{"id":10668,"date":"2011-05-27T00:00:00","date_gmt":"2011-05-27T00:00:00","guid":{"rendered":"https:\/\/www.techopedia.com\/definition\/binary-number-system\/"},"modified":"2024-04-12T11:12:39","modified_gmt":"2024-04-12T11:12:39","slug":"binary-number-system","status":"publish","type":"definition","link":"https:\/\/www.techopedia.com\/definition\/24787\/binary-number-system","title":{"rendered":"Binary Number System"},"content":{"rendered":"

What is the Binary Number System?<\/span><\/h2>\n

The simple binary<\/a> 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<\/a>. This system forms the foundation of all modern computing<\/a> and digital electronics.<\/p>\n

Unlike the decimal system<\/a>, 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.<\/p>\n

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.<\/p>\n

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

This binary representation is great for storing, manipulating, and communicating data<\/a>.<\/p>\n

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

Techopedia Explains the Binary Number System Meaning<\/h3>\n

\"Binary<\/p>\n

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<\/a>, representing the power of 2 values.<\/p>\n

This system efficiently represents and manipulates numbers. This is important for computer<\/a> operations and digital electronics due to its straightforward implementation of on (1) and off (0) states. That\u2019s the meaning of the binary number system.<\/p>\n

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<\/a> brings on the creation of storage, processing, and control mechanisms within computers.<\/p>\n

How Binary Number System Works<\/span><\/h2>\n

The binary number<\/a> 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.<\/p>\n

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.<\/p>\n

For example<\/strong>, the binary number “101<\/strong>” represents the decimal number 5<\/strong> because it’s calculated as (1 \u00d7 2^2) + (0 \u00d7 2^1) + (1 \u00d7 2^0)<\/strong>, or 4 + 0 + 1<\/strong>.<\/p>\n

\"How<\/p>\n

Bits and Bytes<\/h3>\n

\n

<\/span>Bit<\/strong><\/div>
The smallest unit of data in computing<\/a>, 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. <\/div><\/div>\n

<\/span>Byte<\/strong><\/div>
\n

Can represent 256 different values<\/a> (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.<\/p>\n

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

<\/div><\/div>\n

<\/div>\n

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

Binary Numbers Table<\/span><\/h2>\n

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).<\/p>\n

We\u2019ll show you the decimal numbers 0-25 and their binary equivalents.<\/p>\n

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Decimal<\/strong><\/td>\nBinary<\/strong><\/td>\n<\/tr>\n
0<\/td>\n0<\/td>\n<\/tr>\n
1<\/td>\n1<\/td>\n<\/tr>\n
2<\/td>\n10<\/td>\n<\/tr>\n
3<\/td>\n11<\/td>\n<\/tr>\n
4<\/td>\n100<\/td>\n<\/tr>\n
5<\/td>\n101<\/td>\n<\/tr>\n
6<\/td>\n110<\/td>\n<\/tr>\n
7<\/td>\n111<\/td>\n<\/tr>\n
8<\/td>\n1000<\/td>\n<\/tr>\n
9<\/td>\n1001<\/td>\n<\/tr>\n
10<\/td>\n1010<\/td>\n<\/tr>\n
11<\/td>\n1011<\/td>\n<\/tr>\n
12<\/td>\n1100<\/td>\n<\/tr>\n
13<\/td>\n1101<\/td>\n<\/tr>\n
14<\/td>\n1110<\/td>\n<\/tr>\n
15<\/td>\n1111<\/td>\n<\/tr>\n
16<\/td>\n10000<\/td>\n<\/tr>\n
17<\/td>\n10001<\/td>\n<\/tr>\n
18<\/td>\n10010<\/td>\n<\/tr>\n
19<\/td>\n10011<\/td>\n<\/tr>\n
20<\/td>\n10100<\/td>\n<\/tr>\n
21<\/td>\n10101<\/td>\n<\/tr>\n
22<\/td>\n10110<\/td>\n<\/tr>\n
23<\/td>\n10111<\/td>\n<\/tr>\n
24<\/td>\n11000<\/td>\n<\/tr>\n
25<\/td>\n11001<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

<\/div>\n

How to Read and Interpret the Table<\/h3>\n

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.<\/p>\n

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.<\/p>\n

For example, the binary number 1010<\/strong> (representing decimal 10<\/strong>) is calculated as follows: (1 \u00d7 2^3) + (0 \u00d7 2^2) + (1 \u00d7 2^1) + (0 \u00d7 2^0)<\/strong>, which equals 8 + 0 + 2 + 0 = 10 in decimal.<\/p>\n

\"How<\/p>\n

How to Calculate Binary Numbers<\/span><\/h2>\n

Converting Decimal to Binary<\/h3>\n
\n \n \n
\n
    \n \n
  1. \n

    Start with the decimal number<\/strong> you want to convert.<\/p>\n<\/div><\/li>\n

  2. \n

    Divide the number by 2<\/strong> and record the remainder. This remainder is the rightmost digit (least significant bit) of the binary number.<\/p>\n<\/div><\/li>\n

  3. \n

    Divide the number by 2<\/strong> and record the remainder. This remainder is the rightmost digit (least significant bit) of the binary number.<\/p>\n<\/div><\/li>\n

  4. \n

    Use the quotient<\/strong> from the division as the new number to divide by 2.<\/p>\n<\/div><\/li>\n

  5. <\/h3>
    \n

    Repeat the process<\/strong> of dividing by 2 and recording the remainder until the quotient becomes 0.<\/p>\n<\/div><\/li>\n

  6. \n

    Write down the remainders in reverse order.<\/strong> This sequence is the binary representation of the original decimal number.<\/p>\n<\/div><\/li>\n <\/ol>\n \n <\/div>\n<\/div>\n

      <\/ol>\n

      Example<\/strong>: Convert the decimal number 13 to binary.<\/p>\n