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/