# Why commas are placed every third digit

1. Jul 17, 2013

I wanted to add to a previous thread regarding the placement of commas every third digit in our decimal system. Unfortunately, the thread was closed, so I am creating a new thread to mention my pet theory about this practice.

The question is why we choose to place commas every third digit. The simple answer is for ease of readability. This of course begs the question, why not group together digits in sets of 4 or 5 (as we do with tally marks, for instance?) Why 3 specifically?

Some argue that answer has some sort of physiological basis in how our brains process information. I propose a more elegant -- and I think more appealing -- explanation. It boils down to the fact that we have 3 spatial dimensions in our universe (or, at least, 3 that we directly perceive).

Begin with the observation that we have ten fingers on which to count, and this is why we use a base 10 numeral system. If we want to count more than ten but less than one hundred objects, we use two digits, the first indicating the number of groupings of ten (tens place), and the second indicating the remaining ungrouped objects (units place).

Translate, then, this simple counting method into its visual analog.

First suppose we have 7 identical cubes. Since we can count the cubes on our fingers, we simply line them up in a row, in the positive direction along the x-axis, for instance. Done.

Next suppose we have 57 identical cubes. We arrange them into five rows of ten (each row oriented in the positive x-direction again), but side by side (so successive rows are stacked along the positive y-direction), followed by an incomplete row of seven. The digit 5 represents the number of complete rows, and the digit 7 represents the number of cubes leftover.

Further, suppose we want to count 100 identical small cubes. The natural procedure is to arrange the cubes into 10 rows of 10, creating a "square of cubes" that roughly sits in the x-y plane. Following the conventions of early childhood education, I will call this grouping of 100 a "flat." So far so good.

Further, suppose we want to count 357 cubes. To organize these cubes, we first organize three "flats" of one hundred cubes each; we stack these three high along the z-axis in the positive direction. On top of these three flats, we place 5 rows of 10 each, and finally the seven remaining cubes in an incomplete row.

To count one thousand such cubes, we stack 10 flats in the xy plane on top of each other. Done.

But now comes the trick. What if we want to count seven thousand one hundred such cubes? You might suggest that we simply stack seventy flats on top of each other. This seems a decent arrangement, but the problem is that we can only count up to ten objects at a time before abstracting before abstracting into "rows" or "flats" or other groupings. We simply can't count to seventy directly. It is natural, then, that we must now construct a new abstraction of one thousand small cubes, which we will call simply a "10^3 cube." Following our convention, in which we have made a 90 degree turn in our stacking process at each power of ten, we now stack seven 10^3 cubes in the positive x direction in order to count seven thousand.

That is to say, we have begun the process over again, since we starting by arranging unit cubes in the positive x-direction. This process of abstracting a group of one thousand cubes into a single cube, then continuing to count along the x, y, and z axes will repeat indefinitely. Every time we abstract a thousand cubes into a single cube and starting building in the positive x-direction again, we place a comma to indicate the abstraction and repetition. This happens every third power of ten because we have three spatial dimensions in which to count.

Anyone like my pet theory?

2. Jul 17, 2013

### Drakkith

Staff Emeritus
I'd guess because grouping them in threes allows us to easily see which digits are in the hundreds, thousands, millions, etc.

3. Jul 17, 2013

### inotyce

In Asian countries (especially China and Japan) currencies use up to 4 digits per unit

4. Jul 17, 2013

### BobG

While three or four work equally well (five is stretching just a bit, but is still a good option), you have to pick one. As inotyce pointed out, not all cultures picked three. Western culture did and European/American nations happened to be most prolific colonizers.

Also of note, base 10 became a lot more popular with the introduction of symbolic notation for numbers - 2 for two instead of || , 3 for three instead of ||| , etc. Base 5 and base 20 were most popular number bases before symbolic notation (with dual base systems using 5 and 20, such as in your tally marks, being equally common). The choice of base 5 (with some sort of special symbol for 5) was picked for reasons similar to separating long numbers into groups of 3 or 4 or perhaps 5. The human brain can numerate small groups of four or less without having to count them, while humans have to take the time to actually count groups having six or more, with 5 kind of falling on the border.

5. Jul 21, 2013

### wukunlin

Basically this is all there is to it. Number go from ones, tens, hundreds, then you get a first comma for a thousand. After that comma you repeat ones, tens, hundreds, again but in units of a thousands, then you do the same stuff again for millions, billions, etc. Basically each comma represents a -llion.

I think at least for the Brit's it used to be every 2 commas (6 figures) for each -llion with "thousand -llions" in between but I don't see anyone counts that way anymore.

6. Jul 22, 2013

### 1MileCrash

As mentioned, because of I see a number immediately followed by a comma, I know right away that that number has entered into the realm of "thousands, millions, billions" etc.

We have a new name every time it is no longer expressed in terms of hundreds of some other value. Since "hundreds" are 3 digits, it just makes sense.

Of course, as a whole it is arbitrary. But as our other conventions are, it fits the best.