FatNums: An Alternate Representation
for School Arithmetic
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—
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 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.
what each place means (powers 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—
\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
17
0
29
17
2
9
2 FatNums: Another Numeral
This gives us an different way of writing numbers—
2.1 Multiplication
What good are they? Well, let’s consider doing a “longmultiplication” 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 
8
56
48
16
6
42
36
12
0

 8 
 56 
 48 
 16  
 6 
 42 
 36 
 12 
 0  
3 
 21 
 18 
 6 
 0 
 0 

 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.
We never need to stop to carry.
 We can actually do the arithmetic from left to right. This is a subtle point:
I believe this is the natural way many people want to multiply, becuase it’s the order in which we read things; going the other way is something that needs to be taught and reiterated.
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.
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 insightfree. Most students learn it as a rote algorithm, without fully understanding what is happening and why it must happen. FatNums lay this bare.
In general, it’s entirely reasonable to say “there is a count of 17 in the 100’s position”; understanding that that is the same as 1 in the 1000’s and 7 in the 100’s is an equivalence the student should wrestle with and understand, not just skip entirely.
2.2 Normalization
 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 
Exercise
Do we have to normalize righttoleft, or can we also go lefttoright?
If we can, how many passes do we need? Do they all go the same direction?
2.3 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 columnwise 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 shortterm memory, and hence the overall cognitive load. We end up inventing ad hoc notations for carries just because it can get overwhelming and errorinducing 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
17 0 29
17.0.29
17,0,29
17  0  29
17; 0; 29
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
Alan Fekete: G’Day Math’s Exploding dots
Patai Gergely: Tim McLean’s The radix 2^51 trick
Albert Cohen: Algirdas Avizienis’s signeddigit numbers
Mark Engelberg: Frédérique PapyLenger’s and Georges Papy’s Papy minicomputer
Will Crichton: Relates this to Zhang and Norman’s representational analysis of numbers