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### Binary Conversion

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Date: 01/07/98 at 13:31:42
From: Steven Stirling
Subject: Binary conversion

Hey Dr. Math,

I'm enrolled in an adult computer course.  One of the things we are
supposed to look for is converting binary numbers to real numbers and
converting Binary to Hexadecimal.  I have been looking on the net and
really haven't had any luck - I've seen lots of sites offering
programs that will do this but none that will tell me how to do it
manually.  I haven't even been able to find anything about
hexidecimal.  If you could help, the whole class would be grateful.
```

```
Date: 01/07/98 at 14:47:06
From: Doctor Pipe
Subject: Re: Binary conversion

Steven,

To convert a binary number to a decimal number you must first
understand what each digit in the binary number means. To explain this
let's look at the decimal number 247.

The '2' in 247 represents two hundred because it is a two in the
hundreds position (two times a hundred is two hundred). In similar
fashion, the '4' in 247 represents forty because it is a four in the
tens position (four times ten is forty). Finally, the '7' represents
seven because it is a seven in the units position (seven times one is
seven). In a decimal number, the actual value represented by a digit
in that number is determined by the numeral and the position of the
numeral within the number.

It works the same way with a binary number. The right-most position in
a binary number is units; moving to the left, the next position is
twos; the next is fours; the next is eights; then sixteens; then
thirty-twos ...  Notice that these numbers are all powers of two -
2^0, 2^1, 2^2, 2^3, 2^4, 2^5. (The units, tens, hundreds, thousands,
ten thousands of the decimal system are all powers of ten: 10^0, 10^1,
10^2, 10^3, 10^4).

So, to convert the binary number 1001 (don't read that as one thousand
one - read it as one zero zero one) to decimal, you determine the
actual value represented by each '1' and add them together.  The
right-most '1' has a decimal value of 1 (it is in the 2^0, or units,
position) and the left-most '1' has a decimal value of 8 (it is in the
2^3, or eights, position). So the binary number 1001 is equal to
decimal 9.  Here's another way to look at it:

1 0 0 1
^ ^ ^ ^
| | | |_________> 1 x 2^0 = 1 x 1 = 1
| | |___________> 0 x 2^1 = 0 x 2 = 0
| |_____________> 0 x 2^2 = 0 x 4 = 0
|_______________> 1 x 2^3 = 1 x 8 = 8
---
9

Both the decimal system and the binary system are positional number
systems. The hexadecimal system is another positional number system.
The binary system has only two numerals - 0 and 1; the decimal system
has ten numerals: 0,1,2,3,4,5,6,7,8, and 9.  In the hexadecimal (or
hex) system there are sixteen numerals: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,
and F. Zero through nine have the same value as a decimal numeral, and
A is ten, B is eleven, C is twelve, D is thirteen, E is fourteen, and
F is fifteen. After a while you will get used to seeing "letters" used
as numerals!

The decimal number system is also referred to as "base ten" since each
position in a decimal number represents a power of ten - a number that
can be written as 10^n, where n is an integer. The binary number
system is also referred to as "base two" since each position in a
binary number represents a power of two - a number that can be written
as 2^n, where n is an integer. The hex number system is also referred
to as "base sixteen" since each position in a hexadecimal number
represents a power of sixteen - a number that can be written as 16^n,
where n is an integer.

The right-most position in a hexadecimal number is units; moving to
the left, the next position is sixteens; the next is two hundred
fifty-sixes; the next is four thousand ninety-sixes, and so on - all
powers of sixteen - 16^0, 16^1, 16^2, 16^3.

To convert a binary number to a hex equivalent, notice that four
binary digits together can have a value of from 0 to 15 (decimal)
exactly the range of one hex digit. So four binary digits will always
convert to one hex digit!

For example:

10110111 = B7 (hex)

The right-most four digits of the binary number (0111) equal seven, so
the hex digit is '7'. The remaining left-most four digits of the
binary number (1011) equal eleven, so the hex digit is 'B'.  Here is
another way of looking at it:

1 0 1 1 0 1 1 1     from right to left, make four-digit groups
\      /\      /
\    /  \    /
eleven   seven     determine the decimal equivalent of each
|       |       group
V       V
B       7        write the equivalent hexadecimal digit

What is the decimal equivalent of B7 hex?

B 7
^ ^
| |_________>  7 x 16^0 =  7 x  1 =   7
|___________> 11 x 16^1 = 11 x 16 = 176
---
183 decimal

Check that against the decimal equivalent of 10110111 binary:

1 0 1 1 0 1 1 1
^ ^ ^ ^ ^ ^ ^ ^
| | | | | | | |_________> 1 x 2^0 = 1 x   1 =   1
| | | | | | |___________> 1 x 2^1 = 1 x   2 =   2
| | | | | |_____________> 1 x 2^2 = 1 x   4 =   4
| | | | |_______________> 0 x 2^3 = 0 x   8 =   0
| | | |_________________> 1 x 2^4 = 1 x  16 =  16
| | |___________________> 1 x 2^5 = 1 x  32 =  32
| |_____________________> 0 x 2^6 = 0 x  64 =   0
|_______________________> 1 x 2^7 = 1 x 128 = 128
---
183 decimal

Hope this helps.  Good luck in your class!

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

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