# How to Add Binary Numbers

Adding binary numbers is almost identical to adding regular decimal numbers. We start at the right, add those digits and add any carry digits to the next digits to the left.

fact

In binary:
\(0 + 0 = 0\)
\(0 + 1 = 1\)
\(1 + 0 = 1\)
\(1 + 1 = 0\)(carry \(1\))

example

Find \(101 + 001\)

We'll write out our numbers on top of each other to make keeping track of things simpler. $$\begin{align} 101 & \\ +\,\, 001 & \end{align}$$ Starting at the right we have \(1 + 1 = 0\)(carry \(1\)) so we write in our \(0\) and carry that \(1\) to the digits on the left. $$\begin{align} 10^11 & \\ +\,\, 001 & \\ 0 & \\ \end{align}$$ Now our next set of digits are \(0 + 0 + 1 = 1\), the extra \(1\) came from our carry bit. $$\begin{align} 101 & \\ +\,\, 001 & \\ 10 & \end{align}$$ And finally our leftmost digits are \(1 + 0 = 1\). $$\begin{align} 101 & \\ +\,\, 001 & \\ 110 & \end{align}$$ So \(101 + 001 = 110_2 = 6_{10}\) If one of our two numbers has fewer digits than the other just add some padding zeros to the left until they both have the same number of digits.

example

Find \(1011 + 11\)

First we need to pad our second number with leading zeros to make both numbers the same length, then we set up our sum like before: $$\begin{align} 1011 & \\ +\,\, 0011 & \\ \end{align}$$ Starting on the right we get \(1 + 1 = 0\)(carry \(1\)) $$\begin{align} 101^11 & \\ +\,\, 0011 & \\ 0 & \\ \end{align}$$ Then \(1 + 1 + 1 = 1\)(carry \(1\)) remember that carry bit from the last digits. $$\begin{align} 10^111 & \\ +\,\, 0011 & \\ 10 & \\ \end{align}$$ Then \(0 + 0 + 1 = 1\) again remember to include the carry bit. $$\begin{align} 1011 & \\ +\,\, 0011 & \\ 110 & \\ \end{align}$$ And finally \(1 + 0 = 1\) $$\begin{align} 1011 & \\ +\,\, 0011 & \\ 1110 & \\ \end{align}$$ If the left-most digits' addition results in a carry bit then we just write that carry bit as an extra digit to the left of our answer like in the example below:

example

Find \(10 + 11\)

$$\begin{align} 10 & \\ +\,\, 11 & \\ \end{align}$$ \(0 + 1 = 1\) $$\begin{align} 10 & \\ +\,\, 11 & \\ 1 & \\ \end{align}$$ Then \(1 + 1 = 0\)(carry \(1\)) So we write that \(0\) and then add the extra \(1\) to the left (carrying it into the answer) $$\begin{align} 10 & \\ +\,\, 11 & \\ 101 & \\ \end{align}$$
practice problems