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 24); 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:
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:
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:
62516 is equal to 157310
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 1510 or less, only one symbol is used for the sum. A carry will be produced when the sum of two numbers is 1610 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 (78416) and the sum (BDA16) of the previous problem:
Here the sum of 4 and A is E. Adding 8 and D is 1516; 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 1316; 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 1616. Write down the 6 and the carry. In the second column, 1, 1, 9, and 7 equals 1216. 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 1316. 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:
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