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Zeros and Ones
by Dr. Arnie Gotfryd
3:00 pm, Wednesday afternoon. Instead of getting ready to flee school like the vast majority of other Grade 7 classes, this particular group of 12-year-olds was just gearing up for their daily dose of reading, writing, and 'rithmetic.
This was a "cheder," an old-fashioned Jewish day school where kids from Chassidic families spend pretty much all day on Torah studies of various kinds, and only then, at the end of a very long day, are expected to eagerly soak up in less than two hours what takes public school students, more than six.
Now if that sounds overly challenging for the children, imagine how daunting it is for the teachers! I, for one, have no problem imagining what it's like because teaching this kind of class is something I have done a lot of.
That particular Wednesday at 3:00 pm, I was teaching math. "Alright class, today we are going to learn a little about computer language. You know how Hebrew has 22 basic letters and English, l'havdil, has 26? Well, computer language is called binary and it has just two letters - zero and one... Yes, I know zeros and ones are numbers, not letters, but you know what I mean. Now, regular math works on a Base Ten system, but the computer works on what's called Base Two... No, no, not second base - Base Two." 
"Regular math uses 10 digits - 0, 1, 2, etc. up to 9. Then you move over to the tens column and you go all over again until you get to the hundreds, thousands and so on - Zalman - stop yawning. Mendel, put that ball away. As I was saying..."
"Dr. G, could you say that over?"
"Yeah, I have no clue what you're talking about."
"Can I get a drink of water?"
I ramped up my energy level, hoping to magnetize them to the material a little, and tried again.
"Okay, let's look up at the board. You see we've got ones, tens, hundreds, thousands...? So in Base Two you've got ones, two's, fours, eights, and so on. Zero is the same, one is the same, but there is no such numeral as "2" so what you do is, put a 0 in the ones column and a 1 in the two's column, you see? '1 0' is ten in base ten, but it's two in Base Two. Get it?"
But they didn't get it. I took another run at it but no dice. Moishie's now napping on his desk and the other eyes I do see are wavering between glazed and confused.
These kids are smart but they have so much listening to do all day long, how am I going to get through? I thought, okay, if listening is the problem, let's see if I can simply assign them to read the text and do the exercises. Still nothing.
"Alright guys. Who likes pizza?" Hands shoot up. "Okay, I'm not that generous, but Bissli and Coke I could find."
"What do we have to do?" one enterprising young man called out.
"You see those 25 exercises on page 273?"
"No! Impossible! We can't! Are you kidding? We don't even know what's flying!?"
"Take it easy. How about if I give you just one question to solve? If each and every one of you can get that one answer right and show exactly how you got it, I'll bring in the snacks tomorrow - deal?"
"Deal!" The chorus was resounding.
"Fine. Berel, you go to the board and write down any big, long number."
"What number?"
"I don't know. Yisroel, you tell him what to write."
"3 8 7 2 4 0 9 5 6 3 3 2 1..."
"Good enough. Now that's a normal number in Base Ten - three trillion and change. What I want you to do is to write it in Base Two. Show your work and be ready to explain it." 
"Explain what?"
"I don't get it"
"You've got to be kidding."
"You can do it. Divide up into groups. You can use calculators. Work it out together. Use the book for reference. It's not rocket science."
"I wish it was!"
"Don't worry. Rocket science is next." The rest of the class was a sight to behold. It's amazing what the promise of salty crispies and sweet bubblies can do for the motivational state of a bored preteen. Calculators were clacking as they figured out the column values - 1, 2, 4, 8, 16... 4196, 8392, 16784...
"Wait, how do I do this? Hmm. Oh I see. Oy vay! Could this answer be 42 digits long? Are you sure this is how computers work?"
"Kind of," I replied. "Actually they work with logic gates. If the electrons flow through, that's a '1' - if they don't, it's a '0'. It's kind of like an 'on-off' switch."
"Yeah, like the class. We were 'off' until you mentioned the Bissli and Coke", said Zalman. "You know, I wasn't kidding about the 42 digits. Can you check my work?"
I love these kids. They really think I know how to instantly translate some random multi-trillion number into binary. "I'll check it soon. First, let's see if the numbers each group gets match. If not, you've got to figure out who made the mistake and where."
Our Sages say, "Ayn davar omed bifnay haratzon", i.e., If there's a will, there's a way. Within a half hour every kid there not only had the same answer and knew how to get it, they knew the subject well enough to teach it. You just have to know how to push their buttons.
When I got home I had another button to push - the 'on-off' button on my laptop. I needed a binary / decimal converter to check their answer. But before I even got to that, I focused on that button and its icon.
I suddenly had a kabbalistic 'aha' moment. On/off duality is represented by a binary symbol - a '1' and a '0' combined. That binary switch, when turned on, releases a cascade of further binary operations that result in an amazing diversity of processing and display such as any computer can muster.
For the student of the inner dimensions of Torah, there is another level of meaning here. The 'on-off' button of creation, as it were, is the sefira of chochma, a simple point represented by the smallest of letters, yud. The button-yud of chochma is the beginning of creation, a concentrated point (tzimtzum) from which a line extends (kav), and the entire spiritual and material creation unzips. Hence the verse in Psalm 104, "You have made all of them in Wisdom (chochmah)."
Yud-Chochmah is emblematic of duality because it is the point that mediates between yesh and ayin, or being and nothingness, or more precisely no-thing-ness. But for all its inherent duality, the numerical equivalent of yud is not two but rather ten, symbolic of the world in all its fullness. Our sages allude to this in Ethics of the Fathers with the statement, "With ten sayings was the world created."
The 'on-off' button's icon symbolizes this diversity too, for when the '1' and '0' are separated they form the number '10.'  
Still there is something greater than chochmah, greater than the ten sayings by which the world was created and even greater than the duality of being and nothingness - and that is ratzon - will. Without will, nothing goes. It's the impetus that transcends everything and motivates it.
Will is the force that initiates, whether we speak of the will of the student to learn, or the will of the computer operator to push the 'on-off' button, starting the whole cascade of events, which in my case led to an online decimal / binary conversion calculator, that proved in a jiffy that yes, I really did owe those kids their snacks because  3,872,409,563,321 really does equal 111000010110011101100110001011000010111001.
Will can be aroused in a variety of ways - a Bissli, a Coke, or even a good thought, such as Yes! I am in this world for a purpose, and all it takes is to push that button, to activate a little more goodness and kindness, and all the processing and display will show the ultimate result - Moshiach Now!
Dr. Aryeh (Arnie) Gotfryd, PhD is a chassid, environmental scientist, author and educator living near Toronto, Canada. To read more or to book him for a talk, visit his website at


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