Binary Calculator

Use the calculators below to carry out operations such as addition, subtraction, multiplication, and division on binary numbers. They can also convert values between binary and decimal formats in both directions.

About the Binary System

The binary numeral system operates in a way that is very similar to the decimal system most people are familiar with. However, instead of using base 10, binary uses base 2. While decimal numbers use digits from 0 through 9, binary relies only on 0 and 1, with each digit known as a bit. Aside from these distinctions, arithmetic operations like addition, subtraction, multiplication, and division follow the same fundamental principles as in the decimal system.

Binary is the foundation of nearly all modern computing and digital technology because it is easy to implement using electronic circuits and logic gates. Hardware only needs to distinguish between two states (such as on/off or true/false), which simplifies design. In contrast, a decimal-based system would require recognizing ten distinct states (0–9), making it more complex.

Understanding binary becomes easier when recognizing that each position represents a power of 2 (2ⁿ), just as decimal places represent powers of 10 (10ⁿ). For example, the number 8 in decimal form:

8 × 100 = 8 × 1 = 8

For the number 18:

(1 × 101) + (8 × 100) = 10 + 8 = 18

In binary, 8 is written as 1000. Reading from right to left, each position corresponds to increasing powers of 2:

20, 21, 22, 23

Since 2³ = 8, placing a 1 in that position results in 1000.

For example, converting 18 into binary:

18 = 16 + 2 = 24 + 21
10010 = (1 × 24) + (0 × 23) + (0 × 22) + (1 × 21) + (0 × 20) = 18

To convert a decimal number to binary: identify the largest power of 2 less than or equal to the number; subtract this value from the number; repeat the process with the remainder; continue until the remainder is 0; mark 1s for the powers of 2 used, and 0s for the rest.

Example for 18:

2^n:    24 23 22 21 20
Used:    1   0   0   1   0

18 − 16 = 2  →  2 − 2 = 0

To convert binary to decimal, simply add the values of the positions that contain 1s:

10111 = (1 × 24) + (0 × 23) + (1 × 22) + (1 × 21) + (1 × 20)
      = 16 + 0 + 4 + 2 + 1 = 23

Binary Addition

Binary addition follows the same structure as decimal addition, but carrying occurs when a sum reaches 2 rather than 10.

Rules:
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 0 (carry 1 → 10)

Example:
   1 0 1 1 1 1 0 1
 +     1 0 1 1 1
 = 1 0 0 1 0 0

The key difference is that 2 in binary is written as 10. Be careful when adding columns that already include a carried value — this can affect the final result.

Binary Subtraction

Binary subtraction is also similar to decimal subtraction, but borrowing behaves differently due to the limited digits (0 and 1).

Rules:
0 − 0 = 0
0 − 1 = 1, borrow 1, resulting in −1 carried over
1 − 0 = 1
1 − 1 = 0

When borrowing, the column effectively gains a value of 2, while the column it borrows from decreases by 1. If the next column is also 0, borrowing continues from further left until a 1 is found.

EX1:
        -11 20 1 1 1
 −     0 1 1 0 1
 =     0 1 0 1 0

EX2:
        -11 2-10 0
 −      0 1 1
 =      0 0 1

Binary Multiplication

Binary multiplication is often simpler than decimal multiplication since only 0 and 1 are involved. Each step results in either 0 or the multiplicand itself.

Rules:
0 × 0 = 0
0 × 1 = 0
1 × 0 = 0
1 × 1 = 1

Example:
     1 0 1 1 1
 ×       1 1
     1 0 1 1 1
 + 1 0 1 1 1 0
 = 1 0 0 0 1 0 1

As with decimal multiplication, each row shifts left. Placeholder zeros are important and should not be ignored, especially when performing binary addition afterward. Zeros to the right of a binary number are significant, while leading zeros on the left are not.

EX:
   1 0 1 0 1 1 0 0
 = 0 0 1 0 1 0 1 1 0 0
 ≠ 1 0 1 0 1 1 0 0 0 0

Binary Division

Binary division works similarly to long division in the decimal system. The main difference is that subtraction is done using binary rules.

EX:      0 0 1 1 1
       ___________
  1 1 ) 1 0 1 0 1
        1 1
        _____
        1 0 0
          1 1
          _____
            1 1
            1 1
            _____
              0

A solid understanding of binary subtraction is essential for performing division accurately.