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 October 26th, 2013, 07:24 AM #1 Senior Member   Joined: Oct 2012 Posts: 460 Thanks: 0 0.1 binary Hello! Please, as far as there are no problems to turn decimal numbers to binary, I just stuck at turning 0.1 to binary system. 0.1 : 2 = 0. rest 0.05 :2 = 0 rest 0.025 : 2 = o rest ... and the result should be infinite number containing 1s too: 0.00011001100110011 How do they get these 1s at all cause you can all the rational numbers divide with 2... Please, can someone help? Many thanks!
October 26th, 2013, 08:13 AM   #2
Math Team

Joined: Dec 2006
From: Lexington, MA

Posts: 3,267
Thanks: 406

Re: 0.1 binary

Hello, ungeheuer!

Quote:
 $\text{Write }0.1\text{ in binary.}$ $\text{The answer is the infinite decimal: }\,0.0\overline{0011}$

Here is one way . . .

Assume that 0.1 (base ten) equals some decimal in base two.

[color=beige]. . [/color]$0.1_{_{10}} \:=\:0.abcdefghijk...\, _{_{2}}$

$\text{Multiply by 2: }\:0.2 \:=\:a.bcdefghijk...
\;\;\text{Hence: }\,a\,=\,0
\text{And we have: }\:0.2 \:=\:0.bcdefghijk...$

$\text{Multiply by 2: }\:0.4 \:=\:b.cdefghijk...
\;\;\text{Hence: }\,b\,=\,0
\text{And we have: }\:0.4 \:=\:0.cdefghijk...$

$\text{Multiply by 2: }\:0.8\:=\:c.defghijk...
\;\;\text{Hence: }\,c\,=\,0
\text{And we have: }\:0.8 \:=\:0.defghijik...$

$\text{Multiply by 2: }\:1.6\:=\:d.efghijk...
\;\;\text{Hence: }\,d\,=\,1
\text{And we have: }\:0.6 \:=\:0.efghijk...$

$\text{Multiply by 2: }\:1.2\:=\:e.fghijk...
\;\;\text{Hence: }\,e\,=\,1
\text{And we have: }\:0.2 \:=\:0.fghijk...$

$\text{Multiply by 2: }\:0.4\:=\:f.ghijk...
\;\;\text{Hence: }\,f\,=\,0
\text{And we have: }\:0.4 \:=\:0.ghijk...$

$\text{Multiply by 2: }\:0.8\:=\:g.hijk...
\;\;\text{Hence: }\,g\,=\,0
\text{And we have: }\:0.8 \:=\:0.hijk...$

$\text{Multiply by 2: }\:1.6\:=\:h.ijk...
\;\;\text{Hence: }\,h\,=\,1
\text{And we have: }\:0.6 \:=\:0.ijk...$

$\text{Multiply by 2: }\:1.2\:=\:i.jklm...
\;\;\text{Hence: }\,h\,=\,1
\text{And we have: }\:0.2 \:=\:0.ijklm...$

$\text{Multiply by 2: }\:0.4\:=\:i.jklm...
\;\;\text{Hence: }\,i\,=\,0
\text{And we have: }\:0.4 \:=\:0.jklm...$

And we see that we have a repeating pattern.
[color=beige]. . [/color]$0.1_{_{10}} \;=\;0.0\,\!\overline{0011}\,\!\overline{0011}\:\t ext{. . .}$

 October 26th, 2013, 08:14 AM #3 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 461 Math Focus: Calculus/ODEs Re: 0.1 binary $\frac{1}{5}=\frac{3}{15}$ In binary this is: $\frac{11}{1111}=0.\bar{0011}$ Dividing by 2 moves the decimal point to the left one place, thus: $0.1=0.0\bar{0011}_{2}$
 October 26th, 2013, 08:33 AM #4 Global Moderator   Joined: Dec 2006 Posts: 17,466 Thanks: 1312 Or use long division... Code:  0.000110011 __________________ 1010 |1.0000000000 1010 ---- 1100 1010 ---- 10000 1010 ---- 1100 1010 ---- etc.

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