hz

    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

    rpenner Fully Wired

    No. A finite decimal expression [tex]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}[/tex]. m here is the number of digits to the right of the decimal point.

    Multiplying by [tex]10^j[/tex] moves the decimal point j positions to the right.

    So if [tex]b_{\tiny k + m} = a_{\tiny k}[/tex] then it follow that [tex]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[/tex] 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

    Hertz Hz

    So 10.1=10+(10/1)=11?
     
  4. rpenner

    rpenner Fully Wired

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

    [tex]\begin{eqnarray} 10 & = & 10.0 & = & \frac{100}{10} \\ & & 10.1 & = & \frac{101}{10} \\ 11 & = & 11.0 & = & \frac{110}{10} \end{eqnarray}[/tex]

    Similarly, 12.34 = 1234/100
     
  5. Aqueous Id

    Aqueous Id flat Earth skeptic

    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

    Aqueous Id flat Earth skeptic

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

    Emil Valued Senior Member

    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

    Aqueous Id flat Earth skeptic

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

    Emil Valued Senior Member

    2[sup]1[/sup][sub]10[/sub] ≠ 10[sub]10[/sub]
    2[sup]1[/sup][sub]10[/sub] = 10[sub]2[/sub]
     
  10. rpenner

    rpenner Fully Wired

    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: [tex]10.1_{\tiny 2} = \frac{101_{\tiny 2}}{10_{\tiny 2}} = \frac{5}{2}[/tex]
     
  11. Emil

    Emil Valued Senior Member

    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

    Hertz Hz

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

    prometheus viva voce!

    No, 10/1 does not equal 1.

    Feel free to have another go. :)
     
  14. Hertz

    Hertz Hz

    I meant 10/1=10. Sorry.

    10.1=10+(10/1)=20
     
  15. rpenner

    rpenner Fully Wired

    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

    funkstar ratsknuf

    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

    Fraggle Rocker Moderator

    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

    Captain Kremmen All aboard, me Hearties!

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

    funkstar ratsknuf

    Point taken.
     
  20. James R

    James R Just this guy, you know?

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