FatNums: An Alternate Representation
for School Arithmetic🔗

    1 Digital Notation

    2 FatNums: Another Numeral

      2.1 Multiplication

      2.2 In Speech

      2.3 Normalization

      2.4 Other Operations

    3 Operational Consequences

    4 Paper Notation

    5 A Plea

    6 External References

Place notation is one of the great inventions of civilization. As Dan Friedman has quipped, perhaps the Roman Empire fell because of the difficulty with divison. That may be a bit of a stretch, but notations matter, and place notation is a keeper.

A few years ago, my daughter was starting to learn place notation in math. I watched her work through (and sometimes struggle with) the idea of carrying. Even after she’d learned the process, I could see she had no real understanding of the concept. I realized that the notation may in fact be an obstacle, and came up with a different notation that she and I now use.I’m confident that there’s nothing new here, and it’s been well picked over by math education researchers. Pointers to the literature—and, in particular, to a term to look for—would be welcome.

1 Digital Notation🔗

Just about every definition of place notation will tell you that a number is represented as a sequence

\begin{equation*}\cdots d_2 \, d_1 \, d_0\end{equation*}

where the \(i\)th place represents \(10^i\). Therefore, the above is

\begin{equation*}\cdots + d_2 \times 10{}^2 + d_1 \times 10{}^1 + d_0 \times 10{}^0\end{equation*}

The presentation may explicitly tell you that each place is a digit (hence the letter “\(d\)”), or may leave this implicit. Henceforth, we’ll call this the digital notation.

However, a moment’s thought should show that there is no actual reason why the number in each place needs to be a digit. Digital place representation conflates two different concepts:

  • what each place means (multiple of the power of the base)

  • what we can write in each place

Consider, for example, the following way of writing Ramanujan’s number:

\begin{equation*}17 \times 10{}^2 + 0 \times 10{}^1 + 29 \times 10{}^0\end{equation*}

Arithmetically, this comes out to the same value—the same number—as the more conventional

\begin{equation*}1 \times 10{}^3 + 7 \times 10{}^2 + 2 \times 10{}^1 + 9 \times 10{}^0\end{equation*}

The latter, of course, we can write in digital notation as 1729. With the former, we have to be careful to avoid ambiguity. If, for instance, we write it as

17029

that could be misleading; indeed, there are several different numbers that that notation could correspond to. Instead, we need to separate the places: e.g.,

17

  

0

  

29

which makes it unambiguous. Indeed, the same number can be written in multiple different ways: e.g., the above is equivalent to

17

  

2

  

9

and so on (you give up canonicity).

2 FatNums: Another Numeral🔗

This gives us an different way of writing numbers—a different numeral. There are lots of numeral systems around: some may be place-based but use different squiggles for digits (e.g., using Arabic or Hindi numerals), or they may not be placed-based at all (e.g., Roman numerals). These, which are place-based but do not limit each place to digits, my daughter and I call fat numerals, which we truncate to FatNums.This is a source of some domestic dispute. I accidentally called them fat “numbers” initially, and by the time I’d corrected myself, she’d gotten used to the term and rejected “numerals”. We settled on “Nums” to split the difference; this also picks up on a venerable programming languages tradition of BigNums, FixNums, RoughNums, etc.

2.1 Multiplication🔗

What good are they? Well, let’s consider doing a “long-multiplication” problem with digital notation:

\begin{equation*}1762 \times 368\end{equation*}

Remember, every place is a digit, so none can exceed 9. This leads to the dreaded “carry”:

  

1

  

4

  

0

  

9

  

6

1

  

0

  

5

  

7

  

2

  

0

5

  

2

  

8

  

6

  

0

  

0

6

  

4

  

8

  

4

  

1

  

6

Now let’s do it instead using FatNums. To multiply 1762 by 8, we can just write down each of the products in their respective places:

8

  

56

  

48

  

16

Similarly, 1762 multiplied by 6, remembering that the 6 is in the 10’s place in 368, is just:

6

  

42

  

36

  

12

  

0

and so on:

  

  

8

  

56

  

48

  

16

  

6

  

42

  

36

  

12

  

0

3

  

21

  

18

  

6

  

0

  

0

Now we can add up each of the places:

  

  

8

  

56

  

48

  

16

  

6

  

42

  

36

  

12

  

0

3

  

21

  

18

  

6

  

0

  

0

3

  

27

  

68

  

98

  

60

  

16

I know this looks strange to you.
You’re not used to it.
Let it grow on you.

Notice the following:

  • We never need to stop to carry.

  • We can actually do the arithmetic from left to right. This is a subtle point:

    1. I believe this is the natural way many people in left-to-right languages want to multiply, because it’s the order in which we read things; going the other way is something that needs to be taught and reiterated to avoid mistakes.

    2. Multiplying the biggest terms first is useful because it gives us an intuition for magnitudes. That’s what any practicing engineer would do when performing a quick estimate in the field. Starting multiplication with the smallest term is a poor habit to develop.

    3. Nevertheless, you have a choice: FatNums are completely indifferent to which order you choose. You have a grid of locations to fill in, and each location is independent of the others. (Math teachers might appreciate the value this has for creating worksheets.) In contrast, the content of each digit depends on previous operations, and even how many digits there will be in total isn’t known a priori. Therefore, digital notation does not leave a choice.

  • The carrying process is insight-free. Most students learn it as a rote algorithm, without fully understanding what is happening and why it must happen. FatNums lay this bare.

2.2 In Speech🔗

In general, it’s entirely reasonable to say “there is a count of 17 in the 100’s position”. In fact, this is what people say! In American English, people routinely say “seveteen hundred” instead of “one thousand seven hundred”. If you ask people to add “eight hundred” and “nine hundred”, it’s perfectly fine for them to say “eight and nine is seventeen… so seventeen hundred”. That is, they (sometimes) speak and even compute in FatNums, because it can be more convenient for mental computation.

Understanding that the digital and FatNum methods of expression can provide different ways of expressing the same number is a profound idea. This equivalence is something a student should wrestle with and understand, not just skip entirely. Unfortunately, we have taught only one written notation: the digital one. We should teach both!

2.3 Normalization🔗

Returning to the above multiplication, in FatNum terms, we already have our answer! We can now see the digital representation as a normalization of this answer into a canonical representation. Normalization is (the only place) where we perform “carrying”. Proceeding right-to-left:

  

3

  

27

  

68

  

98

  

60

  

16

=

  

3

  

27

  

68

  

98

  

61

  

6

=

  

3

  

27

  

68

  

104

  

1

  

6

=

  

3

  

27

  

78

  

4

  

1

  

6

=

  

3

  

34

  

8

  

4

  

1

  

6

=

  

6

  

4

  

8

  

4

  

1

  

6

Et voilà! We have the same answer as in conventional place notation.

Exercise

  1. Do we have to normalize right-to-left, or can we also go left-to-right?

  2. How many passes do we need? Do they all go the same direction?

2.4 Other Operations🔗

We started with multiplication because it’s conventionally one of the more difficult operations for children to handle. I didn’t even discuss addition, but we did it above in the process of multiplication. Subtraction follows the same way as addition; observe that by liberating each place from having to be a single (almost always left implicit: unsigned) digit, we can have a negative number in a position, to be normalized away later.In contrast, with digital notation, teachers have to introduce the notion of “borrowing” or “regrouping”, which I am certain most students would view as a purely syntactic operation with little to no understanding of what it means. Division is left as an exercise to the reader!

3 Operational Consequences🔗

My daughter asks you to not be misled by the seemingly fewer steps involved in digital multiplication. With digital numerals, there are (seemingly) just two steps: first multiplication, then column-wise addition. With FatNums, there’s (usually) a third step: normalization. However, she points out that the digital method still has the same third step; you’ve just blended it in. In fact, she continues, by that token the digital method has four steps: you have to also account for the carrying during multiplication and the carrying during addition! In contrast, with FatNums, you do all the carrying at once.

With larger numbers, I wonder if there aren’t also consequences for the amount that has to be held in short-term memory, and hence the overall cognitive load. We end up inventing ad hoc notations for carries just because it can get overwhelming and error-inducing to keep track of it otherwise. This feels symptomatic of the burden that digital notation introduces. There’s also the burden that we are constantly switching between operations: multiply, normalize, write/remember the carry, multiply, add (the carry), normalize again, carry again… FatNums have no switching.

I’m not saying that FatNum arithmetic is uniformly easier than digital arithmetic. However, the goal of teaching arithmetic should be to not only enable students to calculate answers, but also to give insight. (Indeed, computers can do the former much better.) I am quite certain that conventional operations over digital notation actively erase insight. That may have been a necessary compromise in an era where all computations had to be performed by hand. But in a (ha) digital era, having students really understand numbers seems far more important than having them mindlessly process them.

4 Paper Notation🔗

There is one last issue to consider. Digital notation has a clean representation (assuming—and that’s not always a safe assumption—a child can write digits perfectly aligned in columns). With FatNums, as we’ve noted above, the notation can easily become ambiguous if we don’t keep the places separate. In the above writeup we’ve had the benefit of computers to place things, a luxury that isn’t available on paper. I’ve spent some time thinking about this:

17 0 29

is problematic because it’s too easy to lose track of the space (though my daughter still uses spaces);

17.0.29

and

17,0,29

are problematic because when there’s only two places, it can look like a decimal (depending on what decimal separator you use), and with more than two places, it can look like grouping (e.g., millions, lakhs, etc.); and

17 | 0 | 29

is problematic because the separator looks too much like a 1. In contrast,

17; 0; 29

seems pretty visually unambiguous and also not onerous to write. There remains the problem that you don’t a priori know how much width to allocate each column, but with a little experience it becomes clear that two digits is wise and three is certainly safe.

5 A Plea🔗

Finally, my daughter asks the math teachers reading this to consider the following. FatNums provide us an alternate way of representing numbers and hence of doing arithmetic over them. If we can usefully perform these operations, there’s no reason to reject these methods just because they aren’t conventional. (In her first year of using this she got a fair bit of resistance from her teacher, but the teacher from her second year was pretty receptive.)

6 External References🔗

As I had hoped, people have kindly sent me references to other work that relates to this. I’ll try to keep up with these, but also see my Tweet:

I also came across this thread by Amit Schandillia. The first 20-odd posts show how the Babylonians had and used this same concept. Let’s make what was ancient new again (but with zeroes)!