Binary Number System

Overview

The binary number system is the foundation of computer science and digital electronics. Unlike the decimal system, which uses ten digits (0–9), the binary system uses only two digits: 0 and 1. These two digits represent the two possible states of digital circuits, often interpreted as off/on, false/true, or low/high voltage.

Computers rely on binary because their electronic circuits can easily distinguish between two states. Each binary digit, or bit, serves as the basic unit of information in computing. Complex data, instructions, and operations are ultimately expressed as sequences of bits, enabling the computer to process information efficiently.

In this article, we will explore how the binary system works, how to convert between decimal and binary, and its practical applications in computers.

1. How Binary Works

1.1 Binary Digits and Powers of Two

In the binary system, each digit represents a power of two, increasing from right to left. The rightmost digit represents 202^020, the next one 212^121, and so on. This positional value system allows us to express any number using only 0s and 1s.

Example

Consider the binary number 1011:

• Rightmost digit: 1×20=11 \times 2^0 = 1 1×20=1
• Next digit: 1×21=21 \times 2^1 = 2 1×21=2
• Next digit: 0×22=00 \times 2^2 = 0 0×22=0
• Leftmost digit: 1×23=81 \times 2^3 = 8 1×23=8

Adding these together: 8+0+2+1=118 + 0 + 2 + 1 = 118+0+2+1=11

Thus, the binary number 1011 represents the decimal number 11.

1.2 Bits and Bytes

A bit is a single binary digit (0 or 1). A byte consists of 8 bits and is the standard unit for measuring data in computers. Multiple bytes can represent larger numbers, text, or complex data types.

• 1 bit: smallest unit, 0 or 1
• 1 byte: 8 bits, can represent 256 possible values (0–255)
• Kilobyte (KB): 1024 bytes
• Megabyte (MB): 1024 KB
• Gigabyte (GB): 1024 MB

Bytes are the basic building blocks for storing data, instructions, and files in digital systems.

1.3 Binary Arithmetic

Computers use binary arithmetic to perform calculations. Basic operations include addition, subtraction, multiplication, and division using binary numbers.

1.3.1 Binary Addition

Binary addition follows simple rules:

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 10 (0 carry 1)

Example:

  1011
+ 1101
------
11000

1.3.2 Binary Subtraction

Binary subtraction uses borrowing similar to decimal subtraction:

• 0 – 0 = 0
• 1 – 0 = 1
• 1 – 1 = 0
• 0 – 1 = 1 (borrow from the next higher bit)

1.3.3 Binary Multiplication

Binary multiplication is simpler than decimal multiplication:

• 0 × 0 = 0
• 0 × 1 = 0
• 1 × 0 = 0
• 1 × 1 = 1

Example:

  101
× 11
------
  101
+1010
------
1111

1.3.4 Binary Division

Binary division follows the same logic as decimal division but uses only 0 and 1.

Example: Divide 1100 by 10:

1100 ÷ 10 = 110

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2. Conversion from Decimal to Binary

Decimal numbers (base 10) can be converted to binary (base 2) using a division-remainder method.

2.1 Division-Remainder Method

1. Divide the decimal number by 2.
2. Record the remainder (0 or 1).
3. Continue dividing the quotient by 2 until the quotient becomes 0.
4. The binary number is formed by reading the remainders from bottom to top.

Example

Convert decimal 23 to binary:

1. 23 ÷ 2 = 11 remainder 1
2. 11 ÷ 2 = 5 remainder 1
3. 5 ÷ 2 = 2 remainder 1
4. 2 ÷ 2 = 1 remainder 0
5. 1 ÷ 2 = 0 remainder 1

Reading from bottom to top: 10111

Thus, decimal 23 = binary 10111

2.2 Conversion Using Subtraction Method

1. Find the largest power of 2 less than or equal to the number.
2. Subtract it from the number and mark 1 in that position.
3. Repeat with the remainder until reaching 0, marking 0 for unused powers of 2.

Example: Convert decimal 18:

• Largest power of 2 ≤ 18 is 16 (2^4), mark 1
• 18 – 16 = 2, largest power ≤ 2 is 2 (2^1), mark 1
• Remaining powers (2^3, 2^2, 2^0) are unused, mark 0

Binary: 10010

2.3 Binary to Decimal Conversion

To convert from binary to decimal:

1. Multiply each bit by its corresponding power of 2.
2. Sum the results.

Example: Convert binary 1101 to decimal: (1×23)+(1×22)+(0×21)+(1×20)=8+4+0+1=13(1 × 2^3) + (1 × 2^2) + (0 × 2^1) + (1 × 2^0) = 8 + 4 + 0 + 1 = 13(1×23)+(1×22)+(0×21)+(1×20)=8+4+0+1=13

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3. Use of Binary in Computers

Binary is the language of computers. All data—text, images, audio, video, and program instructions—are ultimately stored and processed in binary form.

3.1 Binary and Digital Circuits

Digital circuits in computers operate on two voltage levels representing 0 and 1:

• 0: Low voltage, off, false
• 1: High voltage, on, true

Transistors, the building blocks of digital circuits, act as switches that turn on and off based on input signals, making binary representation natural for electronic systems.

3.2 Binary in Data Representation

3.2.1 Text

Text characters are represented in binary using character encoding standards like ASCII or Unicode.

• Example: ASCII code for ‘A’ = 65 decimal = 01000001 binary

3.2.2 Images

Images are converted into binary by representing each pixel as a combination of bits:

• Black and white images: 1 bit per pixel
• Grayscale images: 8 bits per pixel
• Color images: 24 bits per pixel (8 bits each for red, green, and blue)

3.2.3 Audio

Audio signals are sampled and converted into binary using analog-to-digital conversion (ADC). Each sample is stored as a binary number representing amplitude.

3.2.4 Video

Video consists of a sequence of images and audio data, all stored in binary. Compression techniques like MP4 or AVI convert raw binary data into manageable sizes for storage and transmission.

3.3 Binary in Processing

The CPU performs operations directly on binary numbers using logic circuits:

• Arithmetic operations: Addition, subtraction, multiplication, division
• Logical operations: AND, OR, NOT, XOR
• Data movement: Reading and writing memory locations

Every instruction executed by a computer is ultimately converted to machine code, a binary representation that the processor can understand.

3.4 Binary and Networking

Data transmitted over networks is sent as binary signals:

• Electrical pulses in copper wires
• Light pulses in fiber-optic cables
• Radio waves in wireless communication

Binary ensures accuracy and reliability in digital data transmission.

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4. Advantages of Binary System

1. Simplicity: Only two states make design of circuits straightforward.
2. Reliability: Distinguishing between high and low voltage is easy, reducing errors.
3. Compatibility: Suited for digital electronics and modern computer architectures.
4. Efficient Processing: Computers can perform arithmetic and logical operations faster using binary.

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5. Binary Number System in Modern Computing

Binary underpins all aspects of computing:

• Machine Language: Binary instructions directly executed by the CPU.
• File Storage: Data of all types is stored in binary files.
• Encryption: Binary sequences form the basis for cryptographic keys and secure data.
• Artificial Intelligence: AI models process binary-encoded numerical inputs.