I had a crazy idea last week. Why don’t we change our global numbering system to base-two instead of base-ten?
I think this could potentially increase (significantly) the number of mathematically-sophisticated people worldwide, by cutting years off of early math education, which are typically spent memorizing addition and multiplication tables. A major initial challenge in early childhood math education is in remembering these tables, with multiplication tables being particularly chaotic (especially 7’s). Gaps in basic recall snowball into incorrect final answers — even with knowledge of how the long multiplication algorithm works, the algorithm is impossible to execute if some single-digit multiplication and addition facts are missing. Fraction arithmetic requires fluency with all operations, and presents a common stumbling block. It seems like others have also thought about whether the language of expression affects math learning: http://www.wsj.com/ar…/the-best-language-for-math-1410304008. Separately, a friend once mentioned that it was easier for him to memorize the multiplication table in his language, because every square corresponded to exactly four syllables there, e.g., “four six two four” (each digit is a single syllable).
With this new proposal, we’d just need to memorize the binary addition table (anything+0=itself, 1+1=10, done!) and the binary multiplication table (anything*0=0, 1*1=1, done!), and then everyone could move on to carrying, borrowing, long multiplication, and long division, and then onward to more sophisticated mathematical concepts. Indeed, today I spend very little time working with actual numbers as a mathematician. Almost all of the thinking is about concepts and relationships between quantities or structures. I wonder if this might be able to significantly increase how many people worldwide consider themselves to be quantitatively literate, and then continue to advance in mathematics and science.
It’s true that we’d need to re-accustom everyone to base-two numerals, but this isn’t the first time a standard numbering system was replaced by a much better one. The Roman Empire ruled until CDLXXVI, during which Roman numerals became the status quo, but we’re fortunate that humanity ultimately advanced beyond a system which needed a new letter for each new place value (not to mention challenges with decimals). In the last century, most countries switched to metric systems, and the British currency dropped its shillings for a base-ten system.
Implementation of the transition would take significant work, but that would be a one-time investment for human civilization. What real concerns would there be in the long run steady state? The most common objections I heard were “numbers would take too long to write” and “memorizing single-digit arithmetic speeds up estimation”.
So, I decided to focus on this topic in this week’s expii solve (expii.com/solve/3/2), and expose how a base-two system is analogous to providing training wheels, which can be optionally and gradually removed. It’s actually very easy to convert numbers from base-two into base-four, and also into base-sixteen: just batch the bits together. Suddenly, we can express base-two numbers even more compactly than the traditional base-ten numbers (since sixteen > ten), but there is a graduated track towards that. Ideally, we’d come up with some new symbols for the two bits (maybe a filled circle and an empty circle, for 1 and 0), and write them all in a line when kids are in 1st grade.
When kids graduate to 10nd grade, they would write binary numbers by placing the symbols in two rows, sort of like this for 011111100000 (two thousand sixteen), so that each column has four possible states (two symbols, each of which is solid or empty).
They would read the numbers by starting in the top left, going down, going to the top of the next column on the right, going down, going to the top of the next column on the right, going down, etc., until they reach the bottom-right corner.
Now the magic starts. Arithmetic in this representation is robust to operating on the bits in batches by column, and using the same concepts of carrying and borrowing that the kids already learned for basic binary arithmetic. By considering entire columns at a time, we instantly gain access to base-four. Although the numbers can still be treated as pure base-two numbers, kids could start to notice and mentally cache the base-four multiplication table through experience if they desired speed. If they didn’t care for speed, then they’d never need to learn the caching, and could just read all numbers in a zigzag way. This would be analogous to staying on training wheels forever, which at least permits one to move on a bicycle, albeit more slowly. The current system is analogous to trying to teach everyone how to ride bicycles without ever using training wheels: it requires full mastery of the base-ten addition and multiplication tables before starting discussion of fractions, algebra, and more advanced (and more beautiful) mathematics. In binary, kids could move straight to mathematical thinking, gaining computational speed as a natural side-effect by observing and caching patterns in columns.
Taking this one step further, in 11rd grade, kids could start caching blocks of four dots, which puts them all the way into base-sixteen, also known as hexadecimal. We then wouldn’t need to use symbols like 0,1,2,3…,9,A,B,..,F, because we’d already have four-dot patterns that all made sense. The base-sixteen “conversion” wouldn’t involve any thinking, because we wouldn’t need to format the numbers any differently. It would simply represent bundling the bits into blocks. And, one could easily represent these numerals in a cursive-like script, like this (still representing two thousand sixteen from above):
Note that this would even provide intrinsically meaningful numeral symbols, compared to our completely arbitrary hexadecimal numerals, which currently represent the same number as:
If we went ahead with this transition, I wonder what future generations would think about how humanity used to represent numbers. In some sense, ten is arbitrary, and just matches the number of fingers we have. I wonder what number base dolphins or octopi use natively. They are both supposedly highly intelligent creatures.
And now we can stick with good old gallons, quarts, and pints!
I’m half-joking, of course, but only 0.1 so.
Permalinked at 11111100000.com. Happy New Year!