There are lots of ways to represent number – such as the number “ten”. As grown-ups, we’re so used to a particular way of representing “ten” that we don’t often stop to think about what we’re doing – we just write “10” and we move on to something else. And, indeed, our decimal system of writing numbers has enough good things going for it that it may seem strange to even bother with alternatives. And yet, you only have to look at your wallet or your coin purse to see a whole different way of representing number. In your wallet, you have some combination of standard denominations. My wallet contains a single five dollar bill, and eight single-dollar bills. My coin purse contains three quarters, seven dimes, two nickels and seven pennies. Together, they represent the value of an asset called “cash”. In the decimal system of representation, we would write this as $14.62, which combines the unit ($) and the number (14.62). Number can be represented by a set of objects like coins, of standard denominations.
The “coin purse” approach for representing number has its advantages and disadvantages. The most obvious advantage is the ease of addition. To add a number represented by a pile of coins to another number represented by a pile of coins, I simply join to the two piles of coins. Done – I am now left with a pile of coins representing the sum of the two numbers! If this seems like cheating – it really isn’t. In the world of the coin purse representation, the question “how much is in the coin purse?” is as nonsensical a question as “how much is 10?” in our normal way of representing number. The answer to “how much is in the coin purse?” in the world of the coin purse representation is simply showing the coin purse and saying “this much”. However, this whole tale points to a disadvantage of the coin purse representation: the conversion to our normal representation takes some work, and takes some time. However, the primary use for wallets and coin purses is when paying in stores, and there the question isn’t usually “how much money is in your coin purse?” but something more like “do you have two pennies?”. Another common use for coin purses is when dealing with soda machines or the like. Then the question is: “do you have enough quarters?” All of these are easy to answer even without knowing the total make up of what’s in your coin purse. Increasingly, the major use of my coin purse has become accepting coins in change – and not payment at all. I pay with bills, and get coins back, which are put in the coin purse without any counting whatsoever. When coin purse gets full, it is emptied into the coin jar, and once a year or so I find a way to exchange the stuff in the coin jar for bills, which then go into the wallet.
For the kinds of uses that wallets and coin purses are most often used for, that representation of number is quite successful.
The main theme here is that the suitability of a particular representation is very related to what the representation is being used for. In rare cases, a representation can be so well suited for a particular use that we are completely willing to put up with the conversion into and out of that representation from the more familiar representations. (A simple example: if I ask you how much 34 x 1376 is, you may be completely willing to convert the “34” from a representation on paper to one inside of a calculator by pressing the “3” and the “4” button, etc. You would judge the conversion worth it because the calculator is so much better suited to the job of multiplying large numbers than the sheet of paper is.)
As a final example in this blog entry, let’s examine the representation of number used in the picture below. Imagine that you have a friend who counts cars at a particular intersection, to collect evidence that a traffic light might be needed. You relieve her at the agreed-upon time, and she hands you this, saying “sorry – got to run.” You notice that she has been doing the counting differently than you do. You would have used tally marks. How did she do it instead? How does she represent the number of cars on her piece of paper?
You figure that as cars approach the intersection, she wrote down the number of cars she saw. You figure that when three cars approached the intersection at the same time, she simply wrote down “3” instead of “|||”. At the top, on the left, you see such a “3”, and it is followed by a “2”, so you figure she then saw two cars approach the intersection. And then one car, and another, and then three. Or was it eleven cars and then three? You then notice that the top left 3 and 2 are crossed out, and you see a “5” below. You surmise that she must have used periods of relative calm to replace some groups of numbers by their total. The two “1”s have also been crossed out, and there is a “2” below, so you figure your guess that the two “1”s represented two single cars rather than a wave of eleven cars is correct. Below the “3 1 2 4”, which are crossed out, you find a “1” and a “0”, and since 3+1+2+4=10, you figure that this “1 0” represents ten.
So while you have been tally marking the cars that arrived since you took over, you think you have figured out her system, and you decide that it is a workable system. It occurs to you that not only is it a system for marking cars as they arrive at the intersection, it also gives a representation of number. In this system, number is represented as a bunch of smaller numbers, those not crossed out. Like the coin purse, what matters is the total. Unlike the coin purse, the pieces that make up the total don’t come in standard denominations. Yet unlike the coin purse, I never need to worry about making change. In the coin purse, when I need a quarter, and there is no quarter in the coin purse, I need to find somebody to give my two dimes and five pennies to, in return for the quarter. In the car count situation, if I have an “8” and a “7” on my paper, I am free to cross them both out and replace them with a “9” and a “6”, or a “10” and a “5”, or a “15”, all according to my choosing.
Come to think of it, my assets are represented as a bunch of smaller numbers to be totaled up: the cash in my pockets, the coins in my coin jar, the balance in my checking account, the balance in my brokerage account – and this is even before we look at the less easily unlocked value in my house, my car, my collection of CDs and so on. When I withdraw $60 from the ATM, I’ve got $60 more in my wallet, and $60 less in my checking account. It nets out. From a bird’s eye view, from the view of the representation of my assets, nothing has really changed. It wouldn’t make sense for me to look at my wallet and exclaim that I was now richer by $60. Nor would it make sense for me to look at my bank account balance and exclaim that I was now $60 poorer. Until I spend the cash, I’m neither richer nor poorer. Representing number as the total of a bunch of smaller numbers might actually be quite common and quite useful, even if I never actually total up these numbers.