Binary system

In mathematics and computing, the binary numeral system, or base-2 system, represents numeric values using two symbols: 0 and 1. More specifically, the digits of a binary number are typically called bits. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used internally by almost all modern computers and computer-based devices.

In the binary system, each digit or bit has a value of either 0 or 1. These two digits are used to represent all other numbers in the system. For example, the number 10 in the decimal system (which is the system most people are familiar with) is represented as 1010 in the binary system.

The binary system is based on the concept of powers of two. This means that each digit in a binary number is associated with a power of two, with the rightmost digit being the least significant (i.e., having the lowest value) and the leftmost digit being the most significant.

For instance, the binary number 1010 can be interpreted as follows:

1 * 2^3 + 0 * 2^2 + 1 * 2^1 + 0 * 2^0 = 8 + 0 + 2 + 0 = 10

As you can see, the binary number 1010 is equivalent to the decimal number 10.

The binary system is an important part of how computers and other digital devices represent and manipulate data. In these devices, the binary digits 0 and 1 are typically represented by two different electrical states, such as the presence or absence of an electrical voltage. This allows computers to perform a wide variety of operations using simple circuits made up of logic gates, which can combine and manipulate binary values in order to perform complex tasks.

Converting from decimal to binary

To convert a decimal number to binary, you can use the following steps:

  1. Divide the number by 2, and record the remainder.
  2. Divide the quotient by 2, and record the remainder.
  3. Repeat this process until the quotient is equal to 0.
  4. The binary representation of the decimal number is the remainders from the divisions, starting with the last quotient and moving backwards to the first.

For example, to convert the decimal number 27 to binary, you can use the following steps:

  1. 27 / 2 = 13 R 1
  2. 13 / 2 = 6 R 1
  3. 6 / 2 = 3 R 0
  4. 3 / 2 = 1 R 1
  5. 1 / 2 = 0 R 1

The binary representation of 27 is 11011.

You can also use this method to convert a decimal number to any other base system, such as base-3, base-4, etc. Just substitute the appropriate base for the number 2 in the steps above.

Converting from binary to decimal

To convert a binary number to decimal, you will need to use the same algorithm in reverse.

  1. Start with the leftmost digit of the binary number, which is the most significant (i.e., has the greatest value).
  2. For each subsequent digit moving to the right, double the value of the previous digit and add the value of the current digit.
  3. Repeat this process until all digits have been processed. The resulting decimal number is the value of the binary number.

For example, to convert the binary number 10101 to decimal, you can use the following steps:

1 * 2^4 + 0 * 2^3 + 1 * 2^2 + 0 * 2^1 + 1 * 2^0 = 16 + 0 + 4 + 0 + 1 = 21

The decimal representation of the binary number 10101 is 21.

Other number systems

Other numbering systems that are commonly used in computing include the decimal system, which uses ten digits (0 through 9) to represent numeric values, and the hexadecimal system, which uses sixteen digits (0 through 9 and A through F) to represent numeric values. These systems are often used to represent numeric values in a more compact and convenient form, especially when working with large numbers or performing calculations.

Please check also the article about hexadecimal system.

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