About

Computers process instructions and data using the binary system. While humans operate in base-10, hardware logic gates rely on voltage states representing On (1) or Off (0). This converter translates human-readable decimal integers into the machine-readable binary sequence. Accurate conversion is critical for low-level programming, configuring IP subnet masks, and defining bitwise flags in software development.

Understanding binary representation extends beyond simple counting. In systems architecture, data alignment and memory addressing often depend on specific bit widths, such as 32-bit or 64-bit boundaries. Signed integers introduce complexity through Two's Complement representation, where the most significant bit acts as a sign indicator. Misinterpreting a signed binary sequence as unsigned can lead to integer overflow errors or incorrect logic states in control flow.

This tool addresses the readability issues common in raw binary streams. Long strings of zeros and ones are difficult to parse visually. By grouping bits into nibbles (4 bits) or bytes (8 bits), developers can quickly identify patterns, hexadecimal equivalents, and potential off-by-one errors in their code.

Formulas

The primary method for converting a decimal integer N to binary is the Successive Division by 2. For signed integers, the Two's Complement method is applied to represent negative values within a fixed bit width.

1. Successive Division (Unsigned):

Given N = 13:

13 ÷ 2 = 6, rem 1 (LSB)

6 ÷ 2 = 3, rem 0

3 ÷ 2 = 1, rem 1

1 ÷ 2 = 0, rem 1 (MSB)

Result: 1101₂

2. Two's Complement (Signed Negative):

To convert a negative number -A into binary with bit width w:

Val = ( ¬ A ) + 1

Mathematically, for an integer x where x < 0:

Binary = 2^w − |x|

Reference Data

Power	Exponent	Decimal Value	Binary Value	Hex Value	Bit Count
2⁰	0	1	1	0x1	1 bit
2¹	1	2	10	0x2	2 bits
2²	2	4	100	0x4	3 bits
2³	3	8	1000	0x8	4 bits (Nibble)
2⁴	4	16	1 0000	0x10	5 bits
2⁵	5	32	10 0000	0x20	6 bits
2⁶	6	64	100 0000	0x40	7 bits
2⁷	7	128	1000 0000	0x80	8 bits (Byte)
2⁸	8	256	1 0000 0000	0x100	9 bits
2¹⁰	10	1,024	100 0000 0000	0x400	11 bits
2¹⁵	15	32,768	1000 0000 0000 0000	0x8000	16 bits (Short)
2¹⁶	16	65,536	1 0000 0000 0000 0000	0x10000	17 bits
2³¹	31	2,147,483,648	1000... (31 zeros)	0x80000000	32 bits (Int)
2³²	32	4,294,967,296	1 0000... (32 zeros)	0x100000000	33 bits
2⁶³	63	9.22337×10¹⁸	1000... (63 zeros)	0x8000...	64 bits (Long)
MAX	64 (unsigned)	1.84467×10¹⁹	1111... (64 ones)	0xFFFF...	64 bits

Frequently Asked Questions

Two's Complement allows the CPU to perform addition and subtraction using the same logic circuits for both positive and negative numbers. It eliminates the problem of having two zeros (+0 and -0) found in other systems like Sign-Magnitude or One's Complement.

Standard JavaScript integers are safe up to 53 bits (Number.MAX_SAFE_INTEGER). This tool utilizes BigInt, which supports arbitrary-precision integers. However, for display purposes and standard computing contexts, the output is often constrained to typical register sizes (64-bit, 32-bit).

A nibble consists of 4 bits. This specific grouping maps directly to a single Hexadecimal digit (0-F). Grouping binary output into nibbles makes it significantly easier to manually translate binary to hex or debug memory dumps.

In signed arithmetic, the Most Significant Bit (MSB) is the sign bit. If you convert -5 to 8-bit binary, it is 11111011. In 16-bit binary, it is 1111111111111011. The value relies entirely on the container size defined by the architecture.