Binary Calculator

Use this binary calculator to add, subtract, multiply, or divide two binary numbers instantly. Enter your binary values above, pick an operation, and get the result in both binary and decimal. The tool handles carry operations, borrowing, and conversion so you can check homework, verify code logic, or explore how base-2 arithmetic works.

Binary Calculator for Basic Binary Arithmetic

The calculator accepts two binary numbers (strings of 0 and 1). Choose addition, subtraction, multiplication, or division. The result displays in binary along with its decimal equivalent.

You can also use it as a decimal to binary converter. Enter a decimal number and the tool shows the binary representation. This makes it easy to move between number systems without manual conversion.

Want to calculate with decimal operands? Enter your decimal values in the conversion field first, copy the binary output, then paste it into the operation inputs. The calculator handles the rest.

What Is a Binary Number

A binary number uses only two digits: 0 and 1. Each digit is called a bit. Computers store and process all data using the binary system because electronic circuits naturally represent two states (on and off).

Decimal uses ten symbols (0 through 9). Binary uses two. That difference changes how you read place values and perform arithmetic, but the underlying math follows the same rules you already know.

How Binary Numbers Use Base-2 Positional Notation

In decimal (base 10), each position represents a power of 10. The rightmost digit is 10⁰ (1), the next is 10¹ (10), then 10² (100), and so on.

Binary works the same way using powers of 2:

• Rightmost bit: 2⁰ = 1
• Next bit left: 2¹ = 2
• Next: 2² = 4
• Next: 2³ = 8

So the binary number 1010 means (1 × 8) + (0 × 4) + (1 × 2) + (0 × 1) = 10 in decimal. Every position doubles the previous one because the base is 2.

Convert Decimal to Binary Step by Step

To convert a decimal number to binary, repeatedly divide by 2 and record each remainder.

Example: convert decimal 100 to binary.

1. 100 ÷ 2 = 50, remainder 0
2. 50 ÷ 2 = 25, remainder 0
3. 25 ÷ 2 = 12, remainder 1
4. 12 ÷ 2 = 6, remainder 0
5. 6 ÷ 2 = 3, remainder 0
6. 3 ÷ 2 = 1, remainder 1
7. 1 ÷ 2 = 0, remainder 1

Read the remainders from bottom to top: 1100100. Decimal 100 equals binary 1100100.

This method works for any positive integer. For larger numbers the steps multiply, which is why a decimal to binary converter saves time.

Convert Binary to Decimal Value

Multiply each bit by its positional power of 2, then add the products.

Example: convert 1101 to decimal.

• 1 × 2³ = 8
• 1 × 2² = 4
• 0 × 2¹ = 0
• 1 × 2⁰ = 1

Total: 8 + 4 + 0 + 1 = 13. Binary 1101 equals decimal 13.

The calculator displays this decimal value automatically alongside every binary result.

Binary Addition

Binary addition follows the same column-by-column logic as decimal addition. The only difference is that you carry at 2 instead of 10.

How to Add Binary Numbers Bit by Bit

Start at the rightmost bit and move left. At each position, add the two bits plus any carry from the previous column.

The basic rules:

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 10 (write 0, carry 1)
• 1 + 1 + 1 = 11 (write 1, carry 1)

Binary Addition Result and Carry Operations

Example: add 1011 and 1101.

<code> 1 1 1 ← carries 1 0 1 1 + 1 1 0 1 ----------- 1 1 0 0 0 </code>

Step by step, right to left:

1. 1 + 1 = 10. Write 0, carry 1.
2. 1 + 0 + carry 1 = 10. Write 0, carry 1.
3. 0 + 1 + carry 1 = 10. Write 0, carry 1.
4. 1 + 1 + carry 1 = 11. Write 1, carry 1.
5. Carry 1 remains. Write 1.

Result: 11000. In decimal, that is 11 + 13 = 24.

The binary addition calculator above handles multi-bit carry chains instantly, so you can verify your manual work or skip straight to the answer.

Binary Subtraction

Binary subtraction uses borrowing, just like decimal subtraction uses borrowing from the next column.

How to Subtract Binary Numbers

The basic rules:

• 0 − 0 = 0
• 1 − 0 = 1
• 1 − 1 = 0
• 0 − 1 = 1 with a borrow of 1 from the next column

Example: subtract 101 from 1100.

<code> 1 1 0 0 − 0 1 0 1 ----------- 0 1 1 1 </code>

Working right to left:

1. 0 − 1: cannot subtract, borrow from the next column. After borrowing, 10 − 1 = 1.
2. The borrowed column becomes 0. 0 − 0 = 0. But the original was 0, so borrow again. After borrowing, 10 − 0 = 1. (The chain of borrows continues as needed.)
3. Continue until all columns are resolved.

Result: 0111 (decimal 7). Check: 12 − 5 = 7.

Borrowing across multiple columns can get tedious. The calculator tracks every borrow for you.

Binary Multiplication

Binary multiplication is simpler than decimal multiplication because each partial product is either 0 or a copy of the multiplicand.

How to Multiply Binary Numbers

Rules:

• 0 × 0 = 0
• 0 × 1 = 0
• 1 × 0 = 0
• 1 × 1 = 1

Multiply the same way you do long multiplication in decimal:

1. Take one bit of the multiplier at a time, starting from the rightmost.
2. If the bit is 1, copy the multiplicand. If 0, the partial product is 0.
3. Shift each partial product one position left.
4. Add all partial products using binary addition.

Example: 110 × 101.

<code> 1 1 0 × 1 0 1 ----------- 1 1 0 (110 × 1) 0 0 0 0 (110 × 0, shifted left once) 1 1 0 0 0 (110 × 1, shifted left twice) ----------- 1 1 1 1 0 </code>

Result: 11110 (decimal 30). Check: 6 × 5 = 30.

For longer binary numbers the partial products stack up, making a calculator especially helpful.

Binary Division

Binary division follows the long division algorithm. Instead of guessing quotient digits from 0 to 9, each quotient bit is either 0 or 1.

How to Divide Binary Numbers Using Long Division

Steps:

1. Compare the divisor to the leftmost bits of the dividend.
2. If the divisor fits (is less than or equal), write 1 in the quotient and subtract.
3. If the divisor does not fit, write 0 in the quotient.
4. Bring down the next bit and repeat.

Example: 11010 ÷ 110 (decimal 26 ÷ 6).

• 110 does not fit into 1. Quotient bit: 0.
• 110 does not fit into 11. Quotient bit: 0.
• 110 fits into 110. Quotient bit: 1. Subtract to get 0.
• Bring down 1. 110 does not fit into 01. Quotient bit: 0.
• Bring down 0. 110 fits into 010? No. Quotient bit: 0.

Result: quotient 100 with remainder 10 (decimal 4 remainder 2). Check: 6 × 4 + 2 = 26.

Long division in binary is mechanical but repetitive. The calculator handles the full sequence, including remainders.

Binary Arithmetic With Negative Numbers

Computers typically represent negative binary numbers using two's complement. In this system:

1. Start with the positive binary value.
2. Flip every bit (0 becomes 1, 1 becomes 0). This is the one's complement.
3. Add 1 to the result. This gives the two's complement, which represents the negative value.

Example: represent −5 in 8-bit two's complement.

• +5 in binary: 00000101
• Flip bits: 11111010
• Add 1: 11111011

The leading 1 signals a negative number. Addition and subtraction of two's complement numbers use the same binary addition rules. The hardware (or calculator) does not need a separate subtraction circuit.

When working with signed values, always note the bit width (8-bit, 16-bit, 32-bit) because it determines the range and the position of the sign bit.

Binary Calculator Operations With Fractions and Floating-Point Values

Binary fractions work like decimal fractions but with powers of 2 in the denominator.

• The first bit after the binary point (period) is 2⁻¹ = 0.5.
• The next is 2⁻² = 0.25.
• Then 2⁻³ = 0.125, and so on.

Example: binary 101.11 = 4 + 0 + 1 + 0.5 + 0.25 = 5.75 in decimal.

Some decimal fractions (like 0.1) cannot be represented exactly in binary. The result is a repeating binary fraction, similar to how 1/3 repeats in decimal. This is why floating-point arithmetic in computers sometimes produces tiny rounding differences.

The calculator processes fractional binary inputs the same way it handles integers. Enter values with a binary point and the tool computes the result, displaying both the binary and decimal equivalents.

Convert Decimal to Binary for Hexadecimal and Other Number Systems

Binary conversion connects directly to hex (base 16) and octal (base 8) because their bases are powers of 2.

• Each hex digit maps to exactly 4 binary bits.
• Each octal digit maps to exactly 3 binary bits.

Quick reference:

To convert a hex value to binary, replace each hex digit with its 4-bit binary equivalent. To go from binary to hex, group the bits into sets of 4 (starting from the right) and map each group.

This grouping trick is why programmers prefer hex as a compact way to display binary data. The calculator can help you verify these conversions alongside arithmetic.

Binary Calculation Results and Significant Figures

Binary arithmetic produces exact integer results when both inputs are whole numbers. No rounding is involved.

For fractional or floating-point values, precision depends on how many bits you use. More bits mean more decimal places of accuracy, but infinite precision is not possible for repeating fractions.

When you see a result in scientific notation or with many digits, keep in mind that the binary output is an exact representation within the bit length used. The decimal equivalent may be rounded for display. Treat every result as an estimate if your application requires a specific level of significant figures.

Understanding Each Binary Calculator Result and Its Decimal Value

Every result from the calculator includes two parts:

• Binary output: the raw base-2 answer to your operation.
• Decimal value: the base-10 equivalent so you can verify or use the number elsewhere.

This dual display lets you confirm that the binary arithmetic is correct by cross-checking the decimal. If you add 1010 (10) and 0110 (6) and see 10000 (16), both representations confirm the answer.

Use the binary output when working with digital systems, programming, or networking. Use the decimal value for everyday comparisons or when feeding the number into a system that expects base 10.

Results are computed in your browser. The tool does not store your inputs or send them to a server. It is a quick planning aid for binary arithmetic, not a substitute for specialized engineering or scientific software.