Run binary math, check decimal equivalents, and learn base-2 steps in one free tool built for students, coders, and quick accuracy checks.
Enter two binary values to see the answer and steps
Enter only 0s and 1s
Enter a binary value to see the decimal answer and place values
Enter a positive integer
Enter a decimal value to see the binary answer and repeated division steps
You can use this page for three tasks: binary arithmetic, binary to decimal conversion, and decimal to binary conversion. The steps stay simple because the calculator is built around the same place-value rules you use by hand in the base-2 number system.
Choose binary arithmetic when you want to add, subtract, multiply, or divide. Choose a converter tab when you want a fast one-way value change.
For binary fields, enter only 0 and 1. For decimal conversion, enter a whole number like 42, 128, or 214.
Click the math symbol you need, then press the action button. The result box updates instantly and also shows the decimal equivalent for quick error checking.
Compare the binary answer to the decimal answer, then use the guides below to understand carries, borrows, quotient, remainder, and powers of two.
Start with short values if you are learning. Try 1011 + 1101 first, then move to subtraction or division once you feel comfortable reading each column. Binary addition and subtraction are easier to verify when you convert both numbers to decimal before you start.
If you are checking homework, note each carry bit or borrow above the line on paper. Then compare your handwritten answer to the calculator output. That keeps the tool helpful without turning it into a black box.
Use binary to decimal when you see a bit pattern in code, a logic problem, a subnetting example, or a byte value and want the numeric meaning. Use decimal to binary when you need to encode a value, understand memory limits, or confirm that a power of two is stored correctly.
The step breakdown is especially useful for exam prep because it mirrors the manual method: multiply by place value for binary to decimal, or divide by 2 repeatedly for decimal to binary.
A binary answer is more useful when you know what each part means. This section helps you read the result line, the decimal check, and the step-by-step output with confidence.
The bold value is the direct base-2 answer. If you add 1010 and 1100, the result 10110 tells you the total in binary. Every digit is a bit, and each bit has a place value of 1, 2, 4, 8, 16, 32, and so on from right to left.
A longer answer does not mean the value is wrong. It usually means your highest place value increased. For example, when you add 1111 and 1, the answer becomes 10000. That new leftmost 1 shows a carry bit created a new power of two.
The decimal line gives you a quick validation path. It converts both inputs and the final output to base 10 so you can compare the same operation in a familiar format. This is one of the fastest ways to spot a place-value mistake or a missed carry.
For example, if the calculator shows 1011 + 1101 = 11000, the decimal line confirms that 11 + 13 = 24. When both systems agree, you can trust the result.
Binary division can feel harder because you must track the current partial dividend. In long division, you compare the divisor to the active bits, subtract when possible, write a 1 in the quotient, and bring down the next bit. If the divisor does not fit, you write a 0 in the quotient and move on.
The current LiteCalc tool returns the whole-number quotient. That is useful for many class and programming checks. If you also need the remainder, subtract divisor × quotient from the original dividend in decimal or binary.
When a subtraction result goes below zero, this calculator shows a minus sign followed by the binary magnitude. That is easy to read for quick checking. In computer architecture, signed values are often stored with two's complement instead.
If you are working with fixed-width values such as 8-bit or 16-bit data, be careful. A signed result inside a processor may not look the same as a simple negative sign because the bit width controls overflow and two's complement behavior.
You do not need a calculator to understand binary math. These formulas show how to calculate binary values manually and why the tool gives the answers it does.
Multiply each bit by its power of two and add the totals: decimal = Σ(bit × 2^position). The rightmost bit has position 0, the next has position 1, and so on. This is the core place-value rule behind every binary to decimal calculator.
Worked example: 11010110 equals 1×2^7 + 1×2^6 + 0×2^5 + 1×2^4 + 0×2^3 + 1×2^2 + 1×2^1 + 0×2^0. That becomes 128 + 64 + 16 + 4 + 2 = 214.
If you want a fast mental check, group the active powers of two. With practice, you can look at a byte such as 11111111 and know right away that it equals 255 because it contains every value from 1 through 128.
Divide by 2, record the remainder, repeat until the quotient is 0, then read the remainders from bottom to top. This method works because every division strips off the current least significant bit.
Worked example for decimal 214: 214 ÷ 2 = 107 R0, 107 ÷ 2 = 53 R1, 53 ÷ 2 = 26 R1, 26 ÷ 2 = 13 R0, 13 ÷ 2 = 6 R1, 6 ÷ 2 = 3 R0, 3 ÷ 2 = 1 R1, 1 ÷ 2 = 0 R1. Reading the remainders upward gives 11010110.
This same logic helps when you convert decimal values into bytes, permissions, or bit masks in programming and networking tasks.
Binary addition follows four rules: 0+0=0, 0+1=1, 1+0=1, and 1+1=10. The last rule creates a carry bit because two ones equal two in decimal, which is written as 10 in base 2.
Example: 1011 + 1101. Start on the right: 1+1 gives 0 carry 1. Then 1+0+carry gives 0 carry 1. Then 0+1+carry gives 0 carry 1. Then 1+1+carry gives 1 carry 1. Bring down the final carry and the answer is 11000.
Binary subtraction uses borrowing. If the top bit is 0 and you need to subtract 1, borrow from the next column just as you do in decimal subtraction.
Binary multiplication is repeated addition. Because a binary digit can only be 0 or 1, each partial row is either all zeros or a shifted copy of the other number. This makes it easier to spot patterns than decimal multiplication.
Example: 101 × 11 means 101 × 1 + 1010 × 1, which equals 1111. In decimal, that is 5 × 3 = 15, so the answer checks out.
Binary division is repeated comparison and subtraction. If you can keep the divisor, quotient, and remainder straight in decimal long division, you can learn the base-2 version with the same workflow.
Binary math shows up in more places than many people expect. These examples use real numbers so you can see why conversion and arithmetic matter outside a textbook.
Suppose you see 11010110 in a file format or packet example. Converting it to decimal gives 214. That tells you the byte is larger than 128 but below 255, which is a useful sanity check when you are debugging or reading specs.
If your worksheet asks for 101101 + 1110, line up the bits and add column by column. The answer is 111011. Converting to decimal gives 45 + 14 = 59, which confirms the binary sum is correct.
The octal permission 7 means binary 111, which stands for read, write, and execute. The octal permission 5 means 101. Knowing the binary pattern helps you understand why permissions like 755 break into 111 101 101.
Decimal 1024 converts to binary 10000000000. That single 1 in the 2^10 place tells you 1024 is exactly a power of two. This comes up often in memory sizes, array limits, and bitwise operations.
If a mask needs the 8, 4, and 1 positions turned on, combine those powers of two to get 13. Decimal 13 becomes 1101. That makes it easy to see which flags are active before you use the value in code.
Divide 110101 by 101. In decimal, that is 53 divided by 5, so the whole-number quotient is 10 and the remainder is 3. The quotient in binary is 1010. When you know the decimal result first, the binary answer is much easier to verify.
Top-ranking competitors spend more time on place value, grouped bits, and cross-system patterns. This section fills that gap so you can move from raw digits to meaning faster.
An 8-bit byte uses the place values 128, 64, 32, 16, 8, 4, 2, and 1. If a bit is on, include that value. If it is off, skip it. This is why 10000000 equals 128 and 00000001 equals 1.
Four bits make a nibble, and two nibbles make a byte. Grouping bits into nibbles helps you move between binary and hexadecimal quickly. For example, 1101 is 13 in decimal and D in hexadecimal. The byte 11010110 becomes D6 in hex when you split it into 1101 0110.
This pattern matters in programming, memory addresses, color values, and machine-level debugging because hex is shorter to read while still matching binary perfectly in groups of four.
The fastest binary workers memorize the most common powers of two. You do not need a huge list. Knowing 2^0 through 2^10 gets you through most school problems and many practical computing tasks.
Here are the values worth remembering: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024. Once you know them, you can read many binary strings almost on sight.
Unsigned binary treats every bit as part of the value, so an 8-bit range runs from 0 to 255. Signed binary often uses two's complement, which changes the meaning of the leftmost bit.
For example, unsigned 11111111 is 255. In signed 8-bit two's complement, that same pattern is -1. The difference is not a math trick. It comes from the storage rules set by the system.
That is why a classroom binary calculator and a low-level programming environment can show different meanings for the same bit pattern. Always check whether your problem uses signed or unsigned values.
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Quick answers to the binary questions people ask most often.
Add from right to left using the four binary rules. When you get 1 + 1, write 0 and carry 1 into the next column. If you also have a carry, 1 + 1 + 1 becomes 11, so you write 1 and carry 1 again.
Multiply each bit by its matching power of two and add the active values. For 11010110, the decimal value is 128 + 64 + 16 + 4 + 2 = 214.
Divide the decimal number by 2 repeatedly, write down each remainder, and read the remainders from bottom to top. That final sequence is the binary value.
Binary division is long division in base 2. You compare the divisor with the current part of the dividend, subtract when it fits, place 1 in the quotient, and bring down the next bit until you finish.
Two's complement is a signed-number system used by computers. Flip every bit, add 1, and the new pattern represents the negative version of the original value inside a fixed bit width.
Computer circuits are built around two stable states, usually represented as on and off. Binary matches those two states cleanly, so it is reliable for storage, logic, and processing.
A byte contains 8 bits. Four bits make a nibble, and a full byte can represent values from 0 to 255 when treated as unsigned.
Many tools can, but they may show negative values in different ways. This calculator uses a leading minus sign for quick readability, while programming systems often use two's complement.
Convert the inputs and the result to decimal and compare the same operation in base 10. If the decimal math matches, your binary answer is almost certainly correct.