Multi-Base Arithmetic Calculator (Hex / Binary / Octal)
指导
Multi-Base Arithmetic Calculator (Hex / Binary / Octal)
Add, subtract, multiply, divide, and take the modulo of numbers written in 二进制, 八进制, 十进制， 或者 hexadecimal — and see the answer in all four bases at once. The calculator uses arbitrary-precision BigInt arithmetic, so large hexadecimal values stay exact, and it visualises the carry, borrow, shift-and-add, and long-division steps that a typical AI text answer skips. Pick a fixed bit width (8, 16, 32, or 64-bit) to flag signed and unsigned overflow with the corresponding two’s-complement value.
如何使用
- Type the first number. Prefix it with
0bfor binary,0ofor octal,0xfor hex, or leave it bare for decimal. - Pick the operation: addition, subtraction, multiplication, division, or modulo.
- Type the second number using the same prefix rules.
- Optionally switch the bit width from Unlimited (BigInt) to a fixed width to check for overflow.
- The result is shown simultaneously in binary, octal, decimal, and hexadecimal, with the working steps printed below.
特征
- Mixed-base input — combine a hex value with a binary value in the same expression; the parser uses each number’s prefix to detect its base.
- Live results in four bases — every answer is rendered as 0b…, 0o…, decimal, and 0x… so you do not have to copy the result into a separate converter.
- Carry and borrow visualisation — for operands up to 16 bits the tool prints the binary columns, the carry-out row for addition, and the borrow row for subtraction.
- Shift-and-add multiplication — each “1” bit in the multiplier produces a shifted copy of the multiplicand, summed to form the product, exactly the way hardware multipliers work.
- Binary long division — step-by-step table of the running remainder and quotient bits, with the final quotient and remainder in binary and decimal.
- 溢出检测 — selecting 8, 16, 32, or 64-bit reports whether the result fits the signed and unsigned ranges, and shows the two’s-complement wrap value when it does not.
- Arbitrary precision — using JavaScript BigInt, the calculator handles hex values that overflow the standard 53-bit double-precision limit without rounding.
常问问题
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Why do binary, octal, and hexadecimal exist alongside decimal?
Each base maps cleanly to a different layer of computing. Binary represents the two voltage states of a transistor. Octal groups three bits at a time and was widely used on early mainframes such as the PDP-8. Hexadecimal groups four bits, which lines up exactly with a byte's two nibbles, so hex is the standard human-readable way to write memory addresses, color codes, and register values. Decimal lives on for everyday numeracy because humans have ten fingers, not because computers prefer it.
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What is a carry in binary addition?
When two binary digits add to a value of two or more, the lowest bit stays in the current column and the rest is carried into the next-higher column. In binary the only carry value is 1, generated whenever both addends in a column are 1, or when 1+1 lands on top of an incoming carry. This is the same idea as carrying a 1 in decimal arithmetic, but the threshold is two instead of ten.
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How does two's complement represent negative numbers?
Two's complement encodes a negative value by inverting every bit of the positive value and then adding 1. In an N-bit field the most significant bit acts as a sign bit, the range becomes -2^(N-1) to 2^(N-1)-1, and addition or subtraction works without separate logic for signed and unsigned operands. The same bit pattern can be read as a large unsigned number or as a negative signed number, depending on how the program interprets the field.
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What does integer overflow mean?
Overflow happens when an arithmetic result is larger than the largest value a fixed-width integer can hold, or smaller than the smallest. An 8-bit unsigned integer wraps from 255 back to 0; an 8-bit signed integer wraps from 127 to -128. In low-level code this can introduce subtle bugs because the CPU silently truncates the high bits instead of raising an error, so wrapping behaviour has to be reasoned about explicitly.
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Why is hexadecimal preferred over binary when reading memory?
Binary is unwieldy for humans because even small values quickly become long strings of 1s and 0s. Hexadecimal compresses every group of four binary digits into a single character, so a 32-bit register that takes 32 binary digits to write only needs 8 hex digits. The mapping is exact and lossless, which is why debuggers, assemblers, and protocol specifications almost universally show byte values in hex.
