Unit 5: Data Representation
Introduction
Data representation is the process of transforming data into a format that can be easily understood, processed, and communicated by computers and humans. It involves converting raw data into a structured format, such as text, images, or numbers, which can be analyzed and interpreted. Data representation is essential in many areas of computing, including data analysis, machine learning, and artificial intelligence.
There are different methods of data representation, each suited to different types of data and applications. For example, text data can be represented using ASCII or Unicode encoding, while images can be represented using pixel values or vector graphics. Numeric data can be represented using different number systems, such as decimal, binary, or hexadecimal.
Data representation is crucial for anyone working with data, whether they are programmers, data scientists, or business analysts. It enables them to manipulate and analyze data effectively and to communicate their findings clearly to others. In this unit, we will explore some of the key concepts and techniques used in data representation, including binary and hexadecimal number systems, data compression, and data encryption.
Number System
A number system is a mathematical notation used to represent numbers. It consists of a set of symbols, called digits, and rules for combining those symbols to represent numbers. The most common number systems used in computing are the decimal, binary, and hexadecimal systems.
Decimal Number System:
The decimal number system, also known as base-10, is the number system that we use in our everyday life. It consists of ten digits (0 to 9) and uses positional notation, where the value of a digit depends on its position in a number. For example, in the number 1234, the digit 4 represents the ones place, the digit 3 represents the tens place, the digit 2 represents the hundreds place, and the digit 1 represents the thousands place.
Binary Number System:
The binary number system, also known as base-2, is used in computing to represent numbers using only two digits, 0 and 1. It uses positional notation, where the value of a digit depends on its position in a number. Each digit in a binary number is called a bit (short for binary digit). For example, the binary number 1010 represents the decimal number 10.
Hexadecimal Number System:
The hexadecimal number system, also known as base-16, is used in computing to represent numbers using 16 digits, 0 to 9 and A to F. It uses positional notation, where the value of a digit depends on its position in a number. Each digit in a hexadecimal number represents a group of 4 bits in binary. For example, the hexadecimal number 3F represents the binary number 00111111, which represents the decimal number 63.
The different number systems is important in computing because computers operate on binary numbers. Therefore, converting between different number systems is necessary when working with data in computing.
Conversion from Decimal to Binary
Converting a decimal number to binary involves dividing the decimal number by 2 and recording the remainder, then dividing the quotient by 2 and recording the remainder, and continuing this process until the quotient becomes 0. The binary number is then formed by writing the remainders in reverse order.
Here is an example of converting the decimal number 26 to binary:
Step 1: Divide 26 by 2, and record the remainder (0). The quotient is 13. Step 2: Divide 13 by 2, and record the remainder (1). The quotient is 6. Step 3: Divide 6 by 2, and record the remainder (0). The quotient is 3. Step 4: Divide 3 by 2, and record the remainder (1). The quotient is 1. Step 5: Divide 1 by 2, and record the remainder (1). The quotient is 0.
The remainders in reverse order are 11010. Therefore, the binary representation of the decimal number 26 is 11010.
Conversion from Decimal to Binary, Octal, Hexadecimal
Conversion to Binary:
- Divide 137 by 2 and record the remainder (1). The quotient is 68.
- Divide 68 by 2 and record the remainder (0). The quotient is 34.
- Divide 34 by 2 and record the remainder (0). The quotient is 17.
- Divide 17 by 2 and record the remainder (1). The quotient is 8.
- Divide 8 by 2 and record the remainder (0). The quotient is 4.
- Divide 4 by 2 and record the remainder (0). The quotient is 2.
- Divide 2 by 2 and record the remainder (0). The quotient is 1.
- Divide 1 by 2 and record the remainder (1). The quotient is 0.
The remainders in reverse order are 10001001. Therefore, the binary representation of the decimal number 137 is 10001001.
Conversion to Octal:
- Divide 137 by 8 and record the remainder (1). The quotient is 17.
- Divide 17 by 8 and record the remainder (1). The quotient is 2.
- Divide 2 by 8 and record the remainder (2). The quotient is 0.
The remainders in reverse order are 211. Therefore, the octal representation of the decimal number 137 is 211.
Conversion to Hexadecimal:
- Divide 137 by 16 and record the remainder (9, which represents the digit "9" in hexadecimal notation). The quotient is 8.
- Divide 8 by 16 and record the remainder (8, which represents the digit "8" in hexadecimal notation). The quotient is 0.
The remainders in reverse order are 89. Therefore, the hexadecimal representation of the decimal number 137 is 89.
Conversion of Binary, Octal, Hexadecimal to Decimal
Binary to Decimal:
To convert a binary number to decimal, we multiply each digit by the corresponding power of 2 and add up the results. For example, to convert the binary number 1101101 to decimal:
1 * 2^6 + 1 * 2^5 + 0 * 2^4 + 1 * 2^3 + 1 * 2^2 + 0 * 2^1 + 1 * 2^0 = 109
Therefore, the decimal representation of the binary number 1101101 is 109.
Octal to Decimal:
To convert an octal number to decimal, we multiply each digit by the corresponding power of 8 and add up the results. For example, to convert the octal number 753 to decimal:
7 * 8^2 + 5 * 8^1 + 3 * 8^0 = 491
Therefore, the decimal representation of the octal number 753 is 491.
Hexadecimal to Decimal:
To convert a hexadecimal number to decimal, we multiply each digit by the corresponding power of 16 and add up the results. For example, to convert the hexadecimal number AB7 to decimal:
10 * 16^2 + 11 * 16^1 + 7 * 16^0 = 2743
Therefore, the decimal representation of the hexadecimal number AB7 is 2743.
Conversion of Binary to Octal, Hexadecimal
To convert a binary number to octal, we group the binary digits into sets of three starting from the rightmost digit, and then convert each set of three to its equivalent octal digit. Here's an example:
Binary number: 10110110
Grouped into sets of three: 010 110 110
Equivalent octal digits: 2 6 6
Therefore, the octal representation of the binary number 10110110 is 266.
Binary to Hexadecimal:
To convert a binary number to hexadecimal, we group the binary digits into sets of four starting from the rightmost digit, and then convert each set of four to its equivalent hexadecimal digit. Here's an example:
Binary number: 11011010
Grouped into sets of four: 1101 1010
Equivalent hexadecimal digits: D A
Therefore, the hexadecimal representation of the binary number 11011010 is DA.
Conversion of Octal, Hexadecimal to Binary
Octal to Binary:
To convert an octal number to binary, we convert each octal digit to its equivalent three-digit binary number. Here's an example:
Octal number: 625
Each octal digit in order: 6 2 5
Equivalent three-digit binary numbers: 110 010 101
Combine the binary numbers: 110010101
Therefore, the binary representation of the octal number 625 is 110010101.
Hexadecimal to Binary:
To convert a hexadecimal number to binary, we convert each hexadecimal digit to its equivalent four-digit binary number. Here's an example:
Hexadecimal number: A3F
Each hexadecimal digit in order: A 3 F
Equivalent four-digit binary numbers: 1010 0011 1111
Combine the binary numbers: 101000111111
Therefore, the binary representation of the hexadecimal number A3F is 101000111111.
Binary Arithmetic
Binary arithmetic involves performing arithmetic operations using binary numbers, which only have two digits - 0 and 1. The basic binary arithmetic operations are addition, subtraction, multiplication, and division.
Addition: Binary addition works the same way as decimal addition, but we only have two possible digits (0 and 1). Here is an example of binary addition:
10101 (binary)
- 11110 (binary)
110111 (binary)
In this example, we add the digits from right to left, starting with the least significant bit. When the sum of two binary digits is 2 or greater, we write the remainder (either 0 or 1) and carry the quotient (either 1 or 2) to the next column.
Subtraction: Binary subtraction is also similar to decimal subtraction, but we need to borrow 1 from the next higher bit when we subtract a 1 from a 0. Here is an example of binary subtraction:
10101 (binary)
- 11110 (binary)
-0101 (binary)
In this example, we start by subtracting the digits from right to left, starting with the least significant bit. When we subtract a 1 from a 0, we borrow 1 from the next higher bit.
Multiplication: Binary multiplication is similar to decimal multiplication, but we only have two possible digits (0 and 1). Here is an example of binary multiplication:
101 (binary) x 110 (binary)
1010 (binary) 101
10010 (binary)
In this example, we multiply the multiplicand (101) by each digit of the multiplier (110), and then add up the results shifted to the left by the corresponding positions.
Division: Binary division is also similar to decimal division, but we only have two possible digits (0 and 1). Here is an example of binary division:
1010 (binary) / 11 (binary)
101 (binary) 11
Signed and Unsigned Numbers
Unsigned Numbers:
Unsigned numbers are binary numbers that represent non-negative values. In an unsigned binary number, the leftmost (most significant) bit represents the largest value, and the rightmost (least significant) bit represents the smallest value. For example, the binary number 11010 represents the decimal value 26 in unsigned notation. Unsigned numbers are often used in computer systems to represent values such as the number of bytes in a file or the memory location of a variable.
Signed Numbers:
Signed numbers are binary numbers that can represent both positive and negative values. In a signed binary number, the leftmost bit is used as a sign bit to indicate whether the number is positive or negative. If the sign bit is 0, the number is positive, and if the sign bit is 1, the number is negative. The remaining bits represent the absolute value of the number. For example, the signed binary number 10110 represents the decimal value -10 in signed notation. The sign bit is 1 to indicate that the number is negative, and the absolute value is 0110, which represents the decimal value 6. Signed numbers are often used in computer systems to represent values such as temperature, velocity, or balance in a bank account.
In conclusion, the main difference between signed and unsigned numbers is that signed numbers can represent both positive and negative values, while unsigned numbers can only represent non-negative values.
Binary Data Representation
Binary data representation is the method used by computers to represent all types of data, including text, numbers, images, audio, and video. Binary data consists of sequences of 0s and 1s, which can be interpreted by computers as instructions or data.
Binary data representation works by using a combination of bits (0s and 1s) to represent different values. The number of bits used to represent a value determines the range of values that can be represented. For example, an 8-bit binary number can represent values from 0 to 255 (2^8 - 1), while a 16-bit binary number can represent values from 0 to 65,535 (2^16 - 1).
Here are some examples of binary data representation:
- Text: Each character in a text document is represented by a binary code called ASCII (American Standard Code for Information Interchange). For example, the letter 'A' is represented by the binary code 01000001.
- Numbers: Numbers can be represented in binary format using a fixed number of bits. For example, the decimal number 12 can be represented in binary as 00001100 (8 bits) or 1100 (4 bits).
- Images: Images are represented in binary format by assigning a binary code to each pixel. The binary code represents the color of the pixel, which can be red, green, blue, or a combination of these colors.
- Audio: Audio is represented in binary format by sampling the sound wave and assigning a binary value to each sample. The binary value represents the amplitude of the sound wave at that point in time.
- Video: Video is represented in binary format by dividing the screen into pixels and assigning a binary code to each pixel. The binary code represents the color of the pixel, which can change rapidly to create the illusion of motion.
Binary data representation is the method used by computers to represent all types of data in binary format. Each type of data is represented using a different binary code, which allows the computer to interpret the data and perform operations on it.
Binary Coding Schemes
Binary coding schemes are methods of representing data in binary format. There are several binary coding schemes used in computer systems, and each scheme has its own set of rules for representing data.
Here are some common binary coding schemes:
- Binary Coded Decimal (BCD): In BCD, each decimal digit is represented by a 4-bit binary code. For example, the decimal number 12 is represented in BCD as 0001 0010. BCD is often used in electronic devices that require accurate decimal calculations, such as calculators and digital clocks.
- Gray Code: Gray Code is a binary coding scheme in which adjacent values differ by only one bit. For example, the binary codes for 0 and 1 are 00 and 01, respectively, and the binary codes for 1 and 2 are 01 and 11, respectively. Gray Code is often used in digital communication systems to minimize errors in data transmission.
- ASCII: ASCII (American Standard Code for Information Interchange) is a binary coding scheme used to represent text characters. In ASCII, each character is represented by a 7-bit binary code, which can represent up to 128 characters. For example, the letter 'A' is represented in ASCII as 01000001.
- Unicode: Unicode is a binary coding scheme used to represent text characters from multiple languages and scripts. In Unicode, each character is represented by a 16-bit binary code, which can represent up to 65,536 characters. Unicode is widely used on the Internet and in software applications to support multilingual text.
- IEEE 754: IEEE 754 is a binary coding scheme used to represent floating-point numbers in computers. In IEEE 754, a floating-point number is represented by a combination of a sign bit, an exponent, and a mantissa. IEEE 754 is widely used in scientific and engineering applications that require high precision calculations.
In conclusion, binary coding schemes are methods of representing data in binary format, and there are several different schemes used in computer systems. Each scheme has its own set of rules for representing data, and the choice of scheme depends on the type of data being represented and the requirements of the application.
Logic Gates
Logic gates are electronic circuits that perform logical operations on one or more input signals to produce an output signal. These gates are the basic building blocks of digital circuits and are used in computer processors, memory, and other electronic devices.
There are several types of logic gates, including:
- AND Gate: The AND gate produces a high output signal (i.e., 1) only when all of its input signals are high. Otherwise, it produces a low output signal (i.e., 0).
- OR Gate: The OR gate produces a high output signal when any of its input signals are high. It produces a low output signal only when all of its input signals are low.
- NOT Gate: The NOT gate produces an output signal that is the opposite of its input signal. If the input signal is high, the output signal is low, and vice versa.
- NAND Gate: The NAND gate is a combination of the AND gate and the NOT gate. It produces a low output signal only when all of its input signals are high. Otherwise, it produces a high output signal.
- NOR Gate: The NOR gate is a combination of the OR gate and the NOT gate. It produces a high output signal only when all of its input signals are low. Otherwise, it produces a low output signal.
- XOR Gate: The XOR gate produces a high output signal only when the number of high input signals is odd. Otherwise, it produces a low output signal.
- XNOR Gate: The XNOR gate is a combination of the XOR gate and the NOT gate. It produces a high output signal only when the number of high input signals is even. Otherwise, it produces a low output signal.
These logic gates are combined in different ways to create more complex digital circuits. By using logic gates, complex operations can be performed on digital signals, allowing computers and other electronic devices to process and store information.
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