# Finding an analogy to explain the function of a binary adder

I want to find an intuitive analogy to explain how binary addition (more precise: an adder circuit in a computer) works. The point here is to explain the abstract process of adding something by comparing it to something that isn't abstract itself.

In principle: An everyday object or an action that is structured like or functionally resembles an adder.

Think of a thing that can belong to any number of categories x1, x2, x3, x4, x5, x6, x7, x8 for which the property holds that if you put two objects together/perform two actions simultaneously, and both the objects/actions are of the same category you automatically create an object or perform an action that is of the next higher category that the object doesn't yet belong to, the whole thing therefore implementing the basic functionality of an adder.

(Categories are changing here analogous to the bits in the circuit: 00000001 (1) + 00000001 (1) together, adds up to 00000010 (2).)

But I just can't think of such a situation or an object where this pattern would occur. Whatever analogy i create with increasing amount of categories the way these categories transform becomes increasingly harder to explain, and the metaphor becomes overly specific and unhandy.

Hence the question:

What's an everyday object that resembles an adder in it's basic functionality?

• I wonder if this would be better in a computers SE site. Also, check out James Tanton's Exploding Dots, and see if that would be helpful. Commented Apr 20, 2022 at 20:20
• You're probably right. The "Exploding Dot Puzzler" is pretty close to what I wanted, its just missing some context or compelling story to convey it. Commented Apr 20, 2022 at 20:47
• If you watch a few of his videos, you might feel differently (he is usually compelling). Commented Apr 20, 2022 at 20:55
• – JRN
Commented Apr 21, 2022 at 2:04
• @SueVanHattum No, I meant the concept of "Dots that explode". Commented Apr 21, 2022 at 7:58

Why do you need an analogy?

I think just having a bunch of beans to count, and grouping the beans into groups of size 1, 2, 4, 8, 16, etc is intuitive enough without needing an analogy. The comparison with base ten arithmetic, where we group into 1, 10, 100, 1000, etc is clear.

Addition is straightforward: combine the piles and regroup.

• It sounds like OP wants something a bit more specific. But I agree. Commented Apr 20, 2022 at 20:20
• Should have made clear, that my goal here was to find a way to explain the basics of computation solely by reason through analogy instead of inference. Sure, just showing the process would be pretty straightforward. But the reason why most people fail at math is not because they're stupid but because to them doing math does not "feel like" anything, Which is why they struggle to memorize and reason about abstract mathematical stuff. This is why you need analogies. Commented Apr 20, 2022 at 20:34
• @user1976: You don't need analogies for everything, trying for that is not a good use of time. Adding (e.g., 1 + 1 = 2) is so fundamental you need to expect that as a prerequisite. Commented Apr 21, 2022 at 0:52
• @user1976: Analogies are certainly useful to help people grasp a high-level picture, but they can only go so far. If you try to make an analogy cover all the details, either it breaks down, or you have elaborated it enough that it’s not just an analogy but an illustration — like the “beans, grouped in sizes $2^n$” that this answer suggests. Commented Apr 21, 2022 at 13:32

Why would you need an analogy? Binary is just another way of encoding numbers - so the question is, "why do you need something that adds numbers?"

I think what people get hung up on with this issue is they keep wanting to associate "numbers" with decimals.

And to that end, I'd suggest that people should think about the difference between "3", "three", "III", and perhaps a couple of foreign-language words with the same meaning. They're all just symbols for the underlying number that occurs when we measure situations in which there is something, another something, and finally another something, but not any further somethings than that.

And binary numerals are just another kind of symbols, built on a different rule, but they refer to exactly the same numbers. The number "11" binary means the same thing that "III" above does. They're just two different reporesentational systems for the same things.

"So why do you need a binary adder?" Well, why do you need a desk calculator? (Which almost certainly has a binary adder in its circuits, by the way, that does just what you think from the name it would be doing - so you can throw that on top, too.)

Not quite what you want, but there are mechanical binary counters, e.g., this one, video here:

Another wooden version here. You could emphasize the analogy with an odometer.

And here is a Minecraft version, not quite as compelling (to me): YouTube link

Here's a clever marble binary adder:

• I think a mechanical decimal counter is even more instructive, since you can see the 10s slowly turn as the 1s increases, the 100s slowly turn as the 10s increases, etc. Commented Apr 21, 2022 at 21:43

I think if you've played Monopoly you understand that once you get 10 one-dollar bills, you'd rather trade them in for a ten-dollar-bill. And once you get 10 ten-dollar bills, you'd rather trade them in for a hundred-dollar-bill. You're trying to minimize the number of bills you have to manage. It's easier to know how much money you have when you've reduced everything to use the largest bills possible too.

Now suppose the denominations were \$1, \$2, \$4, \$8, \\$16. And there's a rule that says you can't have two of the same kind of bill -- e.g. you must trade your two eight-dollar dollar bills in for a sixteen-dollar-bill.

In this situation, if you have a dollar and the bank is going to pay you a dollar, then you intuitively know NOT to accept another one-dollar bill from the bank. Instead, you give your one-dollar bill to the bank and they give you back a two-dollar bill.

I think people understand the carry mechanism this way too. Imagine doing 3 + 1. If I have 3 dollars (one one-dollar bill and one two-dollar bill). And someone wants to pay me one dollar, I know that I can't accept a second one-dollar bill, so I must trade it in. I give away my one-dollar bill expecting a two-dollar bill, but I can't accept a second two-dollar bill. There's a beautiful moment where the banker has accepted my one-dollar bill and grabbed a two-dollar bill and is about to hand it to me but I look at my own wallet and refuse to accept the two-dollar bill because I already have one and instead, I hand my two-dollar-bill to the bank, which the banker exchanges for a four-dollar bill.

This moment where the banker hasn't finished giving you your change yet but is just holding a bill of some denomination in front of you for you to decide whether to accept or not is the carry mechanism.

Young children can do this hands-on to get a feel for it. You "pay" the child an amount of money by handing them a bunch of bills (being careful not to have two of any one kind of denomination yourself). Then the child begins checking to see if they have two of anything (if they are breaking the "rule" by accepting a second bill of the same kind.) When they hand you an n-dollar bill, you simply take it away and hand them a 2n-dollar bill instead. This can be done in any order. The child doesn't have to start with their lowest denomination. The child is playing a match game - looking to see if the person paying them is offering them a bill of the same type as something they already have. If they do, they just hand it over and are then offered the next bigger denomination of bill in return.

The child can play as the banker too. They will learn that they will be handed a bill of the same denomination as something they were offered, at which time they must return the two bills to their supply of bills and exchange them for one bill of the next higher denomination.

The analog you are looking for is a person. The person is intuitively alerted to the situation that they are about to have too many of a certain denomination of bill and takes action to resolve it.

If you're looking for a metaphor that would instantly click with a modern audience, you might consider mobile merge games. The gameplay is that you have a board that is filled with pieces, and you can combine two identical pieces to yield a single "evolved" piece. For instance, maybe you combine two pieces of thread to get a spool of thread, two spools of thread make a piece of yarn, two pieces of yarn makes a ball of yarn, two balls of yarn makes a piece of rope, and so on. For illustration, here is a video of someone making a notoriously complex item in Merge Mayor from a large number of base units.

Weighing/massing is something that has this aspect. Really any extrinsic property. https://en.wikipedia.org/wiki/Intrinsic_and_extrinsic_properties But I think weighing is easy to understand because you see the output, often in digital form.Seems to satisfy your desire.

I would probably avoid fluid or electrical flow, because of the complexities of series and parallel and driving force. All that said, if they know basic E&M, obviously resistance in series adds (and some basic adders are actually resistance networks). Also, of course head loss of components in series adds linearly for fluid circuits...but I suspect your computer guys will be less comfortable with piping than electronics.