While the binary number system is very important because of its connection with computers, it is not the only other number system of importance. Two other number systems that are frequently seen are octal numbers and hexadecimal numbers. The octal number system is a base-8 number system and uses the digits 0 - 7 to represent numbers. The hexadecimal number system is a base-16 number system and uses the digits 0 - 9 along with the letters A - F to represent numbers. The table below shows a comparison of the first 16 numbers in the binary, octal, decimal, and hexadecimal number systems.



Binary

0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111

Base-2

Octal

0
1
2
3
4
5
6
7
10
11
12
13
14
15
16
17

Base-8

Decimal

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

Base-10

Hexadecimal

0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F

Base-16


Let's look at some larger octal and hexadecimal numbers and analyze these numbers using our relationship between digit, base, and position. Recall the formula we learned from binary to decimal conversion:

DIGIT * BASE POSITION #

Now consider the octal number 1473.28. As you can see in the table below, each column of this number represents a power of eight.

512s 64s Eights Ones   Eighths
83 82 81 80   8-1
1 4 7 3 . 2

Using the information from this table, we can find the decimal value for 1473.28. We will use the same approach we used for converting binary numbers to decimal.


 1*83   =  1*512   =   512.
 4*82   =  4*64    =   256.
 7*81   =  7*8     =    56.
 3*80   =  3*1     =     3.
 2*8-1  =  2*.125  = +   0.25
                       827.25

From our analysis, we see that 1473.28 is equivalent to 827.2510 in decimal.

Now let's look at a hexadecimal number: 33B.416. Since we are using a base-16 number system, the columns of our table are now powers of 16.

256s Sixteens Ones   Sixteenths
162 161 160   16-1
3 3 B . 4

As we did before, we can find the decimal value of 33B.416 by computer the value of each digit and summing. Notice that we substitute 1110 for the hexadecimal digit B in this conversion.


 3*162   =   3*256    =    768.
 3*161   =   3*16     =     48.
 B*160   =  11*1      =     11.
 4*16-1  =   4*.0625  =  +   0.25
                           827.25

Comparing all of our answers, we see that 827.2510 is 1473.28 in octal and 33B.416 in hexadecimal.