About

Binary representation is the fundamental language of computing hardware. While simple text converters exist, understanding the relationship between bit positions and their decimal values is critical for networking (subnet masks), embedded systems, and low-level programming. This tool provides a dual-interface approach: enter a decimal number to see its binary form, or toggle individual bits to observe how they construct the decimal value.

Bit-level manipulation often results in errors regarding padding and signed values. This utility allows forced bit-depth (8, 16, or 32-bit) to align with standard memory registers and visually indicates the sign bit for negative integers, mimicking how a CPU ALU (Arithmetic Logic Unit) processes data.

Formulas

To convert a decimal integer N to binary, we perform repeated division by 2.

N = q × 2 + bit

This process continues until the quotient q is zero. The binary string is the sequence of remainders (bit) read from last to first.

For signed numbers (Two's Complement), if x is negative:

B = (¬|x|) + 1

This inverts all bits of the absolute value and adds 1.

Reference Data

Bit Position	Power of 2	Decimal Value	Example (Byte)
0	2⁰	1	0000 0001
1	2¹	2	0000 0010
2	2²	4	0000 0100
3	2³	8	0000 1000
4	2⁴	16	0001 0000
5	2⁵	32	0010 0000
6	2⁶	64	0100 0000
7	2⁷	128	1000 0000

Frequently Asked Questions

Unsigned integers can only represent non-negative numbers (0 to 255 for 8-bit). Signed integers use the most significant bit (MSB) to indicate negativity, effectively halving the maximum positive value (e.g., -128 to 127 for 8-bit).

Computers process data in fixed-size chunks (words). When programming or analyzing network packets, a binary string like "101" is actually stored as "00000000 00000000 00000000 00000101" in a 32-bit register. Padding ensures you see the full picture.

Each switch represents a power of 2. Toggling the switch at position n adds or subtracts 2^n from the total. This visualizes the additive nature of binary positional notation.