Binary Math

In order to familiarize students with these “strange” numeration systems, I like to begin each day of digital circuit instruction with counting practice. Students need to be fluent in these numeration systems by the time they are finished studying digital circuits!

One suggestion I give to students to help them see patterns in the count sequences is “pad” the numbers with leading zeroes so that all numbers have the same number of characters. For example, instead of writing “10” for the binary number two, write “00010”. This way, the patterns of character cycling (especially binary, where each successively higher-valued bit has half the frequency of the one before it) become more evident to see.

Ask your students to describe what differences exist between manually adding binary numbers and manually adding decimal numbers, if any.

Although this may seem like a trivial question, I’ve met electronics technicians who actually believed that the form of numeration affected the outcome of certain mathematical operations. In particular, I met one fellow who believed the number π was fundamentally different in binary form than it was in decimal form: that a binary “pi” was not the same quantity as a decimal “pi”. I challenged his belief by applying some Socratic irony:

Me: How do you use a hand calculator to determine the circumference of a circle, given its diameter? For example, a circle with a diameter of 5 feet has a circumference of . . . Him: By multiplying the diameter times “pi”. 5 feet times “pi” is a little over 15 feet. Me: Does a calculator give you the correct answer? Him: Of course it does. Me: Does an electronic calculator use decimal numbers, internally, to do math? Him: No, it uses binary numbers, because its circuitry is made up of logic gates . . . (long pause) . . . Oh, now I see! If the type of number system mattered in doing math, digital computers and calculators would arrive at different answers for arithmetic problems than we would doing the math by hand!

Of course, those familiar with computer programming and numerical analysis understand that digital computers can introduce “artifacts” into computed results that are not mathematically correct. However, this is not due to their use of binary numeration so much as it is limited word-widths (leading to overflow conditions), algorithmic problems converting floating-point to integer and visa-versa, and such.