Check your answers by adding the subtrahend and difference for each problem.

HEXADECIMAL (HEX) NUMBER SYSTEM

The hex number system is a more complex system in use with computers. The name is
derived from the fact the system uses 16 symbols. It is beneficial in computer programming
because of its relationship to the binary system. Since 16 in the decimal system is the
fourth power of 2 (or 2^{4}); one hex digit has a value equal to four binary
digits. Table 1-5 shows the relationship between the two systems.

Table 1-5. - Binary and Hexadecimal Comparison

Unit and Number

As in each of the previous number systems, a unit stands for a single object.

A number in the hex system is the symbol used to represent a unit or quantity. The
arabic numerals 0 through 9 are used along with the first six letters of the alphabet. You
have probably used letters in math problems to represent unknown quantities, but in the
hex system A, B, C, D, E, and F, each have a definite value as shown below:

Base (Radix)

The base, or radix, of this system is 16, which represents the number of symbols used
in the system. A quantity expressed in hex will be annotated by the subscript 16, as shown
below:

A3EF_{16}

Positional Notation

Like the binary, octal, and decimal systems, the hex system is a positional notation
system. Powers of 16 are used for the positional values of a number. The following bar
graph shows the positions:

Multiplying the base times itself the number of times indicated by the exponent will
show the equivalent decimal value:

You can see from the positional values that usually fewer symbol positions are required
to express a number in hex than in decimal. The following example shows this comparison:

625_{16} is equal to 1573_{10}

MSD and LSD

The most significant and least significant digits will be determined in the same manner
as the other number systems. The following examples show the MSD and LSD of whole,
fractional, and mixed hex numbers:

Addition of Hex Numbers

The addition of hex numbers may seem intimidating at first glance, but it is no
different than addition in any other number system. The same rules apply. Certain
combinations of symbols produce a carry while others do not. Some numerals combine to
produce a sum represented by a letter. After a little practice you will be as confident
adding hex numbers as you are adding decimal numbers.

Study the hex addition table in table 1-6. Using the table, add 7 and 7. Locate the
number 7 in both columns *X* and *Y*. The point in area *Z* where these
two columns intersect is the sum; in this case 7 + 7 = E. As long as the sum of two
numbers is 15_{10} or less, only one symbol is used for the sum. A carry will be
produced when the sum of two numbers is 16_{10} or greater, as in the following
examples:

Table 1-6. - Hexadecimal Addition Table

Use the addition table and follow the solution of the following problems:

In this example each column is straight addition with no carry. Now add the addend (784_{16})
and the sum (BDA_{16}) of the previous problem:

Here the sum of 4 and A is E. Adding 8 and D is 15_{16}; write down 5 and carry
a 1. Add the first carry to the 7 in the next column and add the sum, 8, to B. The result
is 13_{16}; write down 3 and carry a 1. Since only the last carry is left to add,
bring it down to complete the problem.

Now observe the procedures for a more complex addition problem. You may find it easier
to add the Arabic numerals in each column first:

The sum of 4, E, 1, and 3 in the first column is 16_{16}. Write down the 6 and
the carry. In the second column, 1, 1, 9, and 7 equals 12_{16}. Write the carry
over the next column. Add B and 2 - the sum is D. Write this in the sum line. Now add the
final column, 1, 1, 5, and C. The sum is 13_{16}. Write down the carry; then add 3
and B - the sum is E. Write down the E and bring down the final carry to complete the
problem.

Now solve the following addition problems:

Q.36 Add:

Q.37 Add:

Q.38 Add:

Q.39 Add:

Q.40 Add:

Q.41 Add: