Decimals

**Hertz** · 04-16-12, 02:50 PM

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?

**rpenner** · 04-16-12, 03:11 PM

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.

**Hertz** · 04-23-12, 10:59 AM

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

**rpenner** · 04-23-12, 11:12 AM

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

**Aqueous Id** · 04-23-12, 11:17 AM

Originally Posted by Hertz

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

No. 10.1 = 10 + 1/10

Or

10.1 = 1 x 10¹ + 1 x 10^-1

whereas

11 = 1 x 10¹ + 1 x 10⁰

so

11 - 10.1 = 1 x 10⁰ - 1 x 10^-1

**Aqueous Id** · 04-23-12, 11:21 AM

Originally Posted by Hertz

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?)

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

**Emil** · 04-23-12, 01:15 PM

10.1 = 1 x 2¹ + 0 x 2⁰+ 1 x 2^-1 =10+1/2

**Aqueous Id** · 04-23-12, 01:44 PM

Note: 2¹ ≠ 10₁₀

**Emil** · 04-23-12, 01:52 PM

2¹₁₀ ≠ 10₁₀
2¹₁₀ = 10₂

**rpenner** · 04-23-12, 02:33 PM

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: $10.1_{\tiny 2} = \frac{101_{\tiny 2}}{10_{\tiny 2}} = \frac{5}{2}$

**Emil** · 04-23-12, 03:53 PM

Originally Posted by rpenner

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.

Whoops ... you're right.
edit,
10.1₂ = 1 x 2¹ + 0 x 2⁰+ 1 x 2^-1 =10₂+1₂/10₂

**Hertz** · 04-26-12, 07:33 AM

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

**prometheus** · 04-26-12, 09:14 AM

Originally Posted by Hertz

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

No, 10/1 does not equal 1.

Feel free to have another go.

**Hertz** · 04-26-12, 09:28 AM

I meant 10/1=10. Sorry.

10.1=10+(10/1)=20

**rpenner** · 04-26-12, 09:40 AM

Originally Posted by Hertz

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

As has been said in many ways before, what you have written is incorrect. Wikipedia explains:

Tenth can mean: ... 1/10, a fraction, one part of a unit divided equally into ten parts. It is written 0.1 in decimal notation.

http://en.wikipedia.org/wiki/Tenth
The American dictionary folks at Merriam-Webster write:

Each of the terms for the ordinal numbers excepting first and second is used in designating one of a number of parts into which a whole may be divided (a fourth; a sixth; a tenth) and as the denominator in fractions designating the number of such parts constituting a certain portion of a whole one fourth; three fifths). When used as nouns the fractions are usually written as two words, although they are regularly hyphenated as adjectives (a two-thirds majority). When fractions are written in numerals, the cardinal symbols are used (1/4, 3/5, 5/6).

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?"

**funkstar** · 04-27-12, 05:39 AM

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.

**Fraggle Rocker** · 04-27-12, 05:56 AM

Originally Posted by funkstar

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.

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.

**Captain Kremmen** · 04-27-12, 07:30 AM

Originally Posted by Hertz

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

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

**funkstar** · 04-27-12, 08:08 AM

Originally Posted by Fraggle Rocker

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.

Point taken.