Piles of papers with numbers on them, odd numbers in pile 1, and so on; each pile corresponding to a bit ------------------ Binary numbers: Bisqwit’s first-time experience [camera angle: person + desk] Shalom! I am going to tell you a story how I first came across binary numbers. [video intro] Once upon a time, when I was a child, I came across this little game. In front of me I have six piles of cards. These cards contain integer numbers. The game goes like this. [camera angle: overhead, hands + toys] First, it’s my friend’s turn. They give me the instructions: [Use Melodyne to switch voice, or use speech synth] “Think of a number between 0 and 63. Don’t tell me what that number is, but look through these piles and tell me which of these piles contained your number.” Okay, I answer. I have a number in my mind. So, I will look through these piles. [23: piles 1, 2, 4 and 16] [shuffling through the piles, humming] I found your number in these piles (points). And immediately she declares: “Aha! Your number was 23!” And I go WHAAT, how could you possibly know that?? “Do another!” Allright. This time my number is… [48: piles 16 and 32] It’s in these two piles. And without hesitation, in an instant, she gives the answer: “You were thinking number 48!” And she is absolutely correct. How is that even possible? Here’s the secret. When my number was 23, I chose these four piles. [video edit: color the piles differently] Notice the numbers on top of these piles? 16, 4, 2, 1. She calculated the sum. 16+4+2+1. This comes to 23. When my number was 48, I chose these two piles. And again, the number comes to 32+16 = 48. Why does it work? What is special about the numbers in each of these piles? Let’s look at the piles. [Slide: 64 cards] Here are the cards in the first pile, with the "1" on top. [Overlay slide: Numbers 0..63, found/not found] If we count from 0 to 63, and check if the number is found in this pile, we get the following pattern: No,Yes,No,Yes,No,Yes,No,Yes, and so on. [Scroll to next pile] Then, here is the second pile. We get the following pattern: No,No, Yes,Yes, No,No, Yes,Yes, and so on. [Scroll to next pile] The next pile: No,No,No,No, Yes,Yes,Yes,Yes, No,No,No,No, Yes,Yes,Yes,Yes. [Scroll to next pile] And the next: No,No,No,No,No,No,No,No, Yes,Yes,Yes,Yes,Yes,Yes,Yes,Yes, [Scroll to next pile] And so on! What is the logic of these patterns? [Scroll back to pile 1] Look at the first number in each pile. [Overlay slide: Numbers 0..63, found/not found] Here. Zero was missing, one was found, two was missing, three was found, four was missing, five was found, and so on. It always alternates after one number. In the pile that begins with “2”: Here, zero and one were missing, two and three were found, four and five were missing, six and seven were found, and so on. The pile that begins with “4”: Four numbers were missing: zero, one, two and three. Four numbers were found: four, five, six and seven. Four numbers were missing: eight, nine, ten, eleven. Four numbers were found: twelve, 13, 14 and 15. And this pattern is perfectly consistent. The pile that begins with “32”: 32 consecutive numbers were missing: Zero through 31. 32 consecutive numbers were found: Numbers 32 through 63. In fact, if you have the first number in each pile, you can easily calculate all the other cards that should be in the pile. But how to decide the first number in each pile? Well, the first pile begins with number one. [Slide: ×2, overlay on top in fixed position, reveal more during narration] The first number of the second pile is twice the number in the first pile: Two times one is two. The next pile has twice that number. Two times two is four. The next pile has twice that number. Two times four is eight. Two times eight is sixteen. Two times sixteen is 32. [Slide: Powers of two] This succession of numbers, 1, 2, 4, 8, 16, 32, and so on, is called powers of two. You know, as in exponentiation. Most people learn this in the primary school. [Camera: back on the paper piles] If we use zero to indicate a pile where the number was missing, and one to indicate a pile where the number was found, like this, [ 0 1 0 1 1 1 ] we actually get legitimate binary numbers. One means the number is in the pile, and zero means it’s not. This is the number 23 from earlier. These four piles had the number 23, and these two did not have the number 23. And it reads, 010111. This is the number 23, but in binary. The relationship between the powers of two and these binary numbers is like this. The first digit, on the right, is two raised to the zeroth power. The next digit is two raised to the first power; the next, second power, and so on. [onscreen: 2⁵ 2⁴ 2³ 2² 2¹ 2⁰] [Slide scroll IN from bottom: 9002, 1001] The same principle works with decimal numbers too, by the way. [pause] [Slide scroll OUT to bottom] We could actually have an infinite number of piles; [grow 2⁷, 2⁸, 2⁹ etc] binary numbers are just numbers, they have as many digits as needed. As with decimal numbers, the zeros on the left can be omitted. We don’t need this pile. [discards pile 32] It still reads 23. We can also go the other way around. Let’s say, we have this binary number: 100001. We can just look at the top cards in these two piles, sum them together, and we get the number — 33. I just converted a binary number into decimal! When I learned about this game, I was maybe ten years old. I had never even touched a computer or any electronic device more complex than a pocket calculator, and I knew absolutely nothing about binary numbers. I was dealing with binary numbers without realizing it! Do you have a similar story to share? Post in the comments! By the way, there is one more way to look at binary numbers that, in my experience, is very helpful for teaching. Let’s have a look how we count in decimal numbers. When you have zero of something, how do you write that? You use zero symbol of course. Then, when you have one more, how do you write that? You use the symbol for ONE. [Rest of this chapter, improv, use pen & paper] Hope this help. Have a nice day.