Binary Calculator

You can add, subtract, multiply, and divide binary numbers, convert binary numbers to decimal, convert decimal numbers to binary, and convert binary numbers to hexadecimal.

What Is a Binary Number?

A binary number is a number expressed in base 2. Unlike the decimal system, which uses ten digits from 0 to 9, the binary system uses only two digits: 0 and 1. Each position in a binary number represents a power of 2.

For example:

1011₂ = 1 × 2³ + 0 × 2² + 1 × 2¹ + 1 × 2⁰ = 8 + 0 + 2 + 1 = 11₁₀

Common Binary, Decimal, and Hexadecimal Values

Binary Addition Formula and Method

Binary addition follows base-2 rules:

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 10, which means write 0 and carry 1
• 1 + 1 + 1 = 11, which means write 1 and carry 1

General addition formula:

Binary sum = add each bit from right to left and carry when the sum is 2 or greater.

  10101
+    11
-------
  11000

Decimal check:

10101₂ = 21₁₀, and 11₂ = 3₁₀. Therefore:

21 + 3 = 24, and 24₁₀ = 11000₂.

Binary Subtraction Formula and Method

Binary subtraction follows base-2 borrowing rules:

• 0 - 0 = 0
• 1 - 0 = 1
• 1 - 1 = 0
• 0 - 1 requires borrowing from the next higher bit

When borrowing in binary, 10₂ is equal to 2₁₀. Therefore, after borrowing:

10₂ - 1₂ = 1₂

  10101
-    11
-------
  10010

Decimal check:

10101₂ = 21₁₀, and 11₂ = 3₁₀.

21 - 3 = 18, and 18₁₀ = 10010₂.

Binary Multiplication Formula and Method

Binary multiplication is similar to decimal long multiplication, but each digit is either 0 or 1.

• 0 × 0 = 0
• 0 × 1 = 0
• 1 × 0 = 0
• 1 × 1 = 1

General method:

1. Multiply the first binary number by each bit of the second number.
2. Shift each partial product left according to its bit position.
3. Add all partial products together.

    10101
×     11
---------
    10101
+  101010
---------
   111111

Decimal check:

10101₂ = 21₁₀, and 11₂ = 3₁₀.

21 × 3 = 63, and 63₁₀ = 111111₂.

Binary Division Formula and Method

Binary division uses the same long division method as decimal division. The calculator returns both a quotient and a remainder when needed.

General formula:

Dividend = Divisor × Quotient + Remainder

In base notation:

Dividend₂ ÷ Divisor₂ = Quotient₂ with Remainder₂

Decimal check:

10101₂ = 21₁₀, and 11₂ = 3₁₀.

21 ÷ 3 = 7, and 7₁₀ = 111₂.

Therefore:

10101₂ ÷ 11₂ = 111₂

Binary to Decimal Formula

For a binary number with digits b_nb_n-1...b₀, the decimal value is:

Decimal = b_n × 2ⁿ + b_n-1 × 2^n-1 + ... + b₀ × 2⁰

10101₂ = 1 × 2⁴ + 0 × 2³ + 1 × 2² + 0 × 2¹ + 1 × 2⁰

= 16 + 0 + 4 + 0 + 1 = 21

Therefore:

10101₂ = 21₁₀

Decimal to Binary Conversion Method

Repeated division method:

1. Divide the decimal number by 2.
2. Write down the remainder, either 0 or 1.
3. Continue dividing the quotient by 2 until the quotient becomes 0.
4. Read the remainders from bottom to top to get the binary number.

Decimal to Binary Formula

A decimal number can be represented as a sum of powers of 2:

Decimal = a_n × 2ⁿ + a_n-1 × 2^n-1 + ... + a₀ × 2⁰

where each coefficient a is either 0 or 1.

250 = 128 + 64 + 32 + 16 + 8 + 2

In powers of 2:

250 = 2⁷ + 2⁶ + 2⁵ + 2⁴ + 2³ + 2¹

The powers from 2⁷ to 2⁰ are:

Therefore:

250₁₀ = 11111010₂

Binary to Hexadecimal Conversion Method

1. Start from the right side of the binary number.
2. Group the digits into sets of four bits.
3. Add leading zeros to the left if the first group has fewer than four bits.
4. Convert each 4-bit group into one hexadecimal digit.
5. Combine the hexadecimal digits.

Binary to Hexadecimal Formula

Each group of four binary digits has the value:

Hex digit value = b₃ × 2³ + b₂ × 2² + b₁ × 2¹ + b₀ × 2⁰

The decimal values 10 to 15 are represented by hexadecimal letters:

• 10 = A
• 11 = B
• 12 = C
• 13 = D
• 14 = E
• 15 = F

Convert 10101₂ to hexadecimal:

First, group the binary digits from right to left:

10101 → 0001 0101

Convert each group:

Therefore:

10101₂ = 15₁₆

References

The following references provide useful background information about binary numbers, digital systems, and computing concepts:

• National Institute of Standards and Technology NIST
• NIST Information Technology Laboratory
• U.S. Department of Energy: DOE Explains Binary
• Cybersecurity and Infrastructure Security Agency CISA