Learning mathematics involves a progression to higher and higher concepts, building on the foundations of what we have already learnt. But Andrew Irving and Ebrahim Patel explain that no matter how high your mathematical knowledge reaches you must never lose sight of your foundations, no matter how basic they may seem.

"Education is not the filling of a pail, but the lighting of a fire." (W. B. Yeats)

With physical hobbies such as playing a sport or musical instrument, practice makes us less clumsy and this is how we mark our progress. But what about hobbies of a more cerebral nature?

As we tread further down the path of formal education, we can find newly acquired knowledge so impressive that it actually usurps what has gone before. With physical hobbies, this sort of process is ideal as it replaces an initial clumsiness with smooth(er) results. But can education operate in this way? Well, let us consider the example of mathematics, where one of the first things we learn is how to count.

Counting is soon followed by adding up which, once mastered, lays the foundation for a possible short-hand — multiplication. So is multiplication a less clumsy way of adding up? Well, sort of. Certainly for larger values of n, adding a number, say x, to itself n times instead of simply multiplying x by n would be nothing shy of extravagance! But the key thing here is understanding that both of these are just different ways of counting n lots of x.

In this way, if mathematics is to be understood rather than merely learnt, we must treat it differently to most other hobbies — in our progress we should not leave our clumsier roots behind. For though we may master the greatest techniques of mathematics so that all the clumsiness of our infancy is gone, we must return to the (clumsy) foundations in our search for mathematical understanding.

Nevertheless, frequently discovering less clumsy ways of doing maths can make it hard to remain mindful of what we are really asking and what our answers really mean. For instance, exponentiation teaches us that multiplication too can be short-handed:

2³ x 2² = 2³⁺² = 2⁵ = 32

but are we really asking what two cubed times two squared is? No, rather, we are asking what happens when you count up eight (2³) lots of four (2²). Yes, exponentiation provides a very tidy way of pursuing counting here, but that is all it is. Or is it? Well, it would be only a half-truth to say yes, because there exists a sort of reciprocal relationship between exponentiation and counting. As we shall go on to explain, exponentiation lays the foundation for the orthodoxy of counting.

When we learn to use arithmetic (addition, subtraction, multiplication and division) as short-hand for counting, it is conventional to use the decimal system. In effect, this means that we make our calculations using powers of ten. For example, the number seventy-five thousand two hundred and thirty-four is denoted by 75234 because,

75234 = (7 x 10⁴) + (5 x 10³) + (2 x 10²) + (3 x 10¹) + (4 x 10⁰).

But there are other counting systems too, each with its own unique number, the powers of which are used to express values in associated calculations. This unique number is called the system's base (e.g. the decimal system's base is ten). And, although it may not sound all that exciting, it is something of a microcosm — understanding even the simpler mathematical techniques and ideas does empower us. For instance, let us now consider a sum,

3⁴ + 3³ + 3² + 3¹ + 3⁰

which, whilst not especially difficult, does not provide any obvious shortcut to the answer of one hundred and twenty one. But what is true is that counting up various powers of three using the decimal system (which calculates using powers of ten) does not make a lot of sense when the answer is so clearly expressible in base three, as 11111.

Those of us who enjoy pastimes such as mathematics can suffer from our own progression. With passing years, we grow familiar with ever more grand ideas, allowing us to penetrate even the most complex problems. But all too often, this leads to a counter-productive mind-set, one which blinds us to obvious solutions as we begin to persuade ourselves that all problems require complicated solutions. How can we reconcile our desire to learn with our wish to solve problems effectively? Well, perhaps our answer lies with physical hobbies.

Whether involved in a team or an individual sport, top athletes will (either periodically or continuously) look to improve. Yet, whilst the honing of new skills is a marker of progress, it can take time before those skills are incorporated into a game without hindering an athlete's performance. Thus, an athlete may be encouraged to go back to basics (at least temporarily), that is, to rediscover those skills which allowed them to excel in the first place instead of over-complicating the game. With time, top athletes learn to draw on their most skillful techniques more effectually. This is a lesson which those who enjoy mathematics could borrow — new techniques may be swiftly learned but understanding how they fit in with previously acquired knowledge takes more time.

Perhaps the easiest way to conclude our piece is for us to collectively ask ourselves not why 10 is the answer to all sums, but rather what 10 is at all? 10 is just a notation used to describe precisely one lot of our chosen counting system's base as 10 is just an abbreviated way of writing

(1 x b¹) + (0 x b⁰)

where b denotes our base. In other words, it is the coefficients of descending powers of b (i.e. the 1 and 0 above) which, together, form the notation 10. The choice of b itself is up to us, it could be anything. Generally, we use a base of ten (counting with the decimal system) and so 10 denotes the number ten. But, if we want it to, 10 can denote twelve or twenty seven or any number we wish so that, for example,

2+3=10

as long as we choose to express our answer in base five. Hence, the next time our reader sees the notation 10, we hope they may think: one, zero — a number which might not be ten. And the next time someone asks you a particularly tricky sum, you can tell them, without hesitation, that the answer is obviously one, zero!

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About the authors

Ebrahim Patel and Andrew Irving are part of The Bees, a writing group aiming to promote the understanding and enjoyment of mathematical techniques.

"Aerodynamically, the bumble bee shouldn't be able to fly, but the bumble bee doesn't know it so it goes on flying anyway." (Mary Kay Ash)