Binary Number To Decimal, Hexadecimal, Octal Number Conversion Online Tool

A number system is a collection of various symbols which are called digits. A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner.

When we type some letters or words, the computer translates them in numbers as computers can understand only numbers. The value of each digit in a number can be determined using −

• The digit
• The position of the digit in the number
• The base of the number system (where the base is defined as the total number of digits available in the number system)

What are Number Systems?

Number systems are mathematical systems that represent numbers using a specific set of symbols or digits. The most commonly used number systems are:

1. Decimal number system: This is the most widely used number system and is based on the base-10 system. It uses 10 digits (0-9) to represent numbers, with each digit having a positional value based on its place in the number.
2. Binary number system: This system uses the base-2 system and consists of only two digits (0 and 1) to represent numbers. It is used extensively in computing and digital systems.
3. Octal number system: This system uses the base-8 system and consists of eight digits (0-7) to represent numbers. It is used in some computing systems and applications.
4. Hexadecimal number system: This system uses the base-16 system and consists of 16 digits (0-9 and A-F) to represent numbers. It is used extensively in computing and digital systems, particularly for representing colors and memory addresses.

In addition to these commonly used systems, there are other number systems, such as the base-3 ternary system, duodecimal (base-12), vigesimal (base-20) systems and the base-60 sexagesimal system that have been used historically and in some specialized applications.

Number systems are used in a wide range of applications in computing and mathematics, including digital electronics, cryptography, data storage and transmission, and programming. Understanding and working with different number systems is an important skill for anyone working in these fields.

Decimal Number System

The number system that we follow is called as Decimal Number System where the base is 10.The base 10 indicates that there are 10 different symbols that we use.
The ten symbols are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
These symbols are called as digits.
Any number that we form is a combination of the above digits,
9425 – Combination of 9, 4, 2, 5
504737 – Combination of 5, 0, 4, 7, 3, 7

The decimal number system, also known as the base-10 system, is the most widely used number system in the world. It consists of 10 digits (0-9) and uses positional notation to represent numbers. Each digit has a positional value based on its position in the number.

In the decimal system, the rightmost digit represents units, the next digit to the left represents tens, the next digit represents hundreds, and so on. The positional values of each digit are powers of 10, with the rightmost digit representing 10^0 (1), the next digit to the left representing 10^1 (10), the next digit representing 10^2 (100), and so on.

For example, the number 1234 in decimal notation represents:

1 * 10^3 + 2 * 10^2 + 3 * 10^1 + 4 * 10^0 = 1000 + 200 + 30 + 4 = 1234

The decimal system is used for everyday arithmetic operations, including addition, subtraction, multiplication, and division. It is also used extensively in mathematics, science, finance, and other fields.

Decimal numbers can be converted to other number systems, such as binary, octal, and hexadecimal, using various conversion techniques. For example, to convert a decimal number to binary, the decimal number is repeatedly divided by 2, with the remainder at each step representing the next binary digit. The binary digits are written in reverse order to get the final binary number. Conversely, to convert a binary number to decimal, each digit is multiplied by its corresponding power of 2 and then the resulting values are added together.

Binary Number System

A binary number system is a system which has a base of two. It means that it has only 2 different symbols that make up its number system.
The 2 symbols are 0 and 1
The numbers that are made in Binary Number System are made up of 0 or 1.

The binary number system is a base-2 number system that uses only two digits, 0 and 1, to represent numbers. It is used extensively in computing and digital systems, as these systems operate using binary logic and store data using binary digits.

In the binary number system, each digit represents a power of 2, with the rightmost digit representing 2^0 (1), the next digit to the left representing 2^1 (2), the next representing 2^2 (4), and so on. The value of each digit is determined by multiplying it by its corresponding power of 2 and adding up the resulting values.

For example, the binary number 1011 represents the value of:

1 * 2^3 + 0 * 2^2 + 1 * 2^1 + 1 * 2^0 = 8 + 0 + 2 + 1 = 11

Binary numbers can be used to represent any value that can be represented in decimal (base-10) format, but can be more difficult for humans to read and interpret due to their use of only two digits.

Binary numbers can also be used to represent data in computing systems, where they are often used to represent memory addresses, instructions, and data values. In these systems, binary numbers are usually represented using groups of 8 bits, called bytes, which can represent a range of values from 0 to 255.

In programming languages, binary numbers can be represented using the prefix "0b" followed by the binary digits. For example, in Python, the binary number 1011 can be represented as 0b1011. Similarly, in C and C++, the binary number 1011 can be represented as 0b1011 or 0B1011.

Octal Number System

Characteristics of the octal number system are as follows −

• Uses eight digits,0,1,2,3,4,5,6,7 also called as base 8 number system
• Each position in an octal number represents a 0 power of the base (8). For example 82
• Last position in an octal number represents a x power of the base (8). Example 8^x where x represents the last position - 1

The octal number system, also known as the base-8 system, is a number system that uses 8 digits (0-7) to represent numbers. It uses positional notation, like the decimal and binary systems, but each digit represents a power of 8 instead of a power of 10 or 2.

In the octal system, the rightmost digit represents units, the next digit to the left represents eights, the next digit represents 8^2 (64), and so on. The positional values of each digit are powers of 8, with the rightmost digit representing 8^0 (1), the next digit to the left representing 8^1 (8), the next digit representing 8^2 (64), and so on.

For example, the number 237 in octal notation represents:

2 * 8^2 + 3 * 8^1 + 7 * 8^0 = 128 + 24 + 7 = 159

To convert a decimal number to octal, the decimal number is repeatedly divided by 8, with the remainder at each step representing the next octal digit. The octal digits are written in reverse order to get the final octal number. Conversely, to convert an octal number to decimal, each digit is multiplied by its corresponding power of 8 and then the resulting values are added together.

The octal system is not as commonly used as the decimal or binary systems, but it is still used in some areas of computing and electronics, particularly in older systems or low-level programming. For example, in Unix and Linux systems, file permissions are represented in octal notation.

Hexadecimal Number System

Characteristics of hexadecimal number system are as follows −

• Uses 10 digits and 6 letters, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
• Letters represent the numbers starting from 10. A = 10. B = 11, C = 12, D = 13, E = 14, F = 15 also called as base 16 number system
• Each position in a hexadecimal number represents a 0 power of the base (16). For example, 19A, 30ADE16
• Last position in a hexadecimal number represents a x power of the base (16). For example 16^x where x represents the last position - 1

The hexadecimal number system, also known as the base-16 system, is a number system that uses 16 digits to represent numbers. It uses the digits 0-9 and the letters A-F to represent values from 0 to 15.

In the hexadecimal system, the rightmost digit represents units, the next digit to the left represents 16s, the next digit represents 16^2 (256), and so on. The positional values of each digit are powers of 16, with the rightmost digit representing 16^0 (1), the next digit to the left representing 16^1 (16), the next digit representing 16^2 (256), and so on.

For example, the number A3F in hexadecimal notation represents:

10 * 16^2 + 3 * 16^1 + 15 * 16^0 = 2560 + 48 + 15 = 2623

To convert a decimal number to hexadecimal, the decimal number is repeatedly divided by 16, with the remainder at each step representing the next hexadecimal digit. The hexadecimal digits are written in reverse order to get the final hexadecimal number. Conversely, to convert a hexadecimal number to decimal, each digit is multiplied by its corresponding power of 16 and then the resulting values are added together.

The hexadecimal system is often used in computing and electronics, particularly in low-level programming, because it is a convenient way to represent binary values. Since each hexadecimal digit can represent 4 bits (a binary digit or "bit" is either 0 or 1), a group of 4 bits can be represented by a single hexadecimal digit. This makes hexadecimal notation more compact and easier to read than binary notation.

The hexadecimal system is also used in web development and programming languages such as HTML, CSS, and JavaScript for representing colors. For example, the color white is represented in hexadecimal as #FFFFFF, where each pair of digits represents the red, green, and blue components of the color.

Overall, the hexadecimal system is a useful tool in computing and electronics, allowing for compact and convenient representation of binary values.

Steps for Conversion of Binary Number System to Decimal, Octal, and Hexadecimal Number System Manually

Here are the steps for converting a binary number to decimal, octal, and hexadecimal number systems:

1. Binary to Decimal Conversion:
1. Write down the binary number.
2. Starting from the right-most bit, write down the power of 2 for each bit position. The right-most bit has a power of 2^0, the next bit to the left has a power of 2^1, and so on.
3. Multiply each bit by its corresponding power of 2.
4. Add the products obtained in step c to obtain the decimal number.

For example, let's convert binary number 11001 to decimal:

1 x 2^4 + 1 x 2^3 + 0 x 2^2 + 0 x 2^1 + 1 x 2^0 = 16 + 8 + 0 + 0 + 1 = 25

Therefore, 11001 in decimal form is 25.

2. Binary to Octal Conversion:
1. Group the binary digits into groups of three, starting from the right-most bit. If the left-most group has fewer than three bits, add zeroes to the left until it has three bits.
2. Write down the octal value for each group of three bits.
3. Combine the octal values obtained in step b to obtain the octal number.

For example, let's convert binary number 11001 to octal:

011 001

3 1

Therefore, 11001 in octal form is 31.

3. Binary to Hexadecimal Conversion:
1. Group the binary digits into groups of four, starting from the right-most bit. If the left-most group has fewer than four bits, add zeroes to the left until it has four bits.
2. Write down the hexadecimal value for each group of four bits.
3. Combine the hexadecimal values obtained in step b to obtain the hexadecimal number.

For example, let's convert binary number 11001 to hexadecimal:

0001 1001

1 9

Therefore, 11001 in hexadecimal form is 19.

Note: There are also other methods for converting binary numbers to decimal, octal, and hexadecimal number systems, such as using a calculator or programming language.