I introduced my mother to binary numbers a few weeks ago when I showed her my One Hundred Cheerios in Binary poster. It shows the decimal number 100 in binary — 1100100. She’s not an engineer but she’s good with numbers, so I knew she would get it — if only I could find the right way to explain it. Two days ago, I found the right way.
First Attempt — Place Value
My first attempt to explain binary numbers (just integers, not fractions) was based on place value notation. I reviewed what a decimal number is: a sum of multiples of powers of ten. I explained how each place increases by a factor of 10 as you go from right to left, and then explained how the same principle works in binary, only with factors of 2.
Her first question was why there were only two digits, 0 and 1. This lead me to a discussion of bases: base 10 has 10 digits (0-9), base 5 has 5 digits (0-4), and base 2 — binary — has 2 digits (0-1).
From there I went into a discussion of numbers vs numerals. I explained that “number” is an abstract concept — I told her to think of a pile of three hundred and twenty-one rocks — and that the numeral 321 is only one way to express it. I also talked about Roman numerals and tally marks as other ways to represent numbers.
I hadn’t seen her “aha” yet so I continued by showing her how to count in binary. 0, 1, then — uh oh, we can’t write 2, so now we have to carry. But all that did was muddy the waters with yet another concept — binary arithmetic.
Why I Think This Approach Didn’t Work
Place value is not something people think about much after elementary school, and certainly not if they’ve been out of elementary school for a while (sorry Mom). I think I started out explaining it this way because that’s basically how I did it with my son; he’s in first grade, and place value is fresh in his mind. But I think people forget about place value as they get older, and they think of decimal numerals as numbers themselves.
An associated problem was using the term “powers”; exponentiation is another thing most people leave behind when they leave school.
Finally, I think mentioning Roman numerals and tally marks was a mistake — I should not have gone beyond place value.
Second Attempt — Sum of Powers of Two
My second attempt to explain binary numbers — successful I might add — was a back door approach. I stuck with decimal numbers in my explanation as long as I could, only introducing binary at the end.
I made a list of the first seven nonnegative powers of two — I didn’t call them that, but just said each was double the previous. I said that I could write any number up to 100 just by adding numbers from that list. She said “17” and I wrote down 16 + 1; she said “56” and I wrote down 32 + 16 + 8. We tried a few more like that, as I explained what I was doing — finding the largest number in the list that “fit” in the number, subtracting it, and then repeating this until nothing was left.
Next I rewrote the powers of two in a diagram like this:
I then took 17 — which we had just written as 16 + 1 — and said “OK, does it have a 1 in it? Yes. Let’s write 1 for yes and 0 for no in the blanks. Does it have a 2? No, write down 0. 4? No, write 0. 8? No, write 0. 16? Yes, write 1.” So then the diagram looked like this:
That’s when the lightbulb went on.
We continued with 56 just to make sure she got it:
I said this would work for any number, as long as the list of “doublings” was long enough.
I gave her five examples to “convert” after I left — five numbers under 100. I spelled out each number in words rather than in decimal numerals, although I don’t know if that was necessary. I also left an answer sheet.
She called me after I left to tell me that she got them right, and she wanted me to check two of her own examples: 185 and 349. She had extended her list of doublings to 256 and applied the same conversion algorithm. She got them right!
(Mom, thanks for being persistent, even long after Dad gave up on me!)
In the end, I think the combination of both approaches was probably necessary, although I should have started with the sum of powers of two explanation. Basically, I’m advocating starting with decimal to binary conversion, and then completing the cycle with binary to decimal conversion.
One obstacle every beginner will face is our number words. We “pronounce” the binary number 101000001 as three hundred and twenty-one. Number words have decimal place value built-in, regardless of the numerals behind them.