# Binary Number System

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## What is the Binary Number System?

The simple binary number system definition is a way of representing numbers that use only two digits: 0 and 1. Each digit in this system is a bit, or binary digit. This system forms the foundation of all modern computing and digital electronics.

Unlike the decimal system, which is based on ten different digits (0 through 9) and powers of 10, the binary system operates on powers of 2. In a decimal number, the value of each position represents a power of 10, increasing from right to left. For example, in the number 207, the 7 is in the ‘ones’ place, the 0 in the ‘tens’ place, and the 2 in the ‘hundreds’ place.

In binary, each position represents a power of 2, with the rightmost position being 2^0 (1 in decimal), and each subsequent position to the left increasing in value by powers of 2. Therefore, the binary number “10” represents 2 in decimal, and “100” represents 4 in decimal.

The choice of binary for computer systems is due to its simplicity in representing data electronically. Electrical circuits and devices can easily distinguish between two states (such as high/low, on/off) to represent 0s and 1s.

This binary representation is great for storing, manipulating, and communicating data.

Binary numbers represent all types of data. Text, images, audio, and video are encoded in binary format for processing and storage. Even complex instructions that tell the computer what to do are broken down into binary codes.

### Techopedia Explains the Binary Number System Meaning

The binary number system operates with just two symbols, 0 and 1, making it the simplest form of numeral representation. Each binary digit, or bit, is the basic unit of data in computing, representing the power of 2 values.

This system efficiently represents and manipulates numbers. This is important for computer operations and digital electronics due to its straightforward implementation of on (1) and off (0) states. That’s the meaning of the binary number system.

Binary’s adoption in technology is primarily due to its reliability and simplicity. It fits perfectly with the binary nature of electronic circuits which can easily distinguish between two states. This alignment with electronic logic brings on the creation of storage, processing, and control mechanisms within computers.

## How Binary Number System Works

The binary number system represents numbers using just 0 and 1. Every digit in this system is known as a bit. Unlike the decimal system, where each digit position represents a power of 10, in the binary system, each position represents a power of 2.

Starting from the right, the first position is 2^0 (which equals 1), the next position to the left is 2^1 (which equals 2), then 2^2 (which equals 4), and so on. By combining these 0s and 1s, the binary system can represent any number.

For example, the binary number “101” represents the decimal number 5 because it’s calculated as (1 × 2^2) + (0 × 2^1) + (1 × 2^0), or 4 + 0 + 1.

### Bits and Bytes

Bit
The smallest unit of data in computing, symbolized by a 0 or a 1. Bits are often grouped together to form larger units. For example, 8 bits make a “byte,” which is the standard unit for measuring digital information.

Byte

Can represent 256 different values (2^8), ranging from 00000000 to 11111111 in binary. This capacity allows bytes to encode a wide range of data, from simple numerical values to characters in a text.

Bytes are important in data representation because they serve as the foundation for larger data units, like kilobytes (KB), megabytes (MB), gigabytes (GB), and so forth, each scaling up by a factor of 1024 (or 2^10).

This hierarchy of data units facilitates the organization, processing, and storage of digital information in a manner that’s both efficient and understandable for computers and humans. Through the aggregation of bits into bytes and larger units, complex data and instructions can be represented, manipulated, and stored in digital devices.

## Binary Numbers Table

A binary numbers table is a handy way to understand how decimal numbers (the numbers we use every day) translate into binary numbers (the numbers computers use).

We’ll show you the decimal numbers 0-25 and their binary equivalents.

 Decimal Binary 0 0 1 1 2 10 3 11 4 100 5 101 6 110 7 111 8 1000 9 1001 10 1010 11 1011 12 1100 13 1101 14 1110 15 1111 16 10000 17 10001 18 10010 19 10011 20 10100 21 10101 22 10110 23 10111 24 11000 25 11001

### How to Read and Interpret the Table

To read this table, find the decimal number in the left column and look directly across to the right column to find its binary equivalent. For example, the decimal number 5 translates to 101 in binary.

Each digit in a binary number represents a power of 2, with the rightmost digit representing 2^0, the next digit to the left representing 2^1, and so on. The value of a binary number is the sum of these powers of 2, where the digit is 1.

For example, the binary number 1010 (representing decimal 10) is calculated as follows: (1 × 2^3) + (0 × 2^2) + (1 × 2^1) + (0 × 2^0), which equals 8 + 0 + 2 + 0 = 10 in decimal.

## How to Calculate Binary Numbers

### Converting Decimal to Binary

2. Divide the number by 2 and record the remainder. This remainder is the rightmost digit (least significant bit) of the binary number.

3. Divide the number by 2 and record the remainder. This remainder is the rightmost digit (least significant bit) of the binary number.

4. Use the quotient from the division as the new number to divide by 2.

5. Repeat the process of dividing by 2 and recording the remainder until the quotient becomes 0.

6. Write down the remainders in reverse order. This sequence is the binary representation of the original decimal number.

Example: Convert the decimal number 13 to binary.

• 13 ÷ 2 = 6 remainder 1
• 6 ÷ 2 = 3 remainder 0
• 3 ÷ 2 = 1 remainder 1
• 1 ÷ 2 = 0 remainder 1

Reading the remainders in reverse gives us 1101, so 13 in decimal is 1101 in binary.

### Converting Binary to Decimal

2. Multiply each bit by 2 raised to the power of its position, counting from right to left and starting with 0.
3. Sum all the values obtained from the multiplication. This sum is the decimal representation of the binary number.

Example: Convert the binary number 1101 to decimal.

• (1 × 2^3) + (1 × 2^2) + (0 × 2^1) + (1 × 2^0) = 8 + 4 + 0 + 1 = 13

So, 1101 in binary is 13 in decimal.

## Operations on Binary Numbers

Performing basic arithmetic operations – addition, subtraction, multiplication, and division – in the binary system follows rules similar to those in the decimal system, but just with 0 and 1.

Let’s break down each operation with examples.

1. Add the bits from right to left.
2. If the sum is 2 (10 in binary), carry over 1 to the next higher bit.

Example: Add 1011 (11 in decimal) and 1101 (13 in decimal).

1011 + 1101 = 11000

The result is 11000, which is 24 in decimal.

### Binary Subtraction

1. Subtract the bits from right to left.
2. If you need to subtract a larger bit from a smaller bit, borrow 1 from the next left bit (just like in decimal subtraction).

Example: Subtract 1001 (9 in decimal) from 1101 (13 in decimal).

1101 – 1001 = 0100

The result is 0100, which is 4 in decimal.

### Binary Multiplication

1. Multiply each bit by the entire binary number, one row at a time, shifting one place to the left with each new row (similar to decimal multiplication).
2. Add up all the rows.

Example: Multiply 101 (5 in decimal) by 11 (3 in decimal).

Written out, it would look like this:

The result is 1111, which is 15 in decimal.

### Binary Division

1. Divide the binary numbers from left to right.
2. Subtract and bring down the next bit, just like long division in the decimal system.

Example: Divide 1100 (12 in decimal) by 100 (4 in decimal).

The quotient is 11 (3 in decimal), and the remainder is 0.

## Binary Number System Example

The binary number system is fundamental to the operation of computers and other digital devices. Here are some real-world examples.

### Text Encoding

One of the most common applications of the binary number system is in text encoding, such as in the American Standard Code for Information Interchange (ASCII) system. In ASCII, each letter, number, and symbol is represented by a unique 7-bit binary code. For example, the capital letter “A” is represented by the binary code 0100001, and the lowercase “a” is represented by 0110001.

This binary representation allows computers to store and manipulate text data efficiently. When you type on a keyboard, each keystroke is converted into its corresponding binary code, allowing the computer to understand and display the characters on the screen.

### Digital Data Storage

Every bit of data stored in your computer, smartphone, or any digital device is represented as a series of 0s and 1s. Whether it’s a document, an image, a piece of music, or a video, all are stored using binary code.

For example, a text file is stored as a sequence of binary codes, with each character represented by a specific binary sequence according to a character encoding standard like ASCII or Unicode.

### Network Communications

Data transmitted over the Internet, whether it’s an email, a webpage, or a video stream, is sent as packets of binary data. Network devices, such as routers and switches, use binary addresses (like IP addresses) to route data from its source to its destination.

The binary system facilitates the error detection and correction algorithms that ensure data integrity during transmission.

### Control Systems and Robotics

In control systems and robotics, binary numbers are used to operate and control machinery and equipment. Robots receive operational commands as binary signals, which are then interpreted to perform specific actions, such as moving a limb or turning on a switch.

## The Bottom Line

The binary number system, using only 0s and 1s, is fundamental to all digital technology, from data storage to network communications and robotics. It allows for complex data representation and processing, making it an important part of computing and electronic devices.

The binary system’s principles remain at the center of emerging fields like quantum computing and artificial intelligence (AI).