Binary Calculator

Computers do not count in tens the way people do. They count in binary - a number system with only two digits, 0 and 1 - because every electronic switch inside a chip is simply off or on. This binary calculator does two jobs at once: it converts a value between binary, decimal, hexadecimal and octal, and it performs binary arithmetic (addition, subtraction, multiplication and division) on two binary numbers. Type 1010 into the binary field and you instantly see it is 10 in decimal, 12 in octal, and A in hexadecimal - along with a place-value table that shows exactly why.

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.

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.

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.
3. Read the result. The big number shows the decimal value, the cards show all four bases, and the table breaks the binary number into its place values.
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 and hex equivalents.
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.

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). This calculator uses unsigned numbers, so the first value must be the larger one. 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.

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.

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 everyday arithmetic and number crunching, the Percentage, Fraction, and Average calculators round out the set, and the Square Root Calculator covers roots. Together they cover most of the math you meet beyond a simple four-function calculator.