0.1 binary

ungeheuer · October 26th, 2013, 07:24 AM

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!

soroban · October 26th, 2013, 08:13 AM

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] $0.1_{_{10}} \;=\;0.0\,\!\overline{0011}\,\!\overline{0011}\:\t ext{. . .}$

MarkFL · October 26th, 2013, 08:14 AM

$\frac{1}{5}=\frac{3}{15}$

In binary this is:

$\frac{11}{1111}=0.\bar{0011}$

Dividing by 2 moves the decimal point to the left one place, thus:

$0.1=0.0\bar{0011}_{2}$

skipjack · October 26th, 2013, 08:33 AM

Or use long division...

Code:

      0.000110011
__________________
1010 |1.0000000000
        1010
        ----
         1100
         1010
         ----
           10000
            1010
            ----
             1100
             1010
             ----

etc.

October 26th, 2013, 07:24 AM	#1
ungeheuer 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!
$ungeheuer is offline$

October 26th, 2013, 08:14 AM	#3
MarkFL Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 466 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: $0.1=0.0\bar{0011}_{2}$
$MarkFL is offline$

October 26th, 2013, 08:33 AM	#4
skipjack Global Moderator Joined: Dec 2006 Posts: 18,852 Thanks: 1570	Or use long division... Code: 0.000110011 __________________ 1010 \|1.0000000000 1010 ---- 1100 1010 ---- 10000 1010 ---- 1100 1010 ---- etc.
$skipjack is offline$

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0.1 in b