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Binary Calculator

Binary value:
0

Decimal value:
0
Formula: Binary → Decimal
Decimal = (bn × 2n) + ... + (b0 × 20)

Example: Convert 1011 to Decimal
1011 = 1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 8 + 0 + 2 + 1 = 11
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Understanding the Binary Number System

The binary number system is a method of representing numbers using only two digits: 0 and 1. It works similarly to the decimal system, which uses ten digits (0–9), but binary is based on powers of 2 instead of 10. Each digit in a binary number is called a bit.

Binary is widely used in digital electronics and computing because it’s easy to represent two states—such as on/off or true/false—using 0 and 1. Designing circuits to handle just two values is much simpler than handling ten, making binary ideal for modern technology.

Common Binary to Decimal Conversions

DecimalBinary
00
11
210
311
4100
7111
81000
101010
1610000
2010100

How Binary Numbers Work

Just like decimal uses powers of 10, binary uses powers of 2. For example:

  • In decimal:
    8 = 8 × 10⁰ = 8
    18 = (1 × 10¹) + (8 × 10⁰) = 10 + 8 = 18

  • In binary:
    8 = 1000 = (1 × 2³)
    18 = 10010 = (1 × 2⁴) + (1 × 2¹) = 16 + 2 = 18

    Steps to Convert Decimal to Binary

    1. Identify the largest power of 2 less than or equal to the number.

    2. Subtract that value from the number.

    3. Repeat the process with the remainder.

    4. For each power of 2 used, mark a 1; for those not used, mark a 0.

    Example: Convert 18 to Binary

    2⁴2³2²2¹2⁰
    10010

    Result: 10010

    Converting Binary to Decimal

    To convert binary to decimal, add up all the powers of 2 represented by bits with the value 1.

    Example: 10111

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

    Binary Addition

    Binary addition uses similar rules to decimal addition but with only two digits:

    • 0 + 0 = 0

    • 0 + 1 = 1

    • 1 + 0 = 1

    • 1 + 1 = 10 (which is 0 with a carry of 1)

    Example:

    10111
    + 00011
    ————–
    11010

    Carryover happens when a sum reaches 2 (binary 10), just like in decimal when a sum reaches 10.

    Binary Subtraction

    Binary subtraction also follows simple rules:

    • 0 − 0 = 0

    • 1 − 0 = 1

    • 1 − 1 = 0

    • 0 − 1 = 1 (with borrowing)

    When borrowing, you take 1 from the next significant bit, turning 0 into 2 in binary.

    Example:

    11011
    – 01101
    ————–
    01110

    Borrowing works the same as in decimal subtraction but adapted for binary values.

    Binary Multiplication

    Binary multiplication is straightforward since the only digits are 0 and 1:

    • 0 × 0 = 0

    • 1 × 0 = 0

    • 0 × 1 = 0

    • 1 × 1 = 1

    Just like in decimal multiplication, results are added together, and each row is shifted left.

    Example:

    10111
    × 11
    ———–
    10111
    + 101110
    —————-
    110001

    Multiplication in binary involves repeated additions and left shifts.

    Binary Division

    Binary division is similar to long division in decimal but uses binary subtraction:

    1. Divide the dividend by the divisor using binary subtraction.

    2. Record how many times the divisor “fits” into the current digits.

    3. Continue until all digits are processed.
      Example:

      00111 ← Quotient
      _________
      11 | 101101
      11 ← 11 × 1 = 11
      —–
      100
      11 ← 11 × 1 = 11
      —-
      11
      11 ← 11 × 1 = 11
      —
      0 ← Remainder

    A solid grasp of binary subtraction makes binary division easier to perform.

     

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