Binary to Decimal Converter

The binary to decimal converter and binary to decimal calculator translate any binary number into its decimal equivalent instantly. Enter a binary code to decimal value above to get the result. Read on to learn how binary to decimal conversion works, step by step, with real examples and a full reference table.

What Is the Binary Number System?

The binary number system, also called the base 2 number system, is a positional numeral system that uses only two digits: 0 and 1. Every position in a binary number represents a power of 2, starting from 2⁰ at the rightmost digit. The place value of each bit doubles as you move left. Binary is the foundation of all modern computing because transistors in digital hardware operate in exactly two states: on (1) and off (0).

For example, 1011 in binary: (1 x 8) + (0 x 4) + (1 x 2) + (1 x 1). The sum is 8 + 0 + 2 + 1 = 11 decimal.

What Is the Decimal Number System?

The decimal number system is the base-10 system used in everyday life. It uses ten digits: 0 through 9. Each position in a decimal number represents a power of 10. When computers process and store data internally as binary numbers, they convert results to decimal format for display to users. Understanding the relationship between binary and decimal is essential for anyone working in computing, networking, or digital electronics.

How to Convert Binary to Decimal

To convert a binary number to decimal, multiply each binary digit by its corresponding power of 2 and add all results together. This is called the positional notation method.

Formula: Decimal = d_n x 2ⁿ + d_n-1 x 2^n-1 + ... + d₁ x 2¹ + d₀ x 2⁰

Step-by-step example: Convert 1101 from binary to decimal

• Write the binary digits from right to left: 1, 0, 1, 1
• Assign each digit its power of 2: position 0 = 1, position 1 = 2, position 2 = 4, position 3 = 8
• Multiply each digit by its power: 1x1=1, 0x2=0, 1x4=4, 1x8=8
• Add the results: 1 + 0 + 4 + 8 = 13

Result: 1101 in binary equals 13 in decimal. The same method applies to any binary number, regardless of length. For base 2 to base 10 and base 10 to base 2 conversions, the positional value doubles or halves at each step. To convert base 10 to binary, divide the decimal number by 2 repeatedly and read the remainders from bottom to top.

Binary to Decimal Conversion Table

This binary to decimal conversion table provides a quick reference for the most common binary numbers and their decimal equivalents. A standard byte uses 8 bits. The maximum 8-bit binary value is 11111111, which equals 255 in decimal. The value 1000 binary equals 8 decimal. The decimal 127 equals 1111111 binary, and the decimal 256 equals 100000000 binary.

101 Binary to Decimal

The binary number 101 converts to 5 in decimal.

• 1 x 2² = 4
• 0 x 2¹ = 0
• 1 x 2⁰ = 1
• Total: 4 + 0 + 1 = 5

In computing, 5 appears in Linux file permission settings, where 101 in binary represents read (4) and execute (1) rights with no write permission.

100 Binary to Decimal

The binary number 100 converts to 4 in decimal.

• 1 x 2² = 4
• 0 x 2¹ = 0
• 0 x 2⁰ = 0
• Total: 4 + 0 + 0 = 4

The binary number 100 is the smallest 3-bit number with only its highest bit set, representing the decimal value 4.

110 Binary to Decimal

The binary number 110 converts to 6 in decimal.

• 1 x 2² = 4
• 1 x 2¹ = 2
• 0 x 2⁰ = 0
• Total: 4 + 2 + 0 = 6

In Linux permissions, 110 in binary equals 6 in decimal, representing read (4) and write (2) access without execute rights.

1001 Binary to Decimal

The binary number 1001 converts to 9 in decimal.

• 1 x 2³ = 8
• 0 x 2² = 0
• 0 x 2¹ = 0
• 1 x 2⁰ = 1
• Total: 8 + 0 + 0 + 1 = 9

1010 Binary to Decimal

The binary number 1010 converts to 10 in decimal.

• 1 x 2³ = 8
• 0 x 2² = 0
• 1 x 2¹ = 2
• 0 x 2⁰ = 0
• Total: 8 + 0 + 2 + 0 = 10

The binary number 1010 (10 decimal) is widely used in computer science coursework. It demonstrates how binary counting differs from decimal, since 1010 looks like two tens but represents a single value of 10.

1011 Binary to Decimal

The binary number 1011 converts to 11 in decimal.

• 1 x 2³ = 8
• 0 x 2² = 0
• 1 x 2¹ = 2
• 1 x 2⁰ = 1
• Total: 8 + 0 + 2 + 1 = 11

10101 Binary to Decimal

The binary number 10101 converts to 21 in decimal.

• 1 x 2⁴ = 16
• 0 x 2³ = 0
• 1 x 2² = 4
• 0 x 2¹ = 0
• 1 x 2⁰ = 1
• Total: 16 + 0 + 4 + 0 + 1 = 21

The 10101 binary pattern uses alternating 1s and 0s. It appears in digital signal testing and hardware diagnostics to verify that all bits in a register work correctly.

What Is BCD (Binary Coded Decimal)?

BCD binary coded decimal is an encoding method. Each decimal digit is represented by its own 4-bit binary sequence, rather than converting the full number to one binary value. BCD is widely used in digital clocks, calculators, and financial software where decimal precision is critical.

Example: The decimal number 47 in standard binary is 101111. In BCD, it is encoded as 0100 0111, where 0100 = 4 and 0111 = 7.

BCD avoids the rounding errors that can occur during binary-to-decimal floating-point conversion, making it the preferred encoding in banking software and point-of-sale systems. NIST maintains numerical and measurement standards used in scientific and engineering computation, including guidelines for representing numeric data accurately in digital systems.

Real-World Uses of Binary to Decimal Conversion

Binary to decimal conversion is not only a classroom concept. It appears in many practical computing contexts. IPv4 networking uses 32 binary bits per address, supporting over 4 billion unique addresses. A 64-bit processor holds 64 binary digits per register.

• IP addresses: Each octet in an IPv4 address is an 8-bit binary number converted to decimal. The binary value 11000000 converts to 192 in decimal, as seen in the network address 192.168.1.1.
• File permissions: Linux and Unix systems use binary flags for permissions. The binary value 101 equals 5 in decimal, representing read (4) and execute (1) rights with write access denied.
• Color values: RGB color channels range from 0 to 255. The binary number 11111111 equals 255 decimal, representing full intensity for any color channel.
• Memory addressing: Computer memory addresses are stored in binary. Engineers convert these to decimal or hexadecimal for readable documentation and debugging.
• Data storage: File sizes in bytes, kilobytes, and megabytes are derived from powers of 2. Understanding the binary number system helps interpret storage capacity values correctly.

Binary vs. Other Number Systems

Beyond binary (base 2) and decimal (base 10), two other number systems are common in computing:

• Octal (base 8): Uses digits 0 through 7. Groups of 3 binary bits map to a single octal digit.
• Hexadecimal (base 16): Uses digits 0-9 and letters A-F. Groups of 4 binary bits map to a single hex digit, making hex a compact representation of binary data.

For example, the binary number 11111111 is FF in hexadecimal and 255 in decimal. All three notations represent the same value using different base systems.

Use DigiCalc's decimal to binary converter to reverse this process and convert a decimal number to its binary equivalent. For hexadecimal conversions, the binary to hex converter maps binary directly to base 16. To convert between any number base in a single tool, use the number base converter which handles binary, octal, decimal, and hexadecimal together.

Limitations of This Binary to Decimal Converter

• This tool converts pure binary integers only. Fractional binary values such as 1.101 are not supported.
• Signed binary numbers in two's complement format require a dedicated signed integer converter.
• Binary strings longer than 64 bits may exceed standard integer precision depending on the computing environment.
• IEEE 754 binary floating-point representation requires a specialized floating-point converter.

For more number system conversions, use DigiCalc's hex to decimal converter to convert hexadecimal values into their decimal equivalents.