Binary Math: How to Add, Subtract, and Multiply 1s and 0s

Instantly calculate massive binary equations without manual base-conversion using our Binary Calculator, or read on to learn how to do the math on paper.

In the Base-10 decimal system that humans use, we have ten digits (0 through 9). When we add numbers and hit the maximum digit of 9, we carry a "1" over to the next column.

Computers use the Base-2 binary system, which only has two digits: 0 and 1. This means the mathematical rules for addition and subtraction trigger the "carrying" rule much faster, but the logic remains exactly the same.

Binary Addition

There are only four possible combinations when adding two single binary digits (bits):

1. 0 + 0 = 0
2. 1 + 0 = 1
3. 0 + 1 = 1
4. 1 + 1 = 0 (and you carry a 1 to the next column).

Wait, why does 1 + 1 equal 0? Because the number 2 does not exist in binary. In decimal, 1 + 1 = 2. In binary, the decimal number 2 is written as 10. Therefore, you write down the 0, and carry the 1.

Example: Adding 1011 and 1101

Let's add these vertically from right to left. 1 0 1 1 (Decimal 11) + 1 1 0 1 (Decimal 13)

Column 1 (Right): 1 + 1 = 0 (Carry 1) Column 2: 1 + 0 + (carried 1) = 0 (Carry 1) Column 3: 0 + 1 + (carried 1) = 0 (Carry 1) Column 4 (Left): 1 + 1 + (carried 1) = 1 (Carry 1 to a new 5th column)

Result: 11000 (Decimal 24).

Binary Subtraction

Binary subtraction is slightly trickier because it requires "borrowing," just like in decimal math when you try to subtract a larger number from a smaller one.

There are four basic rules for subtracting bits:

1. 0 - 0 = 0
2. 1 - 1 = 0
3. 1 - 0 = 1
4. 0 - 1 = 1 (but you must borrow a 1 from the next column to the left).

When you borrow a 1 from the next column, the current column temporarily becomes 10 (which is 2 in decimal). 2 - 1 = 1.

Example: Subtracting 101 from 110

1 1 0 (Decimal 6) - 1 0 1 (Decimal 5)

Column 1 (Right): 0 - 1. You cannot do this. You borrow from Column 2. The 0 becomes 10 (2). 2 - 1 = 1. Column 2: The 1 was borrowed, so it is now 0. 0 - 0 = 0. Column 3 (Left): 1 - 1 = 0.

Result: 001 (Decimal 1).

Binary Multiplication

Binary multiplication is surprisingly easy because you are only ever multiplying by 0 or 1.

• Multiplying by 0 equals 0.
• Multiplying by 1 equals the number itself.

It works exactly like standard long multiplication, but the addition at the end is much simpler!

FAQ

How do computers subtract if they only have adding circuits? Modern computer processors (CPUs) do not actually have subtract, multiply, or divide circuits. They only have adder circuits. To subtract, the CPU converts the second number into a negative using a method called "Two's Complement," and then simply adds the negative number to the first number!

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