Binary Converter

Binary is the language computers speak natively — not because engineers chose it arbitrarily, but because transistors have two stable states (on and off, 1 and 0), and a number system built on two symbols maps perfectly onto physical hardware. Every decimal number you type, every character you see on screen, every color in an image, and every instruction a processor executes exists somewhere in the system as a sequence of binary digits (bits). Understanding how to convert between binary and decimal — and understanding why the conversion works the way it does — is the foundation of understanding how computers represent any information at all.

Binary Arithmetic

Binary addition follows the same rules as decimal with different carry triggers. Decimal: 1 + 1 = 2. Binary: 1 + 1 = 10₂ (zero, carry 1). The full binary addition rules: 0+0=0, 0+1=1, 1+0=1, 1+1=0 carry 1, 1+1+1 (with carry)=1 carry 1.

Add 1011₂ + 0110₂: rightmost column: 1+0=1. Second column: 1+1=0 carry 1. Third column: 0+1+1(carry)=0 carry 1. Fourth column: 1+0+1(carry)=10₂ (0, carry 1). Fifth column: carry 1. Result: 10001₂. Verify: 11₁₀ + 6₁₀ = 17₁₀. Binary 10001₂ = 16+1 = 17₁₀. Correct. CPU addition circuits implement exactly this process at the bit level, billions of times per second.

Two's Complement: How Computers Store Negative Numbers

Positive binary numbers are straightforward. Negative numbers use two's complement representation — a specific encoding that lets the CPU add and subtract without special case logic. To find the two's complement of a number: flip all bits (find the one's complement) then add 1.

To represent -13 in 8-bit two's complement: 13₁₀ = 00001101₂. Flip all bits: 11110010. Add 1: 11110011. So -13 is stored as 11110011 in two's complement. To verify: -13 + 13 should = 0. 11110011 + 00001101 = 100000000 (9 bits, but the 9th bit overflows and is discarded in 8-bit storage) → 00000000. Zero. Correct. This elegant trick means CPUs can perform subtraction using their addition circuits — subtraction is just addition of the two's complement representation. It's why computer hardware can be simplified dramatically: one circuit type handles both operations.

The Positional Value System

Decimal is base-10: each digit's value depends on its position, multiplying by 10 for each place from right to left. In 347: the 7 is in the ones position (10^0 = 1), the 4 is in the tens position (10^1 = 10), the 3 is in the hundreds position (10^2 = 100). Total: 3 × 100 + 4 × 10 + 7 × 1 = 347.

Binary is base-2: identical structure but powers of 2 instead of 10. Binary 1011: reading right to left, positions represent 2^0 = 1, 2^1 = 2, 2^2 = 4, 2^3 = 8. So 1011₂ = (1 × 8) + (0 × 4) + (1 × 2) + (1 × 1) = 8 + 0 + 2 + 1 = 11₁₀. The subscript 2 indicates binary; subscript 10 indicates decimal. Binary 11001₂ = (1 × 16) + (1 × 8) + (0 × 4) + (0 × 2) + (1 × 1) = 16 + 8 + 0 + 0 + 1 = 25₁₀.

Converting Decimal to Binary: The Division Method

To convert decimal to binary, repeatedly divide the number by 2, recording the remainder (always 0 or 1) at each step. Read the remainders from bottom to top — they form the binary number.

Convert 45₁₀ to binary: 45 ÷ 2 = 22 remainder 1. 22 ÷ 2 = 11 remainder 0. 11 ÷ 2 = 5 remainder 1. 5 ÷ 2 = 2 remainder 1. 2 ÷ 2 = 1 remainder 0. 1 ÷ 2 = 0 remainder 1. Reading remainders bottom to top: 101101₂. Verify: 32 + 8 + 4 + 1 = 45. Correct. The number of binary digits (bits) needed to represent n in decimal is approximately log₂(n) + 1. log₂(45) ≈ 5.49, so 6 bits — confirmed.

Converting Binary to Decimal: Powers of 2

The reverse conversion multiplies each bit by its positional power of 2 and sums. A useful power-of-2 reference: 2^0=1, 2^1=2, 2^2=4, 2^3=8, 2^4=16, 2^5=32, 2^6=64, 2^7=128, 2^8=256, 2^9=512, 2^10=1024.

Convert 10110101₂ to decimal: 8 bits. Position values right to left: 1, 2, 4, 8, 16, 32, 64, 128. Multiply each bit by its position value: (1×1) + (0×2) + (1×4) + (0×8) + (1×16) + (1×32) + (0×64) + (1×128) = 1 + 0 + 4 + 0 + 16 + 32 + 0 + 128 = 181₁₀.

Bytes, Nibbles, and Common Data Sizes

In computing, bits group into standard units. A nibble is 4 bits — enough to represent 0 through 15 (0000 through 1111 in binary, or 0 through F in hexadecimal). A byte is 8 bits — enough to represent 0 through 255 (0000 0000 through 1111 1111). A kilobyte is 1,024 bytes (2^10). A megabyte is 1,048,576 bytes (2^20). A gigabyte is 1,073,741,824 bytes (2^30).

These powers-of-2 sizes explain why computer storage always comes in 8, 16, 32, 64, 128, 256, 512, 1024 GB variants — never 100 GB or 250 GB. Hardware designers work in powers of 2 because the addressing is binary. A 32-bit address space can address 2^32 = 4,294,967,296 bytes = 4 GB maximum. This was the fundamental barrier that required the transition to 64-bit computing when programs and data exceeded 4 GB in size.

Practical Conversions Encountered in Programming

Alexandra, 27, in Seattle, Washington works as a software developer and encounters binary frequently. IP addresses in IPv4 are 32-bit binary numbers written in dotted decimal notation. 192.168.1.1 means four 8-bit numbers: 11000000.10101000.00000001.00000001 in binary. Subnet masks like 255.255.255.0 in binary are 11111111.11111111.11111111.00000000 — 24 consecutive 1s followed by 8 zeros, which is why it's often written as /24 in CIDR notation. File permissions in Unix systems (chmod 755) use octal (base-8), which converts directly from binary groups of 3 bits: 755₈ = 111 101 101₂ = read-write-execute for owner, read-execute for group and others. And color values in CSS (#4A90D9) are hexadecimal — each pair of hex digits represents one byte of RGB color value.