Question

Convert the following binary numbers to hexadecimal. a. 10101001 b. 11100111 c. 01101110 d. 01111111

Short answer

a. A9, b. E7, c. 6E, d. 7F

Step-by-step solution

Understanding Binary and Hexadecimal

Binary numbers are in base 2 and use only the digits 0 and 1, whereas hexadecimal numbers are in base 16 and use the digits 0-9 and letters A-F (where A-F represent 10-15). To convert binary to hexadecimal, group the binary digits into sets of four, starting from the right.

Grouping Binary Digits

For each binary number, split the digits into groups of four, as this is how they directly translate to hexadecimal digits. Add leading zeros if necessary to complete the groups: - a. 10101001 → 1010 1001 - b. 11100111 → 1110 0111 - c. 01101110 → 0110 1110 - d. 01111111 → 0111 1111

Converting Binary Groups to Hexadecimal

Now convert each 4-digit binary group to its corresponding hexadecimal digit. Use the binary to hexadecimal table: - 0000 = 0, 0001 = 1, 0010 = 2, 0011 = 3, 0100 = 4 - 0101 = 5, 0110 = 6, 0111 = 7, 1000 = 8, 1001 = 9 - 1010 = A, 1011 = B, 1100 = C, 1101 = D, 1110 = E, 1111 = F Apply the conversions: - a. 1010 1001 → A9 - b. 1110 0111 → E7 - c. 0110 1110 → 6E - d. 0111 1111 → 7F

Combining Hexadecimal Digits

Combine the hexadecimal digits from each group to get the final hexadecimal number: - a. 10101001 in binary is A9 in hexadecimal. - b. 11100111 in binary is E7 in hexadecimal. - c. 01101110 in binary is 6E in hexadecimal. - d. 01111111 in binary is 7F in hexadecimal.

Key concepts

Understanding Binary Numbers

Binary numbers are numerical values expressed in base 2. This means they only utilize two digits: 0 and 1. Think of it as a series of switches—each digit is a switch that can either be off (0) or on (1).
Binary is the foundational language of computers, as they use it to process all kinds of data and instructions. Each binary digit represents a power of 2.
For example:

• The binary number 101 represents the decimal number 5, because it is calculated as:
  \(1\times2^2 + 0\times2^1 + 1\times2^0 = 5\).

To express larger numbers, you simply increase the number of binary digits (bits) used. It's a very efficient way of representing information in computing.

Hexadecimal Numbers Explained

Hexadecimal numbers are expressed in base 16, which means they utilize sixteen distinct symbols. These include the ten digits from 0 to 9 and six additional letters from A to F.
The letters A through F represent the numbers 10 through 15 in decimal.
Hexadecimal is often used in computing because it provides a more compact representation of data compared to binary.

• For example, the hexadecimal number 1A represents the decimal number 26.
  This is because: \(1\times16^1 + 10\times16^0 = 26\).

Hexadecimal numbers are particularly useful in programming and digital electronics, making it easier to understand binary-coded data by grouping every four binary digits into a single hexadecimal digit.

Base 2 and Base 16 Conversion

Understanding the conversion between binary (base 2) and hexadecimal (base 16) can simplify many computing tasks. The relationship between the bases is straightforward, as four binary digits (bits) align perfectly with one hexadecimal digit.

• To convert from binary to hexadecimal, first group the binary number into sets of four.
  Pad the number with additional zeros if necessary to make complete groups.
• Assign each group a hexadecimal digit using a conversion chart or table.
  For instance, the binary group 1010 converts to the hexadecimal digit A.

This alignment makes hexadecimal an excellent shorthand for expressing larger binary values, making data easier to read and manipulate.

Binary to Hexadecimal Table

A binary to hexadecimal table is a handy reference tool that simplifies the conversion process. By using the table, you can easily match binary groups to their corresponding hexadecimal symbols.
This is particularly helpful when dealing with large amounts of binary data.

• Here's a brief overview of the table:
  Binary 0000 equals hexadecimal 0, 0001 equals 1, 0010 equals 2, and so on up to 1111, which equals F.
• For example, if you encounter the binary 1101, consult the table to convert it to its hexadecimal equivalent D.

By memorizing common binary to hexadecimal conversions, you can enhance your efficiency and fluency with these numerical systems.