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### Converting Numbers: Binary to Decimal

```
Date: 11/20/97 at 02:40:32
From: Linda Davidson
Subject: Converting numbers from binary to decimal

I am taking Computer Science 105 at a local college. I just cannot
get how to convert from binary to decimal and back again. Is there a
simple formula?

Linda
```

```
Date: 11/20/97 at 08:26:14
From: Doctor Jerry
Subject: Re: Converting numbers from binary to decimal

Hi Linda,

The easiest half is to convert from binary to decimal. Just remember
that the first place to the left of the "decimal" point is 2^0 = 1,
the second place is 2^1, and so on; to the right of the "decimal"
point is 2^(-1) = 1/2 in the first place, 2^(-2) = 1/4 in the second
place, and so on.

So, 10011.01  is 1*2^4+0*2^3+0*2^2+1*2^1+1*2^0+0*2^(-1)+1*2^(-2),
which is 8+2+1+1/4 = 11.25.

Going the other way is slightly harder. Take the base 10 number 57,
for example.  57 = d_0*2^0+d_1*2^1+d_2*2^2+... Read d_0 as d sub 0 and
so on.  The numbers d_0, d_1,... are the binary digits in the binary
expansion of 57. They are 0 or 1. If we divide 57 by 2 we get 28.5.

Looking at d_0*2^0+d_1*2^1+d_2*2^2+...divided by 2 we get
d_0/2+d_1+d_2*2^1+...  So, 2*(d_0/2) is the first digit.  In 28.5, if
we get .5, then 1 is d_0; if we get 0, then 0 is d_0.

Now, in your calculator containing 28.5, subtract the 0.5, leaving 28.
Again divide by 2. You get 14.0.  So, d_1=0. Again divide by 2. You
get 7. So, d_2 = 0. Similarly, d_3 = 1, d_4 = 1, and d_5 = 1.

Let's check.  111001 is 1*2^5+1*2^4+1*2^3+1*2^0 = 32+16+8+1 = 57.

If you have 57.325, you can do the 57 and the 0.375 parts separately.
Using similar ideas, just multiply 0.375 by 2 repeatedly to get the
digits. For example, 0.375*2 = 0.75. So the first digit to the right
is 0; 0.75*2 = 1.5 so the second digit is 1.  0.5*2 = 1 so 0.375 is
.011.

You can also look at

http://mathworld.wolfram.com/BaseNumber.html

-Doctor Jerry,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Number Theory

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