10. Hexadecimal Numbers

The one main disadvantage of Binary Numbers is that the binary equivalent of a large decimal number can be quite long, which makes it difficult to both read or write without producing errors especially when working with 16 or 32-bit numbers. One common way of overcoming this problem is to arrange the binary numbers into groups of four as Hexadecimal Numbers, starting with the least significant digit at the right hand side. This Hexadecimal or simply "Hex" numbering system uses the Base of 16 systems. Hence, it uses 16 (sixteen) different digits with a combination of numbers from 0 to 9 and the capital letters A to F to represent its Binary or Decimal equivalent.

We can make life easier by splitting these large binary numbers up into even groups to make them easier to write down and understandable. For example, the following group of binary digits 1101   0101   1100   1111₂  are much easier to read and understand than  1101010111001111₂   when they are all bunched up together. In the everyday use of the decimal numbering system we use groups of three digits or 000's from the right hand side to make a very large number such as a million or trillion easier to understand and the same is true in digital systems.

Hexadecimal Numbers are a more complex system than using just binary or decimal and is mainly used when dealing with computers and memory address locations. By dividing a binary number up into groups of 4 bits, each group or set of 4 digits can now have a possible value of between "0000" (0) and "1111" (8+4+2+1 = 15) giving a total of 16 different number combinations from 0 to 15. Don't forget that "0" is also a valid digit. We remember from the first tutorial about Binary Numbers that a four-bit group of digits is called a "nibble" and as four-bits are also required to produce a hexadecimal number, a hex digit can also be thought of as a nibble, or half-a-byte. Then two hexadecimal numbers are required to produce one full byte from 00 to FF. Also, since 16 in the decimal system is the fourth power of 2 (or 2⁴), one hex digit has a value equal to four binary digits so now q = "16".

The numbers 0 to 9 are still used as in the original decimal system, but the numbers from 10 to 15 are now represented by capital letters of the alphabet from A to F inclusive and the relationship between binary and hexadecimal is shown below.

a. Hexadecimal Numbers

Decimal	4-bit Binary	Hexadecimal
0	0000	0
1	0001	1
2	0010	2
3	0011	3
4	0100	4
5	0101	5
6	0110	6
7	0111	7
8	1000	8
9	1001	9
10	1010	A
11	1011	B
12	1100	C
13	1101	D
14	1110	E
15	1111	F
16	0001 0000	10 (1+0)
17	0001 0001	11 (1+1)
Continuing upwards in groups of four

Using the original binary number from above 1101 0101 1100 1111₂ this can now be converted into an equivalent hexadecimal number of D5CF₁₆   which is much easier to read and understand than a long row of 1´s and 0´s that we had before. Similarly, converting Hex based numbers back into binary is simply the reverse operation.

Then the main characteristics of a Hexadecimal Numbering System is that there are 16 distinct counting digits from 0 to F with each digit having a weight or value of 16 starting from the least significant bit (LSB). In order to distinguish Hexadecimal numbers from Denary numbers, a prefix of either a "#", (Hash) or a "$" (Dollar sign) is used before the actual Hexadecimal value, #D5CF or $D5CF.

As the base of a hexadecimal system is 16, which also represents the number of individual symbols used in the system, the subscript 16 is used to identify a number expressed in hexadecimal. For example, D5CF₁₆

b. Counting using Hexadecimal Numbers

So we now know how to convert 4 binary digits into a hexadecimal number. But what if we had more than 4 binary digits how would we count in hexadecimal beyond the final letter F. The simple answer is to start over again with another set of 4 bits as follows.

0...to...9, A,B,C,D,E,F, 10...to...19, 1A, 1B, 1C, 1D, 1E, 1F, 20, 21....etc

Do not get confused, 10 or 20 is NOT ten or twenty it is 1 + 0 and 2 + 0 in hexadecimal. In fact twenty does not even exist in hex. With two hexadecimal numbers we can count up to FF which is equal to decimal 255. Likewise, to count higher than FF we would add a third hexadecimal digit to the left so the first 3-bit hexadecimal number would be 100_16, (256₁₀) and the last would be FFF_16, (4096₁₀). The maximum 4-digit hexadecimal number is FFFF₁₆ which is equal to 65,535 in decimal and so on.

This adding of additional hexadecimal digits to convert both decimal and binary numbers into a Hexadecimal Number is very easy if there are 4, 8, 12 or 16 binary digits to convert. But we can also add zero's to the left of the most significant bit, the MSB if the number of binary bits is not a multiple of four. For example, 11001011011001₂ is a fourteen bit binary number that is to large for just three hexadecimal digits only, yet too small for a four hexadecimal number. The answer is to ADD additional zeros to the left most bit until we have a complete four bit binary number or multiples thereof.

c. Adding of Additional 0's to a Binary Number

Binary Number	0011	0010	1101	1001
Hexadecimal Number	3	2	D	9

The main advantage of a Hexadecimal Number is that it is very compact and by using a base of 16 means that the number of digits used to represent a given number is usually less than in binary or decimal. Also, it is quick and easy to convert between hexadecimal numbers and binary.

Example No1

Convert the following Binary number 1110 1010₂ into its Hexadecimal number equivalent.

111010102
Group the bits into four's starting from the right hand side
=	1110	1010
=	14	10	(in decimal)
=	E	A	(in Hex)
Then, the hexadecimal equivalent of the binary number 1110 10102  is   #EA16

Example No2

Convert the following Hexadecimal number #3FA7₁₆ into its Binary equivalent, and also into its Decimal or Denary equivalent using subscripts to identify each numbering system.

#3FA716
	= 0011 1111 1010 01112
	= (8192 + 4096 + 2048 + 1024 + 512 + 256 + 128 + 32 + 4 + 2 + 1)
	= 16,29510

Then, the Decimal number of 16,295 can be represented as:-

#3FA7₁₆   in Hexadecimal

or

0011 1111 1010 0111₂   in Binary.

Next topic will be on Octal Numbers