Binary Hexidecimal Representations
Binary hexadecimal representations involve two distinct number systems: binary, which is a base-2 system, and hexadecimal, a base-16 system. The binary system uses only two symbols, 0 and 1, to represent values, making it fundamental for digital devices such as computers, which utilize binary data to store information and perform operations. Each binary digit, known as a bit, can represent one of two states, and combinations of bits can express larger numbers, with groups of four bits forming a nibble and eight bits forming a byte.
Hexadecimal simplifies binary representation by encoding larger values into a more compact form. It employs the digits 0 to 9 and the letters A to F to represent values ranging from 0 to 15, allowing a single hex digit to represent a four-bit binary group. This makes hexadecimal particularly useful for humans when interacting with binary data, as it reduces the complexity of reading long sequences of binary digits. For example, the binary representation of one million is lengthy, while its hexadecimal counterpart is concise, demonstrating how hexadecimal enhances human readability and efficiency in computing contexts. Understanding these systems is essential for those working in computer science and software engineering, as they play a critical role in data representation and manipulation.
On this Page
- ABSTRACT
- Understanding the Binary Number System
- Position 8
- Position 7
- Position 6
- Position 5
- Position 4
- Position 3
- Position 2
- Position 1
- Position 3
- Position 2
- Position 1
- Position 8
- Position 7
- Position 6
- Position 5
- Position 4
- Position 3
- Position 2
- Position 1
- The Importance of the Binary Number System
- Sample Problem
- Answer:
- Using Hexadecimal to Simplify Binary Numbers
- Position 8
- Position 7
- Position 6
- Position 5
- Position 4
- Position 3
- Position 2
- Position 1
- Position 5
- Position 4
- Position 3
- Position 2
- Position 1
- Bibliography
Subject Terms
Binary Hexidecimal Representations
- FIELDS OF STUDY: Computer Science; Computer Engineering; Software Engineering
ABSTRACT
The binary number system is a base-2 number system. It is used by digital devices to store data and perform mathematical operations. The hexadecimal number system is a base-16 number system. It enables humans to work efficiently with large numbers stored as binary data.
Understanding the Binary Number System
A mathematical number system is a way of representing numbers using a defined set of symbols. Number systems take their names from the number of symbols the system uses to represent numbers. For example, the most common mathematical number system is the decimal system, or base-10 system. Deci- means "ten." It uses the ten digits 0 through 9 as symbols for numbers. Number systems can be based on any number of unique symbols, however. For example, the number system based on the use of two digit symbols (0 and 1) is called the binary or base-2 system.
Both the decimal and binary number systems use the relative position of digits in a similar way when representing numbers. The value in the rightmost, or first, position is multiplied by the number of digits used in the system to the zero power. For the decimal system, this value is 100. For the binary system, this value is 20. Both 100 and 20 are equal to 1. Any number x raised to the zero power is equal to 1. The power used increases by one for the second position, and so on.
Position 8
Seventh Power
Decimal 10,000,000 or 107
Binary 128 or 27
Position 7
Sixth Power
Decimal 1,000,000 or 106
Binary 64 or 26
Position 6
Fifth Power
Decimal 100,000 or 105
Binary 32 or 25
Position 5
Fourth Power
Decimal 10,000 or 104
Binary 16 or 24
Position 4
Third Power
Decimal 1,000 or 103
Binary 8 or 23
Position 3
Second Power
Decimal 100 or 102
Binary 4 or 22
Position 2
First Power
Decimal 10 or 101
Binary 2 or 21
Position 1
Zero Power
Decimal 1 or 100
Binary 1 or 20
Using the decimal number system, the integer 234 is represented by placing the symbols 2, 3, and 4 in positions 3, 2, and 1, respectively.
Position 3
Decimal 100 or 102
Digits 2
Position 2
Decimal 10 or 101
Digits 3
Position 1
Decimal 1 or 100
Digits 4
In the decimal system, 234 = (2 × 100) + (3 × 10) + (4 × 1), or (2 × 102) + (3 × 101) + (4x100). The binary system uses the relative position of the symbols 0 and 1 to express the integer 234 in a different manner.
Position 8
Binary 128 or 27
Bit 1
Position 7
Binary 64 or 26
Bit 1
Position 6
Binary 32 or 25
Bit 1
Position 5
Binary 16 or 24
Bit 0
Position 4
Binary 8 or 23
Bit 1
Position 3
Binary 4 or 22
Bit 0
Position 2
Binary 2 or 21
Bit 1
Position 1
Binary 1 or 20
Bit 0
In the binary system, 234 = (1 × 128) + (1 × 64) + (1 × 32) + (0 × 16) + (1 × 8) + (0 × 4) + (1 × 2) + (0 × 1), or 234 = (1 × 27) + (1 × 26) + (1 × 25) + (0 × 24) + (1 × 23) + (0 × 22) + (1 × 21) + (0 × 20).
The Importance of the Binary Number System
The binary number system is used to store numbers and perform mathematical operations in computer systems. Such devices store data using transistors, electronic parts that can each be switched between two states. One state represents the binary digit 0, and the other, the binary digit 1. These binary digits are bits, the smallest units of data that can be stored and manipulated. A single bit can be used to store the value 0 or 1. To store values larger than 1, groups of bits are used. A group of four bits is a nibble. A group of eight bits is a byte.
Sample Problem
To work with binary numbers in digital applications, it is important to be able to translate numbers from their binary values to their decimal values. Translate the following binary byte to its decimal value: 10111001
Answer:
The decimal value of the binary byte 10111001 is 185. The decimal value can be determined using a chart and then calculating.
128 or 27
Bit 1
64 or 26
Bit 0
32 or 25
Bit 1
16 or 24
Bit 1
8 or 23
Bit 1
4 or 22
Bit 0
2 or 21
Bit 0
1 or 20
Bit 1
= (1 × 27) + (0 × 26) + (1 × 25) + (1 × 24) + (1 × 23) + (0 × 22) + (0 × 21) + (1 × 20)
= (1 × 128) + (0 × 64) + (1 × 32) + (1 ×16) + (1 × 8) + (0 × 4) + (0 × 2) + (1 × 1)
= 185
Using Hexadecimal to Simplify Binary Numbers
The hexadecimal number system is a base-16 system. It uses the digits 0 through 9 and the letters A through F to represent numbers. The hexadecimal digit, or hex digit, A has a decimal value of 10. Hex digit B equals 11, C equals 12, D equals 13, E equals 14, and F equals 15. In hexadecimal, the value 10 is equal to 16 in the decimal system. Using hexadecimal, a binary nibble can be represented by a single symbol. For example, the hex digit F can be used instead of the binary nibble 1111 for the decimal value 15. Sixteen different combinations of bits are possible in a binary nibble. The hexadecimal system, with sixteen different symbols, is therefore ideal for working with nibbles.
Position 8
Seventh Power
Decimal 10,000,000 or 107
Hexadecimal 268,435,456 or 167
Position 7
Sixth Power
Decimal 1,000,000 or 106
Hexadecimal 16,777,216 or 166
Position 6
Fifth Power
Decimal 100,000 or 105
Hexadecimal 1,048,576 or 165
Position 5
Fourth Power
Decimal 10,000 or 104
Hexadecimal 65,536 or 164
Position 4
Third Power
Decimal 1,000 or 103
Hexadecimal 4,096 or 163
Position 3
Second Power
Decimal 100 or 102
Hexadecimal 256 or 162
Position 2
First Power
Decimal 10 or 101
Hexadecimal 16 or 161
Position 1
Zero Power
Decimal 1 or 100
Hexadecimal 1 or 160
One disadvantage of using binary is that large numbers of digits are needed to represent large integers. For example, 1,000,000 is shown in binary digits as 11110100001001000000. The same number is shown in hex digits as F4240, which is equal to (15 × 65,536) + (4 × 4,096) + (2 × 256) + (4 × 16) + (0 × 1).
Position 5
Hexadecimal 65,536 or 164
Hex digits F or 15
Position 4
Hexadecimal 4,096 or 163
Hex digits 4
Position 3
Hexadecimal 256 or 162
Hex digits 2
Position 2
Hexadecimal 16 or 161
Hex digits 4
Position 1
Hexadecimal 1 or 160
Hex digits 0
Computers can quickly and easily work with large numbers in binary. Humans have a harder time using binary to work with large numbers. Binary uses many more digits than hexadecimal does to represent large numbers. Hex digits are, therefore, easier for humans to use to write, read, and process than binary.
Bibliography
Cheever, Erik. "Representation of Numbers." Swarthmore College, www.swarthmore.edu/NatSci/echeeve1/Ref/BinaryMath/NumSys.html. Accessed 28 Jan. 2025.
Glaser, Anton. History of Binary and Other Nondecimal Numeration. Rev. ed., Tomash Publishers, 1981.
Govindjee, S. "Internal Representation of Numbers." University of California Berkeley, Department of Civil and Environmental Engineering, Spring 2013.
Lande, Daniel R. "Development of the Binary Number System and the Foundations of Computer Science." Mathematics Enthusiast, 1 Dec. 2014, pp. 513-40.
Reselman, Rob. “Binary and Hexadecimal Numbers Explained for Developers.” TechTarget, 10 Jan. 2022, www.theserverside.com/tip/Binary-and-hexadecimal-numbers-explained-for-developers. Accessed 28 Jan. 2025.