# Decimals

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

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1. ### HertzHzRegistered 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. ### rpennerFully WiredValued Senior Member

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3,683
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. ### HertzHzRegistered Senior Member

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60
So 10.1=10+(10/1)=11?

4. ### rpennerFully WiredValued 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 Idflat Earth skepticValued Senior Member

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6,033
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 Idflat Earth skepticValued Senior Member

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

7. ### EmilValued 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

8. ### Aqueous Idflat Earth skepticValued Senior Member

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6,033
Note: 2[sup]1[/sup] ≠ 10[sub]10[/sub]

9. ### EmilValued 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. ### rpennerFully WiredValued Senior Member

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3,683
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. ### EmilValued Senior Member

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2,789
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. ### HertzHzRegistered Senior Member

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60
If .1 is one tenth then 10/1=1 so 10.1=10+10/1=11.

13. ### prometheusviva voce!Moderator

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2,045
No, 10/1 does not equal 1.

Feel free to have another go.

14. ### HertzHzRegistered Senior Member

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60
I meant 10/1=10. Sorry.

10.1=10+(10/1)=20

15. ### rpennerFully WiredValued Senior Member

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3,683
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. ### funkstarratsknufValued Senior Member

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1,390
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 RockerStaff Member

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23,552
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 KremmenAll aboard, me Hearties!Valued Senior Member

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12,739
Exactly right.
Every time I give you 10 dollars and 10 cents, you give me 11 dollars.

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1,390
Point taken.

20. ### James RJust this guy, you know?Staff Member

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27,967
Surely the opening post is either an example of breathtaking inanity or trollbait.