As an adolescent I loved number theory, particularly prime numbers; but that did not stop me looking into some other, apparently mundane aspects of numbers. And finding them fascinating.
Take decimal fractions as an example. I must have been around 14 when I first came across the concept of “circulating” decimals. I understood that some common (or “vulgar”) fractions were easy to express in decimal form: 1/2 was 0.5, and 3/4 was 0.75, and so on. I understood that some common fractions were a little harder to express in decimal form: 2/3 was 0.66666… and 5/6 was 0.833333…. and so on. But it took me a while to appreciate why. That all this was related to the base we worked on, 10, which led to our using the term “decimal”. That common fractions “terminated” only when the denominator consisted of factors of the base, factors of the number 10. That whenever the denominator contained factors other than 2 or 5, the fraction could not “terminate”. It “recurred”. That “recurring” decimals came in two forms, classic recurring (as in 1/3) and “circulating” (as in 1/11 or 1/13, where a sequence of numbers repeats rather than just a number).
And as I got deeper into circulating decimals, there was a growing sense of magic. As I learnt that circulating decimals had a “period of recurrence”, the length of the sequence of numbers that repeat. So 7 and 13 had periods of recurrence of 6: for example 1/7 was 0.142857142857142857…. where the numbers 142857 repeated in sequence. [Note: there’s a lot more to 1/7 represented in decimal form, but that’s for a different post some other time…. 1/7 is .142857 recurring, 2/7 is .285714, 3/7 is 428571, and so on, the same six digits but with different start points; as triples e.g. 285 and 714 they always add to 999, as doubles e.g. 28 and 57 and 14 they always add to 99…]
The magic began when my teacher let slip that 1/97 had a period of recurrence of 96. That every 96 digits, the same sequence started off again. It continued when the teacher followed it up by saying “Think about it. The period of recurrence of a circulating decimal can never equal the denominator, the maximum is one less than the denominator”. And then he asked me to go and prove it.
As a class we were just beginning to learn about modular arithmetic at the time, so the teacher knew precisely what he was doing by asking me for the proof. The aha moment was not long in coming, in realising that, regardless of the value of the numerator in the fraction, there were a finite number of unique residues possible. That the maximum number of unique residues is one less than the denominator itself. That 1/97 could not have more than 96 different residues.
The joy was not in being told that 1/97 had a period of recurrence of 96; it was in realising that I could work out why. For myself.
Maybe that’s what Richard Feynman was trying to tell all of us when he said “What I cannot create I do not understand“. Or what Albert Einstein meant when he said “If I can’t picture it I can’t understand it“.
We all have our aha moments, and that’s when real learning takes place. And good teachers make that possible. I’m grateful that I spent time at a school where they understood this. Thank you St Xavier’s!
Recently I had reason to revisit some of the work of W. Brian Arthur, one of my favourite economists; while doing that, I couldn’t help but re-read his excellent The Nature of Technology. In that book, he makes the point that invention happens for two reasons: someone identifies a need to be filled, as in building jet engines rather than propellor-driven ones; or someone observes an effect that can be used for something else, as in the discovery of penicillin.
While reading the book, I began to realise that learning itself is a form of invention: as I learn something, I create something inside me. I began to realise that the learning that takes place follows the same path as Brian Arthur’s classification of inventions: sometimes I learn by identifying a need and then trying to meet that need, and sometimes I learn by observing an effect and figuring out how to put that effect to good use.
That then took me down a different route, reminding me of something I’d heard George Gilder say: Every economic era is characterised by new abundances and new scarcities. Successful companies will recognise the abundances as well as the scarcities.
I couldn’t resist conflating these ideas, mashing them up. Could I start considering Brian Arthur’s “needs” as Gilder’s “scarcities”, and his “observed effects” as Gilder’s “abundances”? Could I go further and consider “learning” to be a specific instance of “invention”? Then I reminded myself, I live in the age of the web. I could blog about it, share my ideas, and get feedback. Refine and enrich my understanding. Learn. Hence this post.
Learning as a form of invention. Abundances and scarcities as stimuli for learning. Now I was getting somewhere, even if I didn’t know precisely where this was heading. I then began to think about all this in the context of enterprise software, particularly social software. And came to the following conclusions:
- There are an increasing number of knowledge workers in the workplace.
- With the passage of time, more and more of those knowledge workers will be drawn from the Maker Generation.
- Worker productivity will be influenced heavily by the propensity of the work environment to enable and encourage learning.
- Learning will be stimulated by the discovery of scarcities as well as abundances.
- The environment will therefore have to be designed to help surface those scarcities and abundances.
- The likelihood of this happening will follow Linus’s Law.
Most of you know by now that I work for Salesforce.com. Most of you know that I’m fascinated by that strange space where enterprise software meets collaboration and learning. Most of you will understand why I chose to work there.
Companies that learn to connect their staff, their customers, their distribution networks, their supply webs, their products, to each other…… companies that learn to do this on an infrastructure that helps them identify the new scarcities, discover the new abundances…… companies that realise that there’s a very powerful community effect in place here…. companies that understand how to harness all this for invention and innovation….. those are the companies that will succeed in the economic era we’re entering.
An era where connectivity and compute power and mobility are abundant. An era where customers use this power to create other, newer abundances: the ability to act in groups, the ability to share activity and intention.
An era where privacy and trust are scarce, where there are many demands on a customer’s attention. Where openness and transparency form the foundations of a new, sustainable, trust.
The era of the Social Enterprise.