How to compare two binary numbers

Comparing binary numbers is one of the most frequently performed operations inside a computer.

It is necessary to compare numbers to perform many arithmetic operations, but also for evaluating conditionals, executing loops, and… in many cases.

So, just as it is necessary for a computer, we also need to learn to do it “by hand.” Fortunately, comparing two binary numbers is super easy (even more so than in decimal).

Comparison of unsigned variables

If our binary number represents an unsigned value, the comparison is very very easy.

Example of comparing unsigned variables

Let’s consider the binary numbers 1101 and 1011 to compare:

• 1101 (13 in decimal)
• 1011 (11 in decimal)

Let’s start by comparing the most significant bits:

• For 1101, the most significant bit is 1.
• For 1011, the most significant bit is also 1.

Both numbers have the same most significant bit, so we move to the next:

• The second bit for 1101 is 1.
• The second bit for 1011 is 0.

Therefore, 1101 is greater than 1011.

Comparison of signed variables

When working with signed variables, things get a bit more complicated. In signed variables, the leftmost bit indicates the sign of the number. If this bit is 1, the number is negative; if it is 0, the number is positive.

So to compare, we need to do the following:

If one is positive and the other is negative, the positive one wins. If both are positive, it’s a comparison as we did before. Both cases are easy.

The only “mini difficulty” (but very mini) is if both are negative. In two’s complement, negative numbers are inverted. So to compare them we have to,

• Either invert them and compare
• Or simply compare from left to right (as in the case of positives), but now the one with a 0 is greater