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.

## Her Homework

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!)

## Summary

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.

I learned binary numbers in a math class and it is very confusing. I think you did a very good job on this.

Thank you

Michael

Thanks You ,very very helpful

>I then took 17 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.”

How do you know it doesn’t have a 2 if you start at the bottom, from 1? I always start at the top, 16 in this case. Then once I’ve taken the 16, all that’s left is a 1.

Sue,

Yes, starting at “the top” is the way the conversion from decimal to powers of two works. That’s the way I explained the conversion of 17 and 56 (notice how I wrote the powers of two in descending order — 17 = 16 + 1 and 56 = 32 + 16 + 8 — reflecting starting at “the top”).

At the point of the text you quoted, we were just trying to build the binary number from the already computed list of powers of two. I could have built it in any order, but it seemed natural — at the time at least — to build it from right to left.

So this is a two-step process: conversion from decimal to powers of two, which must proceed “top down,” and conversion from powers of two to binary, which can be done in any order. I should have wrote more explicitly about the algorithm of the first step (I just edited the article as such); as for the second step, the next time I teach someone I’ll think about which order makes more sense, if any (I can’t recall if I put any thought into different orders at the time).

Thanks for the feedback!

Rick — Thanks for taking the trouble to write this up. Very interesting. I’m mainly impressed by your evident commitment to effective pedagogy and its priceless rewards: “the light bulb went on :-)” Nice!

Thank you. I finally think I understand this. Thank you again.

Ha! I am STILL trying to explain this to my mom, too! My kid gets it (as you said, place values) but my brilliant scientist mom doesn’t. This will make a great Christmas discussion this year. THanks!

Cat,

Please let me know how it turns out!

My mother isn’t good with numbers at all so I could never explain it, but I’m glad it worked for you!

thank you took all of 5 min. to grasp= you rock

Thanks for this! I just had a lecture where they were moving at lightning speed and I had no idea what was going on. Reading this and practicing on my own really helped.

I get it sorta now lol just I don’t understand how is 101000001 considered as 321 ??? Btw you did a great job !!

@kelleythelegend,

101000001 = 256+64+1 = 321.

Never mind I understand now how 321 is 101000001 thank you Rick !

this method really helped me thanks…

Thanks for the very clear explanation, I was able to grasp this after reading and a bit of practice. The best explanation I’ve come across.