Problem 23 - Discrete Mathematics with Applications

Question

In Exercises \(21-28,\) perform the indicated operations. $$ \begin{aligned} & 3076_{\text { sixteen }} \\\\+& 5776_{\text { sixteen }} \end{aligned} $$

Step-by-Step Solution

Answer

The addition of the given hexadecimal numbers is 3076\(_{sixteen}\) + 5776\(_{sixteen}\) = 87D\(_{sixteen}\).

Step 1: Write down the given numbers in an addition format.

Write the two given hexadecimal numbers in an addition format, where 3076 is on top and 5776 is below it. $$ \begin{array}{c@{}c@{}c@{}c@{}c} & & 3 & 0 & 7\\ & & 5 & 7 & 6\\ \cline{1-5} \end{array} $$

Step 2: Add the least significant digits and carry if needed.

Add the least significant digits (7 and 6) and carry any value greater than base 16 (F). Here, 7+6=13 in base 10, which is equal to D in base 16. So, we will write D as the sum and will not carry any value. $$ \begin{array}{c@{}c@{}c@{}c@{}c} & & 3 & 0 & 7\\ & & 5 & 7 & 6\\ \cline{1-5} & & & & D\\ \end{array} $$

Step 3: Add the next least significant digits and carry if needed.

Add the next least significant digits (0 and 7) and carry any value greater than base 16 (F). Here, 0+7=7 in base 10, which is equal to 7 in base 16. So, we will write 7 as the sum and will not carry any value. $$ \begin{array}{c@{}c@{}c@{}c@{}c} & & 3 & 0 & 7\\ & & 5 & 7 & 6\\ \cline{1-5} & & & 7 & D\\ \end{array} $$

Step 4: Add the most significant digits and carry if needed.

Add the most significant digits (3 and 5) and carry any value greater than base 16 (F). Here, 3+5=8 in base 10, which is equal to 8 in base 16. So, we will write 8 as the sum and will not carry any value. $$ \begin{array}{c@{}c@{}c@{}c@{}c} & & 3 & 0 & 7\\ & & 5 & 7 & 6\\ \cline{1-5} & 8 & 7 & D\\ \end{array} $$

Step 5: The final result

We obtained the sum as 87D in base 16, which is the final answer for the addition of the given hexadecimal numbers. $$ \begin{array}{c@{}c@{}c@{}c@{}c} & & 3 & 0 & 7 & \text { sixteen } \\ & + & 5 & 7 & 6 & \text { sixteen } \\ \hline & 8 & 7 & D & \text { sixteen }\end{array} \ $$

Key Concepts

Discrete Mathematics and Its Applications

Discrete mathematics forms the mathematical foundation of computer and information science. It is concerned with the study of discrete objects and offers a wide array of tools for solving problems with a finite or countable number of elements. Among the topics covered in discrete mathematics, one finds algorithms, graph theory, and symbol manipulation, which directly applies to the operations of hexadecimal numbers.

Understanding hexadecimal addition is vital in areas like digital electronics and computer science, where binary representations are often grouped into hexadecimals for a more human-readable format. Mastery of hexadecimal arithmetic can also aid in logical design, encryption algorithms, and coding theory, all of which are key subjects within discrete mathematics.

Base 16 Arithmetic (Hexadecimal)

Base 16 arithmetic, also known as hexadecimal arithmetic, is a number system consisting of sixteen symbols: 0-9 and the letters A-F which represent the values 10-15. In contrast to the more familiar base 10 system, which has only ten unique digits, hexadecimal numbers can represent larger values in a more compact form, making it particularly useful for computing applications.

For example, the decimal number 255 can be represented as FF in hexadecimal. This compactness facilitates easier manipulation of large numbers, such as memory addresses in a computer, and maps nicely to two bytes in binary format (11111111). Transitioning between base 10 and base 16 is an essential skill for those working with digital systems.

Hexadecimal Operations

Hexadecimal operations function similarly to arithmetic in any other number base system, but with a few additional rules to handle the 16 different digit values. When adding hexadecimal numbers, as in the provided exercise, we add each column as in base 10, remembering to proceed from right to left. If the sum in any column is greater than 15 (F in hexadecimal), you carry the overflow to the next column on the left.

This arithmetic process may seem daunting, but it's akin to adding numbers in base 10, with the carry-over occurring every 16th value instead of every 10th. Remembering the hexadecimal to decimal equivalents (A=10, B=11, C=12, D=13, E=14, F=15) is crucial for smooth operation. These conversion skills are not only intellectual curiosities but have practical uses in technologies such as IPv6 address notation, color codes in HTML, and more.