Binary Calculator

A binary calculator converts a number between binary (base 2), decimal, hexadecimal and octal and performs binary arithmetic. Binary uses only the digits 0 and 1. For example, binary 1010 equals 10 in decimal, because (1×8) + (0×4) + (1×2) + (0×1) = 10, and the same value is 12 in octal and A in hex. Type any value below, whole number or fraction, to convert or compute instantly.

For hexadecimal arithmetic, the Hex Calculator adds and multiplies in base 16 directly; to read or build signed bit patterns of a chosen width, use the Two's Complement Calculator; and for the remainder of a division on its own, the Modulo Calculator.

What "binary" actually means

A number system's base (or radix) is how many distinct digits it uses. Decimal is base 10 because it uses ten digits, 0 through 9. Binary is base 2: the only legal digits are 0 and 1, called bits (short for "binary digits"). Just as each position in a decimal number is worth ten times the one to its right, each position in a binary number is worth twice the one to its right. So the binary places, reading from the right, are 1, 2, 4, 8, 16, 32, and so on: the powers of two. After a binary point the places continue as halves, quarters and eighths (2⁻¹ = 0.5, 2⁻² = 0.25, 2⁻³ = 0.125), so binary 101.101 is 4 + 1 + 0.5 + 0.125 = 5.625.

Powers of two table: 2ⁿ from 2⁰ to 2¹⁶

Every binary place value is a power of two, so this table is the backbone of all binary math. For example, 2⁸ = 256, a 1 followed by eight zeros, and 2¹⁰ = 1,024, which is why 1,024 bytes are called a kibibyte (KiB); in SI units a kilobyte (kB) is exactly 1,000 bytes.

The conversion formula

To read any number in any base, multiply each digit by its place value and add. Formally, a number with digits d in base b has the value:

For binary, b = 2. So the binary number 1011 equals (1×2³) + (0×2²) + (1×2¹) + (1×2⁰) = 8 + 0 + 2 + 1 = 11. The calculator above prints exactly this breakdown, bit by bit, so the answer is never a black box.

Worked example: binary to decimal

Take the byte 11001010. Line up the place values 128, 64, 32, 16, 8, 4, 2, 1 under the bits. Wherever there is a 1, add that place value: 128 + 64 + 8 + 2 = 202. Wherever there is a 0, add nothing. That is the entire trick to reading binary: it is just addition of selected powers of two.

What is 128 + 64 + 32 + 16 + 8 in binary?

Adding a run of powers of two sets one bit each. The sum 128 + 64 + 32 + 16 + 8 = 248, which is 11111000 in binary (hex F8, octal 370). The step-by-step breakdown below shows each place value and whether its bit is on.

Worked example: decimal to binary

Going the other way, use repeated division by 2 and collect the remainders. Convert 156: 156 / 2 = 78 r 0, 78 / 2 = 39 r 0, 39 / 2 = 19 r 1, 19 / 2 = 9 r 1, 9 / 2 = 4 r 1, 4 / 2 = 2 r 0, 2 / 2 = 1 r 0, 1 / 2 = 0 r 1. Read the remainders from bottom to top: 10011100. Typing 156 in the decimal field returns the same result without the longhand.

Worked example: binary addition

Binary addition uses the same columns as decimal, but you carry as soon as a column reaches 2 instead of 10. The four rules are 0+0=0, 0+1=1, 1+0=1, and 1+1=0 with a carry of 1. Add 1010 (10) and 110 (6): the rightmost columns produce carries that ripple left, and the answer is 10000, which is 16, exactly 10 + 6. The Binary math tab shows both the binary and decimal forms so you can check yourself.

How to use this calculator

1. Pick a mode at the top: "Convert" to translate one number across bases, or "Binary math" to add, subtract, multiply or divide.
2. To convert, type into any of the four fields: binary, octal, decimal or hexadecimal. The other three update the instant you type, so you can start from whichever base you know. Fractions (101.101 or 0.1), a minus sign, prefixes such as 0x or 0b, and spaces between bit groups are all fine.
3. Read the result. The big number shows the decimal value, the cards show all four bases (repeating digits in parentheses), and the table breaks the binary number into its place values. A negative number also gets its two's complement bit pattern.
4. To do math, switch to Binary math, enter two binary numbers (only 0s and 1s), tap an operator, and read the result in binary plus its decimal, octal and hex equivalents. A negative difference is shown with a minus sign and in two's complement.
5. Watch for errors. If you type a digit that is illegal for the chosen base (say an 8 in a binary field), the calculator flags it instead of guessing.

Who this tool is for

• Students learning number systems in a computer-science, electronics or math class who need to check homework and see the steps.
• Programmers reading hex color codes, bitmasks, memory addresses, or file permissions and wanting a fast cross-base translation.
• Hardware and networking folks working with registers, subnet masks, and IP addresses, where binary and hex are everywhere.
• Anyone curious who has wondered what "01000001 is the letter A" really means and wants to play with the math directly.

Key terms explained

• Bit: a single binary digit, either 0 or 1. The smallest unit of data.
• Byte: a group of 8 bits. A byte holds 2⁸ = 256 patterns, the values 0 to 255.
• Nibble: 4 bits, exactly one hexadecimal digit. Two nibbles make a byte.
• Base / radix: the number of digits a system uses: 2 for binary, 8 for octal, 10 for decimal, 16 for hex.
• Most/least significant bit (MSB/LSB): the leftmost bit carries the largest place value; the rightmost carries the smallest (the ones place).
• Place value: the worth of a digit based on its position, a power of the base.
• Two's complement: the usual way computers store negative integers in a fixed number of bits; the width must always be stated, because −10 is 11110110 in 8 bits but 1111111111110110 in 16 bits.

Quick reference: common conversions

The table below lists frequently used values across all four bases. Each binary entry is padded to a tidy width so the bit patterns line up.

Why hex and octal are binary shorthand

Long bit strings are hard to read, so programmers compress them. Because 16 = 2⁴, every hexadecimal digit maps to exactly four bits: 1111 is F, 1010 is A, and so on. The byte 11001010 splits into 1100 and 1010, which is CA in hex. Similarly, because 8 = 2³, each octal digit stands for three bits. That tight mapping is why color codes (#FF8800), memory addresses, and Unix file permissions (chmod 755) are written in hex or octal rather than raw binary.

Binary subtraction, multiplication and division

Subtraction works column by column with borrowing, mirroring decimal: 1010 − 110 = 100 (10 − 6 = 4). When the second number is larger, the result is negative: 110 − 1010 = −100 (−4), which the calculator also shows as 11111100 in 8-bit two's complement. Multiplication is a series of shifts and adds; multiplying by a power of two just appends zeros, so 101 × 10 = 1010 (5 × 2 = 10). Division is integer division: 1011 / 10 = 101 remainder 1 (11 / 2 = 5 r 1). The tool always shows the remainder separately so nothing is silently dropped.

Binary math examples: add, subtract, multiply, divide

This matrix runs every operation on the same pairs of binary numbers so you can check the pattern. Each cell is the result in binary with its decimal value in parentheses; division shows quotient and remainder.

Tips for working in binary

• Memorize the powers of two (1, 2, 4, 8, 16, 32, 64, 128, 256...). Almost all binary mental math leans on them.
• Group bits in fours from the right when reading long numbers; it lines them up with hex and is far easier to scan.
• A leading zero never changes the value: 0101 and 101 are both 5. Padding is just for alignment.
• Doubling = a left shift. Appending a 0 on the right multiplies by 2, just as adding a 0 to a decimal number multiplies by 10.

Common pitfalls

The mistakes below trip up almost everyone learning binary; the calculator is built to catch them.

• Using illegal digits: a 2 in binary or an 8 in octal is invalid. The tool flags these instead of producing a wrong answer.
• Reading bits in the wrong direction: the rightmost bit is the ones place, not the leftmost.
• Forgetting the carry in addition when a column hits 1+1.
• Mixing up octal and hex: both compress binary, but in groups of three versus four bits.
• Expecting every decimal fraction to end in binary: only fractions whose denominator is a power of 2 do; 0.1 becomes 0.0(0011), repeating forever.

Binary in computing: text, color and permissions

Once a value is a string of bits, the same bits can mean very different things depending on how software interprets them. Text uses character encodings: in ASCII the letter A is the number 65, which is 01000001 in binary and 41 in hex; lowercase a is 97 (01100001). Color on the web is three bytes, one each for red, green and blue, so the hex code #FF8800 means red 255, green 136, blue 0, and each channel runs from 0 to 255 precisely because a byte holds 256 values. Unix file permissions use three octal digits, where each digit is three permission bits (read = 4, write = 2, execute = 1); chmod 755 grants the owner read+write+execute (7) and everyone else read+execute (5). Reading these everyday codes is just base conversion, which is exactly what the calculator above does.

Binary mental math: shortcuts worth learning

You can do a surprising amount of binary work in your head once a few shortcuts stick. Powers of two double each step (1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024), and 2¹⁰ = 1,024 is why memory is counted in kibibytes of 1,024 bytes rather than SI kilobytes of 1,000. One less than a power of two is all ones: 2⁴ is 16, so 15 is 1111; 2⁸ is 256, so 255 is 11111111. That pattern is why the largest unsigned value in n bits is always 2ⁿ − 1. To halve, shift right; dropping the rightmost bit divides by two and the dropped bit is the remainder, so an odd number simply ends in 1. These tricks turn most conversions into recognition rather than long division.

Limitations and assumptions

This calculator is a learning and conversion aid, not a full bit-manipulation environment. Keep these boundaries in mind:

• Negative numbers are written with a minus sign in every base; two's complement is shown separately at the smallest standard width that fits, not as a fixed-width register that can overflow.
• Fraction digits that never end are cut off in the fields (32 binary, 12 octal, 10 hex and 40 decimal places), with the repeating block shown in the result card. Floating-point formats such as IEEE 754 are not modeled.
• The Binary math tab takes non-negative whole numbers; a subtraction may still return a negative result.
• Division is integer division with a separate remainder, not exact decimal division.
• Values of any length are exact; only the place-value table is limited to 64 binary digits.

Related concepts and calculators

Binary is one corner of a larger toolkit for working with numbers. To raise a number to a power, which is exactly what computing place values does, use the Exponent Calculator. The Scientific Calculator handles logarithms (log base 2 tells you how many bits a value needs) and more advanced functions. For base-16 arithmetic the Hex Calculator is the dedicated tool, for signed bit patterns the Two's Complement Calculator, and for integers too long for an ordinary calculator the Big Number Calculator. Together they cover most of the number-base math you meet beyond a simple four-function calculator.

Sources

• National Institute of Standards and Technology, Prefixes for binary multiples: kibi (Ki) = 2¹⁰ = 1,024 and mebi (Mi) = 2²⁰, approved by the International Electrotechnical Commission in December 1998, while the SI prefix kilo (k) means 1,000.
• ECMAScript Language Specification, §6.1.6.2 The BigInt Type: integers of any size, with bitwise operations defined on two's complement binary strings. This is the exact integer arithmetic the calculator uses.