Problem 14 - Basic Engineering Mathematics

Question

Convert the following decimal numbers into their hexadecimal equivalents: (a) \(37_{10}\) (b) \(108_{10}\)

Step-by-Step Solution

Answer

37 in decimal is 25 in hexadecimal; 108 in decimal is 6C in hexadecimal.

Step 1: Convert Decimal to Binary for 37_10

First, we convert the decimal number 37 to binary. Divide the number by 2 and record the remainder:\(37 \div 2 = 18\) remainder \(1\)\(18 \div 2 = 9\) remainder \(0\)\(9 \div 2 = 4\) remainder \(1\)\(4 \div 2 = 2\) remainder \(0\)\(2 \div 2 = 1\) remainder \(0\)\(1 \div 2 = 0\) remainder \(1\)Writing the remainders from bottom to top, we get the binary: \(100101_2\).

Step 2: Convert Binary to Hexadecimal for 37_10

Convert the binary number \(100101_2\) into hexadecimal by breaking it into groups of 4 bits from right:Group 1: \(1001\) - which is \(9\) in decimal.Group 2: \(0101\) - which is \(5\) in decimal.Thus, \(100101_2\) is \(25_{16}\) in hexadecimal.

Step 3: Convert Decimal to Binary for 108_10

Convert the decimal number 108 to binary:\(108 \div 2 = 54\) remainder \(0\)\(54 \div 2 = 27\) remainder \(0\)\(27 \div 2 = 13\) remainder \(1\)\(13 \div 2 = 6\) remainder \(1\)\(6 \div 2 = 3\) remainder \(0\)\(3 \div 2 = 1\) remainder \(1\)\(1 \div 2 = 0\) remainder \(1\)Writing the remainders from bottom to top, we get the binary: \(1101100_2\).

Step 4: Convert Binary to Hexadecimal for 108_10

Convert the binary number \(1101100_2\) into hexadecimal by padding with zeros on the left if necessary and then breaking into 4 bits groups:Binary: \(0110 1100\) (We added a 0 at the start to make a full group of 4)Group 1: \(0110\) - which is \(6\) in decimal.Group 2: \(1100\) - which is \(C\) in hexadecimal.Thus, \(1101100_2\) is \(6C_{16}\) in hexadecimal.

Key Concepts

Decimal to Binary Conversion

Understanding how to convert a decimal number to binary is foundational in working with number systems. Decimal numbers are what we use in our everyday life. But computers use binary, a system based entirely on the numbers 0 and 1.

Here's how to do it:

• Start by dividing the decimal number by 2.
• Write down the remainder (this will be 0 or 1).
• Continue dividing the quotient obtained by 2, writing down the remainder each time, until the quotient is 0.
• The binary number is then constructed by reading all the remainders from bottom to top.

For example, to convert 37 to binary:

• Divide 37 by 2 gets a quotient of 18 and a remainder of 1.
• Divide 18 by 2 gets a quotient of 9 and a remainder of 0.
• Keep going until you divide by 2 and get a quotient of 0.

By stringing together these remainders from the last to the first, the binary representation is formed. Thus, 37 in binary becomes \(100101_2\).

Binary to Hexadecimal Conversion

After converting a decimal to binary, the next step could be converting the binary number into a hexadecimal format. Hexadecimal represents numbers using a base-16 system, which is often used in computing for its simplicity in representing binary-coded values.

Here's how you convert binary to hexadecimal:

• Break down the binary number into groups of four bits, starting from the right. Add leading zeros if necessary to complete the last group.
• Convert each group of four into a single hexadecimal digit.
• The hexadecimal system uses numbers 0-9 and letters A-F (where A=10 and F=15) as digits.

Taking \(100101_2\) and breaking it into groups gives you \(1001\) and \(0101\).

• \(1001\) transforms to 9 in decimal.
• \(0101\) corresponds to 5 in decimal.
• The combined hexadecimal representation becomes \(25_{16}\).

This is a quick way to switch from binary, which can be lengthy, to a more compact number in hexadecimal.

Understanding Number Systems Conversions

Converting numbers between different systems is essential when working with computers or digital electronics. Whether you are programming or engaged in system design, fluently traversing between decimal, binary, and hexadecimal systems is key.

Here’s a brief outline of why these conversions matter:

• **Decimal System (Base 10)**: Used in everyday normal counting and mathematics.
• **Binary System (Base 2)**: All digital electronics, including computers, operate in binary, as it represents an off (0) or on (1) state of a switch.
• **Hexadecimal System (Base 16)**: Offers a concise representation of binary numbers, which can be quite long. One hex digit efficiently represents four binary digits.

By understanding the processes of converting, for example, from decimal to binary and then to hexadecimal, you gain a fundamental comprehension of how data is truly represented and manipulated within computer systems.

This set of skills not only helps in educational settings but also lays the groundwork for complex problem-solving in technological fields.