A Guide to Binary Calculation

The binary numeral system, or base-2 number system, represents numeric values using only two symbols: 0 and 1. This system is the fundamental language of computers and digital electronics. Understanding how to perform calculations in binary is essential for anyone in the fields of computer science and engineering.

Understanding Binary Place Values

Just like the decimal system (base-10) has place values like ones, tens, and hundreds, the binary system (base-2) has place values based on powers of 2. You can learn more about powers using our Exponent Calculator.

For the binary number 10110:

• 1 x 2⁴ = 16
• 0 x 2³ = 0
• 1 x 2² = 4
• 1 x 2¹ = 2
• 0 x 2⁰ = 0

Adding these up (16 + 0 + 4 + 2 + 0) gives us the decimal equivalent: 22.

How Binary Arithmetic Works

Our calculator handles these operations for you, but here’s a quick overview of the logic.

Binary Addition (+)

Binary addition follows the same principles as decimal addition, but the rules are simpler:

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 0, carry over 1

Binary Subtraction (−)

Subtraction can be performed using borrowing, similar to decimal subtraction. For example, when calculating 10 - 1, you "borrow" from the left, turning the 10 into 02. Then 2 - 1 = 1. A more common method in computing is using "two's complement" to turn subtraction into an addition problem.

Binary Multiplication (×)

Multiplication is also very similar to its decimal counterpart. You perform long multiplication, where you multiply the top number by each digit of the bottom number and then add the results. Since the digits are only 0 or 1, it's very simple: you either write down a row of zeros or a copy of the top number.

Binary Division (÷)

Division uses long division, just as in decimal math. You compare the divisor with a part of the dividend. If the dividend part is larger or equal, you put a 1 in the quotient, subtract, and bring down the next digit. If it's smaller, you put a 0 in the quotient and bring down the next digit.

For more general math problems not restricted to binary, feel free to use our powerful Scientific Calculator or explore other tools on our Math page.