Binary to Decimal Converter & Number Base Calculator

Convert numbers between Binary (Base 2), Decimal (Base 10), Hexadecimal (Base 16), Duodecimal (Base 12), and any radix from 2 to 36 with an interactive 8/16/32-bit matrix and step-by-step mathematical proofs.

From Base:

To Base:

Quick Presets:

Binary Number (Base 2) BASE 2

Allowed digits: only 0 and 1 (bits / zeros and ones)

Decimal Number (Base 10) BASE 10

Allowed digits: 0 to 9 (standard decimal numbers)

Bit Length 8 bits

Byte Size 1 byte

Parity (1s Count) Even Parity

Hex Value 0xD6

Step-by-Step Mathematical Derivation

🖐️ Base 5 System (Quinary / One-Hand Finger Counting)

Based on one human hand with 5 fingers. Historically used by various indigenous cultures across Africa and the Americas.

🕛 Base 12 System (Duodecimal / Dozens)

Counted using the thumb to touch the 12 finger bones (phalanges) of the other 4 fingers. Highly divisible by 2, 3, 4, and 6 (giving us 12 months, 12 hours on clock faces, and dozens).

🏛️ Base 20 System (Vigesimal / Mayans & Celts)

Based on all 20 fingers and toes. Used by the Mayans, Aztecs, and remnants still exist in languages like French ('quatre-vingts' = 4×20 = 80).

⏳ Base 60 System (Sexagesimal / Time & Angles)

Originated in ancient Sumer and Babylonia 4,000 years ago. Has 12 factors, which is why today an hour has 60 minutes, a minute has 60 seconds, and a circle has 360 degrees.

1. Binary ➔ Decimal: Positions and Powers of 2 Multiplication

Each position from right to left (0, 1, 2, 3, 4...) corresponds to an increasing power of 2. Multiply each bit and sum the active values!

Enter any binary number (only 0 and 1):

Step 1: Binary Expansion Diagram (Positions and Powers of 2)

Each bit expands to its corresponding power of 2 (2⁰ = 1, 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16...). The values on the right sum up to the decimal result:

Step 2: Bit Multiplication Table (Bit × 2^Position)

Position (from right)	Bit	Power of 2	Multiplication	Value

Sum only the positions where the bit is 1 (light is ON 💡):

32 + 8 + 4 + 1 = 45

Resulting Decimal Number: 45

💡 How Does This Principle Work in Other Bases (Hexadecimal, Base 32)?

Hexadecimal (Base 16): Each position from right to left is multiplied by powers of 16 (16⁰ = 1, 16¹ = 16, 16² = 256, 16³ = 4096...). For example, 2F₁₆ = (2 × 16¹) + (15 × 16⁰) = 32 + 15 = 47.
Base 32 (Base 32): Multiplied by powers of 32 (32⁰ = 1, 32¹ = 32, 32² = 1024, 32³ = 32768...). Symbols A–V represent 10–31.
Octal (Base 8): Multiplied by powers of 8 (8⁰ = 1, 8¹ = 8, 8² = 64, 8³ = 512...). For example, 144₈ = (1 × 64) + (4 × 8) + (4 × 1) = 100.

2. Decimal ➔ Binary: Successive Division by 2 Column Method

Divide the decimal number repeatedly by 2, write down the remainders, and read them from bottom to top!

Enter any decimal number (0 to 65535):

🟢 Even number divides evenly ➔ Remainder = 0

🔴 Odd number leaves a remainder ➔ Remainder = 1

Operation	Quotient	Remainder (Bit)	Explanation / Proof

💡 How to Divide into Other Number Bases?

Dividing into Hexadecimal (Base 16): Divide repeatedly by 16. Remainders 10–15 become letters: 10=A, 11=B, 12=C, 13=D, 14=E, 15=F. E.g., 254 ÷ 16 = 15 (remainder 14 = E), 15 ÷ 16 = 0 (remainder 15 = F) ➔ FE₁₆.
Dividing into Octal (Base 8): Divide repeatedly by 8, remainders 0–7. E.g., 100 ÷ 8 = 12 (remainder 4), 12 ÷ 8 = 1 (remainder 4), 1 ÷ 8 = 0 (remainder 1) ➔ 144₈.

The Architecture of Positional Number Systems: From Binary to Hexadecimal

Human civilizations have developed countless numeric representations throughout history—from the base-60 sexagesimal system of ancient Sumer and Babylon (which still defines our 60-minute hours and 360-degree circles) to the base-20 vigesimal calendar of the Maya. Yet, modern computing and human commerce have converged on two primary pillars: Base 10 (Decimal), shaped by our ten biological fingers, and Base 2 (Binary), dictated by the physical reality of silicon transistors switching between electrical voltages (high and low, on and off).

In 1703, German polymath Gottfried Wilhelm Leibniz published his seminal treatise "Explication de l'Arithmétique Binaire", documenting how every conceivable number and logical operation could be expressed using only two symbols: 0 and 1. Leibniz was inspired by the binary hexagrams of the ancient Chinese I Ching (Book of Changes). More than two centuries later, in 1937, Claude Shannon published his master's thesis proving that Boolean algebra and binary arithmetic could directly map onto electromechanical relays, laying the mathematical foundation for every digital computer in existence today.

Mathematical Foundations: How Number Base Conversions Work

Every standard positional numeral system relies on a radix (or base) b. In any positional system, the value of a numeric string d_nd_n-1...d₁d₀ is computed as the polynomial sum of each digit multiplied by the radix raised to its positional index:

Value = d_n × bⁿ + d_n-1 × b^n-1 + ... + d₁ × b¹ + d₀ × b⁰

1. Base X to Decimal (Polynomial Expansion)

To convert any non-decimal number into base 10, each digit is evaluated according to its power of the base. For example, converting the binary byte 11010110₂ to decimal:

(1 × 2⁷) + (1 × 2⁶) + (0 × 2⁵) + (1 × 2⁴) + (0 × 2³) + (1 × 2²) + (1 × 2¹) + (0 × 2⁰)
= 128 + 64 + 0 + 16 + 0 + 4 + 2 + 0 = 214₁₀

2. Decimal to Base Y (Successive Division / Modulo Method)

Converting from decimal to any target base b utilizes the repeated integer division and remainder collection algorithm:

Divide the decimal integer by the target base b.
Record the integer quotient and the remainder (the remainder represents the current least significant digit).
Replace the integer with the quotient and repeat the process until the quotient reaches 0.
Read the collected remainders in reverse order (from bottom to top / last remainder to first) to construct the final number.

Why Duodecimal (Base 12), Hexadecimal (Base 16), and Base 32 Matter

While decimal dominates human currency and metric measurements, mathematicians and engineers frequently leverage alternative radices for their unique structural properties:

Duodecimal / Dozenal (Base 12): Base 12 has four non-trivial factors (2, 3, 4, 6), compared to decimal's meager two (2, 5). This makes common fractions like 1/3 = 0.4₁₂ and 1/4 = 0.3₁₂ clean terminating numbers rather than infinite recurring fractions (1/3 = 0.333...₁₀). Many advocates, such as the Dozenal Society of America, consider base 12 the optimal base for human computation.
Hexadecimal (Base 16) & Octal (Base 8): Because 16 = 2⁴ and 8 = 2³, hex and octal directly cluster binary bits. Exactly 4 binary bits (a nibble) correspond to 1 hex digit (e.g., 1111₂ = F₁₆), and 8 bits (a byte) correspond to exactly 2 hex digits (0x00 to 0xFF). This makes Hex the universal shorthand for memory addresses, IPv6, and Web color codes.
Base 32 & Base 36: Base 32 (2⁵) uses 32 case-insensitive alphanumeric symbols (often avoiding ambiguous characters like 0, O, 1, I) for compact data transfer (e.g., TOTP two-factor authentication secrets, geohashes), while Base 36 (0–9 and A–Z) is ideal for human-readable URL shorteners and compact database identifiers.

Two's Complement: Representing Negative Integers in Binary

Modern CPU hardware implements negative numbers using Two's Complement. In an N-bit system, the most significant bit (MSB) acts as a negative weight (-2^N-1):

In 8-bit unsigned format: 11111111₂ = 255
In 8-bit signed two's complement: 11111111₂ = -128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = -1
To invert the sign of a binary number: invert all bits (one's complement) and add 1.

❓ FAQ (Frequently Asked Questions)

How do I convert binary to decimal manually?

Multiply each binary digit by 2 raised to the power of its position index (starting with 2^0 on the far right), and sum the results. For example: 1011 in binary is (1*8) + (0*4) + (1*2) + (1*1) = 8 + 0 + 2 + 1 = 11 in decimal.

Why do computers operate exclusively on binary (ones and zeros)?

Computer processors are composed of billions of microscopic silicon transistors acting as physical switches. Transistors have two reliable states: conducting electrical current (high voltage = 1) or blocking it (low voltage = 0). This two-state design provides maximum immunity against electrical noise and voltage fluctuations.

What is the largest number an 8-bit binary register can hold?

An 8-bit unsigned register holds 256 distinct values, from 0 (00000000) to 255 (11111111). In signed Two's Complement representation, an 8-bit register holds values from -128 (10000000) to +127 (01111111).

Why is Hexadecimal (Base 16) used so widely in computer programming?

Because 16 is an exact power of 2 (2^4 = 16), every 4 binary bits (a nibble) maps directly to exactly 1 hexadecimal character (0-9, A-F). A full 8-bit byte is always represented by exactly 2 hex characters (e.g., 11111111_2 = 0xFF), making memory dumps, IP addresses, and byte arrays clean and human-readable.

What is the difference between Signed (Two's Complement) and Unsigned integers?

Unsigned integers treat all bits as positive magnitude values. Signed integers use the most significant bit (MSB) as a sign bit with a negative weight (-2^(N-1)). This allows computer arithmetic units (ALUs) to perform addition and subtraction using the exact same hardware circuitry without requiring separate subtraction units.

How does this converter handle custom bases up to Base 36?

Bases 2 through 36 use standard alphanumeric characters: digits 0-9 for values 0-9, and uppercase letters A-Z for values 10-35. The converter accurately parses and calculates arbitrary-precision integer conversions using native JavaScript BigInt arithmetic.

How does repeated division by 2 work for converting decimal to binary?

Divide the decimal number repeatedly by 2, recording each remainder (0 if even, 1 if odd). Take the quotient and divide again until reaching 0. Reading all recorded remainders in reverse order (bottom-to-top / last-to-first) gives the final binary number.

How many bits are in a byte and a nibble?

There are 8 bits in a standard Byte and 4 bits in a Nibble (half-byte). Since a nibble represents values from 0 to 15 (0000 to 1111), exactly one hexadecimal character matches one nibble, and two hex characters match one full byte.