Analog Thinking in maths – an example

David Coulson

2016

dtcoulson@gmail.com

Not long ago I had cause to consider the following problem:

000,000,1 xx

In words, “What number raised to the power of itself becomes one million?”

Now this problem is not hard to solve with a calculator and a bit of paper, but if you’ve been following some of my work you will know that I am interested in taking mathematics out of the calculator as well as lifting it off bits of paper and putting it back into the heads of the people who ask questions, mathematical or not.

000,000,1 xx

A lot of questions run through my mind when I am walking down a country lane or driving in my car, and because I am a mathematician it is easy for me to mathematise these questions. It’s easy to do and it’s fun to do. But if I had to stop my car or trot up that country lane back to my house to solve the question, I wouldn’t bother. Mathematics, like a chemical reaction has an activation energy. If the effort required to start the investigation is too high, the investigation won’t happen, and mathematical curiosity dies on the spot. That applies mostly to those arbitrary things we wonder about, the questions we are not paid to solve, the questions that we discuss for fun.

There are mathematical wizards that can find the cube of a number by a clever analysis of the digits that make up that number, or the cube root of it, or can tell you the weekday on which you were born by a clever analysis of your birth year, month and date. I am very keen to point out that I am not one of those people and I am not interested in promoting circus tricks, as clever as they are. My mission, if I can call it such, is to get people thinking of numbers as quantities, not simply as symbols on a page, using a process that I call analog mathematical thinking. Four percent discounted off a pricetag is a squidgeon less than the original price. That’s how most of us think when we are not concerned about the actual value. The perimeter of a circle is three-and-a bit times its diameter, so a ten-foot hole needs a thirty-something-foot fence around it. The square root of a number is normally much smaller than the number. The Sine of a number is a fraction. And so it goes....

Commonly it is helpful to transform the problem into a geometric analogy (hence the name) and use our common-day sense of size to steer our mathematical thinking quickly towards an answer, albeit a rough one. If we can solve mathematical problems easily and quickly, we will solve them more often. We will even solve problems recreationally, and if that sounds bizarre consider further that the mathematics will be done almost unconsciously, so slightly impacting on our attention that we won’t even think of it as mathematics at all but simply as thinking.

So how can I solve the problem 000,000,1 xx

in a way that seems intuitive and does not wreak of mathematics, nor requires me taking my hands off the steering wheel as I drive?

First, a discussion of where the problem comes from. I was thinking about the question, What does it mean to be one in a million? I wanted to know at a gut level how difficult it is to be so different from the mainstream of society that a person would be regarded as one-in-a-million. In a society of four million people (where I live) it sounds like there are only four people who have the same interests as me (if I am that one-in-a-million person).

But looked at another way, one million is 106. If you can define six categories for yourself in which you would place yourself in either the top ten percent or the lowest ten percent, then you are literally one-in-a-million. The categories have to be independent of one another for the analysis to be perfectly correct, and of course no two categories are completely independent. But if they are nearly so then your answer is nearly so.

For me, that would be pretty easy to do. My attitude towards sport, my religiosity, my career, my education easily make up the first four categories, so I am very quickly one in ten-thousand. Throw in a political ideology, a second language spoken, a set of diverse cultural interests and you have filled out the six categories very easily, so that many of you reading this will now see yourselves as one in a million too.

In fact, more of us are one-in-a-million than people realise. It would not surprise me if another clever mathematician was to come along and prove that people who are one-in-a-million actually outnumber people who are less so. The world could well be dominated by (nearly) unique individuals.

So that’s the start of the problem. 106 = 1,000,000. It then occurred to me that people might not see themselves as being in an upper or lower ten percent but might simply see themselves as being more or less than average on some scales, in which case the problem becomes a matter of raising 2 to the power of something to make 1,000,000. That of course means a lot more categories (20), but the principle still applies.

000,000,1 BA

So then I started to wonder about an optimal level in between halves and tenths that would balance the number of categories with the size of that category. For example a person who was in the lower fifth of any category needs to have about 9 categories for him/her to be one-in-a-million. The problem quickly turned into consideration of XX = 1000000 ... where X is the number of categories as well as the number of partitions in each category. .... And hence the question I am trying to solve in an analog sort of way.

Trying to geometrify XX directly is not going to be very helpful because the exponent implies an unworldly dimensionality that I can’t visualise. Three dimensions limits me to 33 which is obviously way too small. So I have to transform the problem into its baby brother, namely the log representation.

6 10 xLogx

Now there is an interesting implication of a problem like this in which the unknown is stated inside of a transcendental function as well as outside of it, and that is that the problem cannot be solved except by trial and error. So even if I had a calculator and a piece of paper with me as I was driving the car or walking down the country lane, I would have to resort to some longwinded approximation technique, and stop the approximation once I considered the answer good enough. So if I’m going to approximate, why not really approximate and just do the estimation in my head, using largely sensory-based representations?

In a problem where two quantities are multiplied, it is helpful sometimes to see those two quantities as the sides of a rectangle, and the product of the multiplication as an area.

xLog10

6x

The nature of the original problem helps us further. We know that x and Log(x) are both positive numbers and that the shorter side of the box is the Log side.

So what two numbers would multiply to make 6?

xLog10

6x

If the numbers were truly 6 and 1 then 1 would be the Log of 6. But 1 is the Log of 10, a bigger number. So I know that the log has to be a fraction, which means the box has to be skinnier and therefore taller to contain the same area.

X and Log(x) are not independent of each other but I can pretend for the moment that they are. So I will choose 6 x 1 and see what that tells me.

I’m going to make the box taller and skinnier and see what happens.

xLog10

6x

What sort of a fraction should the log be? I don’t know, but the good thing is that it doesn’t really matter. The log function is pretty flat between 0 and 1, so I can grab any value and see what it does.

5.0 10 xLog

If the skinny side of the box is halved then the box has to be twice as tall to contain the same area. That implies a height of 12.

5.0

612But wait! If the height of the box is

bigger than 10 then the log of that number has to be greater than 1. That only gets worse as the box gets taller.

A box that was 100 units tall would have to have a width greater than 2. A box that was 1000 units tall would have to have a width greater than 3. Clearly, to get a value of 6, the box has to be less than 10 units tall.

xLog10

6x

Already I have closed the fence around the answer very tightly, especially if the answer has to be an integer. 7, 8 or 9. That’s it!

From here I solved this problem very quickly with a tidbit of information I memorised when I was a kid at school.

3.0 210 Log

This has proven to be a very useful little thing for reminding me of other log values I didn’t memorise.

3.010 2 Therefore 6.010 2 x 2 4

and 9.010 2 x 2 x 2 8

also 7.010 5 ( because 5 x 2 = 10)

So if I want to test the box with the number 8, I have 7.2 0.9 x 8

This answer is clearly too big, so the only integer left in consideration is 7. DONE! The answer is obtained easily because of the times table, an understanding of logs and a little extra thing I memorised at the age of 16.

xLog10

6x

With regard to the initial problem that launched this investigation, I know now that the optimal way of categorising people is to have seven categories each with seven partitions. That sounds very much like a 7-point Likeart scale.

From here we could go into a big long philosophical debate as to whether people fall evenly into all categories of a 7-point scale (the distribution is probably bell-shaped rather than uniform) but that’s a discussion that takes me away from the point I want to make, which is that CRUNCHING NUMBERS IS NOT HARD.

never

rarely

Less than average

average

More than average

Most of the time

always

6 xx Sin

Had I been confronted with the question

I might have treated it in the same analogous way, treating the problem as the area of a box and using some basic knowledge of the properties of Sine waves to steer my thinking.

xSin

6x

And in about five seconds I would have seen that the required value has to be something bigger than 6 because the Sine of a number is always a fraction. Furthermore, if I limit x to being a positive number then the Sine value also has to be positive. And from there I would stretch and squash the box until I found an answer that made sense.

If the answer is bigger than 6 radians, then the answer has to be in the second cycle of the Sine wave, in which case the answer might well be 7 or 8 radians. We have already made good progress.

If I brazenly decide that the Sine value has to be ¾ to make any sense, then I can deduce that the x value itself has to be about 8, reinforcing my earlier opinion. (Correct answer 7.3)

xSin

6x

Another example: 6 xx Cos

Again, the trig value is necessarily a fraction, so the value of x has to be bigger than 6. Again, the pattern of thinking follows the pattern I used for a Sine wave. But in this case the Sine wave reaches a peak earlier than the standard Sine wave, so I am thinking that x ought to be a smaller number than in the previous case. Instead of 8 radians I will guess that it is 7 radians. (Correct answer 6.1 radians)

These are not recipes to be written into a mathematical cookbook. Rather, I am trying to impart a kind of philosophy when it comes to processing numbers, a feeling first of all that no problem we are likely to see at home is so large that it needs technology to solve; second, that if you understand how functions behave then you will know how to pick out a representative value that gets you in the right neighbourhood of an answer quickly; thirdly, that understanding a problem at a sensory level ( weight, height, width etc) helps you to deduce almost intuitively how big an answer ought to be; fourthly that some abstract problems can be represented by an analogy that appeals to the senses, usually vision; fifthly that sometimes the thinking can be so intuitive that you can arrive at an answer before you know how you got there and may spend more time verifying it than deriving it. We do not seem to be processing vector equations when we descend a flight of stairs, yet something in our heads is balancing forward propulsion with the force of gravity and the dimensions of the staircase. We have made a symbolic art out of mathematics, and long may that be so, but we should not abandon the parallel processing that goes on in the back of our heads when we try to nut out a problem in the absence of numbers.

[END]