Binary is a base-2 number system that uses only two digits: 0 and 1. It is the fundamental language of computers and digital electronics. Each digit (bit) represents a power of 2. For example, binary 1010 = 1×8 + 0×4 + 1×2 + 0×1 = 10 in decimal.

How do you convert binary to decimal?

Multiply each binary digit by its positional power of 2 (starting from 2^0 on the right) and sum the results. For example, 1101 = 1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 8 + 4 + 0 + 1 = 13.

How do you convert decimal to hexadecimal?

Repeatedly divide the decimal number by 16 and record remainders. Remainders 10-15 are represented as A-F. For example, 255 ÷ 16 = 15 remainder 15, so 255 in hex is FF.

How do you add binary numbers?

Binary addition follows the same rules as decimal but with base 2: 0+0=0, 0+1=1, 1+0=1, 1+1=10 (write 0 carry 1), 1+1+1=11 (write 1 carry 1). For example, 1011 + 1101 = 11000.

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Understanding Number Systems and Binary Math

Computers process all data as binary (base-2) numbers, using only 0s and 1s. Each binary digit (bit) represents a power of 2. The decimal system we use daily is base-10, using digits 0-9. Hexadecimal (base-16) uses 0-9 and A-F, making it a compact way to represent binary data since each hex digit corresponds to exactly 4 binary digits. Octal (base-8) uses digits 0-7 and maps to 3 binary digits.

Converting between bases is fundamental to computer science. To convert binary to decimal, multiply each bit by its positional power of 2 and sum: 1010 = 1×8 + 0×4 + 1×2 + 0×1 = 10. To convert decimal to binary, repeatedly divide by 2 and record remainders from bottom to top. Hexadecimal is widely used in web colors (#FF5733), memory addresses, and debugging.

Binary arithmetic follows the same principles as decimal arithmetic but with only two digits. Addition: 0+0=0, 0+1=1, 1+1=10 (carry 1). Subtraction borrows from the next higher bit. Multiplication works like long multiplication with simpler partial products (only 0 or the multiplicand). This calculator handles all conversions instantly and performs binary addition, subtraction, and multiplication, displaying results in both binary and decimal for easy verification.

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Binary Arithmetic

Understanding Number Systems and Binary Math

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