Add, subtract, multiply, or divide base-2 integers with this binary calculator, with exact binary, decimal, octal, hex, and quotient/remainder checks.
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Base-2 arithmetic
Binary calculator for exact add, subtract, multiply, and divide work
Enter whole binary values, choose the operation, and compare the answer as binary, decimal,
octal, and hexadecimal. Spaces, underscores, apostrophes, and matching 0b prefixes are ignored so pasted programming examples are easier to check.
Exact integer binary arithmetic The calculator uses integer-safe arithmetic for base-2 values. A leading minus sign is treated as a signed-magnitude shortcut, not as a fixed-width two's-complement pattern.
Quick examples
Start with a common binary addition, subtraction, multiplication, division, or large-value check.
This page is for integer arithmetic on base-2 numerals. Use the bitwise calculator for AND, OR, XOR, NOT, or fixed-width shifts.
Binary calculator for exact base-2 arithmetic and conversion checks
The binary calculator performs addition, subtraction, multiplication, and integer division on binary numbers while showing the same result in decimal, octal, and hexadecimal. Use it as a binary arithmetic calculator for programming, digital electronics, classroom number-system work, or any situation where you need to verify base-2 values without losing integer precision.
What this binary calculator does
A binary number is a base-2 numeral that uses only the digits 0 and 1. The calculator reads each operand as a whole binary integer, performs the selected arithmetic operation, and then shows the answer in binary, decimal, octal, and hexadecimal so you can check the same value from several common number-system views.
This page focuses on binary addition, binary subtraction, binary multiplication, and binary division. It accepts readability separators such as spaces, underscores, and apostrophes, and it also strips a matching 0b prefix when you paste a programming-style literal. That makes the tool practical for short textbook examples as well as longer bit patterns copied from code, logs, or digital-design notes.
The implementation uses exact integer arithmetic rather than ordinary floating-point numbers. That matters because binary work often involves values that are naturally wider than everyday decimal examples. A result that looks plausible but has silently rounded would be worse than an error, so the calculator keeps the integer model explicit.
value = Σ(bit_i × 2^power_i)
Each binary digit contributes its digit value multiplied by the matching power of 2, starting from the rightmost bit at 2^0.
dividend = divisor × quotient + remainder
Integer binary division is reported as a quotient and remainder so the answer can be checked directly.
How binary arithmetic works
Binary arithmetic follows the same positional rules as decimal arithmetic but carries or borrows at 2 instead of 10. In binary addition, 1 + 1 produces 10, so the current column writes 0 and carries 1 into the next column. If another carried 1 is already present, the column result changes again in the same way a decimal column changes when it crosses 9.
Subtraction borrows from the next higher bit when the minuend digit is smaller than the subtrahend digit. Multiplication works by multiplying by each bit in the second operand and shifting the partial products. Division uses the same long-division idea as decimal division, but each step only needs to decide whether the shifted divisor fits or does not fit.
The calculator returns negative subtraction results with a leading minus sign. That is a signed-magnitude display convention for the answer, not a claim that the output is a fixed-width two's-complement bit pattern. If a programming language, CPU register, or file format interprets negative values by width, sign bit, or overflow rules, those rules need to be checked separately.
Worked examples: addition, multiplication, and division
For binary addition, 1010 + 1101 equals 10111. The decimal check is 10 + 13 = 23, so the answer is also 27 in octal and 17 in hexadecimal. Seeing all four forms helps catch transcription mistakes because a wrong carry usually changes more than one representation.
For multiplication, 101 × 11 equals 1111. In decimal that is 5 × 3 = 15. The binary method can be viewed as one copy of 101 plus another copy shifted one place left, which is the same structure used by many low-level multiplication explanations.
For integer division, 1011 ÷ 10 gives quotient 101 with remainder 1. In decimal that is 11 ÷ 2 = 5 remainder 1, and the identity check is 1011 = 10 × 101 + 1. A binary division calculator should show the remainder instead of hiding it, because the remainder is often the value that matters in indexing, packing, and divisibility checks.
Decimal, octal, and hexadecimal checks
Binary is precise but visually dense, so the calculator also returns decimal, octal, and hexadecimal. Decimal is useful when you need the everyday integer value. Octal groups binary digits in sets of three. Hexadecimal groups binary digits in sets of four, which is why it is common in programming, debugging, memory addresses, color values, and compact bit-pattern notes.
The binary result is grouped from the right in four-bit chunks for readability. A leading group may be shorter than four bits because the grouping is only a display aid; it does not add leading zeros or change the value. If you need a fixed-width word such as 8 bits, 16 bits, or 32 bits, use the bitwise calculator or a width-aware two's-complement tool so the padding and sign interpretation are explicit.
The bit-length output tells you how many binary digits are needed to write the magnitude of the result without optional leading zeros. This helps when deciding whether a value fits into a nibble, byte, word, or custom field, while still keeping this page focused on ordinary integer arithmetic rather than storage-format decoding.
This is an integer binary calculator. It does not model fractional binary points, repeating binary fractions, IEEE floating-point encodings, signed overflow, endianness, byte order, or CPU-specific flag behavior. Those topics depend on representation rules outside ordinary base-2 arithmetic.
A common mistake is mixing up binary arithmetic with bitwise logic. Adding 1010 and 1101 is an arithmetic operation; ANDing or XORing them is a bitwise operation that compares positions instead of place values. The two workflows can use the same characters but answer different questions.
Another common mistake is assuming every negative-looking binary result is a two's-complement value. Two's complement only becomes meaningful when you know the bit width. This page keeps negative arithmetic answers readable with a minus sign and leaves width-specific signed interpretation to tools designed for that purpose.
Frequently asked questions
What is a binary calculator used for?
A binary calculator is used to perform arithmetic on base-2 numbers and verify the answer in other number systems. It is useful for computer science, digital electronics, networking, firmware notes, and classroom exercises because binary values are how many low-level systems represent integer data.
How do I add binary numbers?
Add from right to left just as you would in decimal, but carry when a column reaches 2. The key facts are 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, and 1 + 1 = 10 in binary. If a carried 1 also reaches the column, include it before deciding whether another carry is needed.
How does binary subtraction borrow?
Binary subtraction borrows from the next higher bit when you need to subtract 1 from 0. The borrowed value is worth 2 in the current column, so 0 − 1 becomes 1 after the borrow and the higher column is reduced. Chains of zeros require borrowing across multiple positions, just like decimal subtraction can require borrowing across several zeros.
Does the calculator show binary division remainders?
Yes. Binary division is shown as an integer quotient with a remainder. That is usually more useful than forcing a decimal fraction because the quotient-and-remainder identity lets you verify the result directly and matches many programming and number-theory workflows.
Can I use 0b prefixes or grouped binary digits?
Yes. The calculator accepts matching 0b prefixes and ignores readability separators such as spaces, underscores, and apostrophes. For example, 0b1111_0000 and 1111 0000 are read as the same binary integer.
Is a negative result the same as two's complement?
No. A negative result on this page is displayed with a leading minus sign as an arithmetic answer. Two's complement is a fixed-width representation system, so the same visible bits can mean different signed values depending on whether the width is 4 bits, 8 bits, 16 bits, or something else.
Why show octal and hexadecimal on a binary arithmetic page?
Octal and hexadecimal are compact ways to inspect binary values. One octal digit corresponds to three binary bits, and one hexadecimal digit corresponds to four binary bits. Showing these alongside decimal makes the result easier to verify and easier to reuse in programming or electronics contexts.
When should I use the base calculator instead?
Use the base calculator when either operand can be in a base other than binary or when you want to choose an output base from 2 through 36. Use this binary calculator when both inputs are base-2 values and you want a focused binary arithmetic worksheet.
When should I use the bitwise calculator instead?
Use the bitwise calculator when the task is AND, OR, XOR, NOT, left shift, right shift, masking, or fixed-width bit inspection. Those operations compare or transform bit positions directly, while this page performs ordinary arithmetic on the integer values represented by binary numerals.
