Binary Calculator

What is Binary?

Binary is a numeral system that uses only two symbols, typically "0" and "1," to represent numbers and information. It is the foundation of all digital computing and is widely used in various electronic devices and computer systems.

In the binary system, each digit, or "bit" (short for binary digit), can have one of two values: 0 or 1. These binary digits are used to represent data and information in a way that can be easily processed by electronic circuits, such as those found in computers, calculators, and other digital devices.

Binary is often used to represent numerical values, where each digit in a binary number corresponds to a power of 2. For example, in the binary number "1010," the rightmost digit represents 2^0 (which is 1), the next digit to the left represents 2^1 (which is 2), the next one represents 2^2 (which is 4), and the leftmost digit represents 2^3 (which is 8). When you add up these values, you get the decimal equivalent of the binary number, which in this case is 10.

Binary is essential in computer programming and digital electronics because it allows for efficient storage and manipulation of data using electronic circuits, which can easily distinguish between two voltage levels or states (usually represented as 0 and 1). This binary representation forms the basis for all computer operations, including arithmetic, logic, and data storage.

How to do binary calculations?

Binary calculations are performed using the base-2 numeral system, which only includes two symbols: 0 and 1. Here’s a quick guide on how to perform basic binary arithmetic: addition, subtraction, multiplication, and division.

• Binary Addition
  • Binary addition works similarly to decimal addition, with the exception that the highest digit in binary is 1 instead of 9. Here are the basic rules:
  • 0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10 (which is binary for 2; you write down 0 and carry 1)
  • Example of Binary Addition:
    1011 (this is 11 in decimal)
    + 1101 (this is 13 in decimal)
    ------
    11000 (this is 24 in decimal)
  • You add each column, starting from the rightmost column, and carry over any values as you would in decimal addition.
• Binary Subtraction
  • Binary subtraction works similarly to decimal subtraction, but there are fewer digits. Here are the basic rules:
  • 0 - 0 = 0
1 - 0 = 1
1 - 1 = 0
0 - 1 = 1 (with a borrow of 1 from the next higher bit)
  • Example of Binary Subtraction:
  • You subtract each column, borrowing as necessary from the next column to the left, just like in decimal subtraction.
• Binary Multiplication
  • Binary multiplication is similar to decimal multiplication. The multiplication rules are:
  • 0 * 0 = 0
0 * 1 = 0
1 * 0 = 0
1 * 1 = 1
  • Example of Binary Multiplication:
  • 101 (this is 5 in decimal)
    x 11 (this is 3 in decimal)
    ------
    101
+ 101
    ------
    1111 (this is 15 in decimal)
  • You multiply each digit of the second number by each digit of the first number, right to left, and add the results, taking care to shift the results left by one digit (equivalent to multiplying by 2) for each new row.
• Binary Division
  • Binary division is the process of dividing binary numbers and is similar to long division in the decimal system.
• Important Notes
  • When performing binary calculations, especially with pen and paper, it’s often helpful to write out the place values above the numbers (e.g., 8 4 2 1 for the first four places from the right in binary).
  • Just as in decimal arithmetic, when performing binary calculations with large numbers, care must be taken with the alignment of the digits, especially when carrying or borrowing.
  • The binary number system is the basis for all binary code and computer programming, as it represents the two states of digital electronics: off (0) and on (1).

You can perform these operations using pen and paper, a calculator that has a binary mode, or even with programming languages like Python that can handle binary numbers directly.

Binary/Decimal Conversion table