checking your sums

20 07 2010

At the algebraic level, you can check your answer by substituting in your numerical answer for the unknown, and then finding out whether the left and the right side are equal, as initially proposed.

There are other, simpler examples of this process. Times tables. 4 x 8 is 32. How can you tell? Well:

1. it sounds right :)

2. 4 x 10 is 40, so two fours less than this is 32

3. count up in eights: 8, 16, 24, 32… the fourth item is 32

4. count up in fours: 4, 8, 12, 16, 20, 24, 28, 32… the eighth item is 32

Two different ways of getting the same answer suggests you’ve got the right answer. Like the general rule of thumb that if you find a fact in two “independent” sources, they greatly increase the chance of them being right.

This should be provable simply with Bayes’ Theorem.

One instance, me thinking there is another side to maths, simply makes it peculiar or unique. Two instances, and we have some validity, and so on. The jump from one to two is huge. This goes for a lot things. 2020worldpeace springs to mind…

two ways of thinking about division

20 07 2010

I may have covered this, but while tutoring it became crystal clear. Consider:

8 / 2 = 4

Actually, when I look at it like this, it immediately smacks of fraction. However, if we were using the more normal divide sign, there are two ways of interpreting it. At least, perhaps.

1. “how many two’s are in eight?”

2. “what is eight divided by two?”

The difference becomes particularly graphic when we consider fractions:

2 and 1/2 divided by 1/2… is 5

This is a bit of a jump, mentally, for most people. Which of the above interpretations fits? ie

1. “how many halves are there in two and a half?”

2. “what is two and a half divided by half?”

In my mind, the first makes sense. It’s to do with shapes, or words even.

“how many half-slices of pizza are in two full pizzas and one half-slice?”

“how many pairs in eight?”     :  :  :  :

This is very different from thinking about division as cutting, ie

“what is eight divided by two?” or ” what is eight divided into two groups?”   : :      : :

Now consider a bunch of kids in a room, and at one time they may be thinking of division one way, and then there’s another way being explained to do something. The way of thinking must be contextualised, rather than there being a rule that is true for all time and for all things.

Of course, this is simply the difference between 4×2 versus 2×4. Wow. I thought these were the same. But they aren’t. At least, not in terms of division. In the first way, we have four groups of pairs, and in the second way we have two groups of four. In terms of XQ, and the two different ways of thinking about addition (the action being on the plus, the action being on the equal sign), the action here is in terms of the mind. The divisor determines the number within each group, or, the number of groups. Again, counting in terms of things, and counting more in terms of a higher order, cuts, divisions.

OMG. Can we teach primary teachers to have this sensitivity? Why not? It’s pretty simple really. It’s a matter of listening to what a kid is doing. Noticing, that’s all. And the best way to show that you understand is by giving them more that they can do well, which gives them confidence, and then giving them things that they can’t seem to get with the same thinking methodology. Or perhaps getting two kids who happen to be doing the two different ways confidently to engage and see what they make of it.


I hate stats

5 07 2010

Ok, I don’t really hate them. I am deeply suspicious of their use, especially when applied to eliciting aspects of the human condition. So, it was with great relief that I came across an article by Karl who quotes Jung:

“If, for instance, I determine the weight of each stone in a bed of pebbles and get an average weight of 145 grams, this tells me very little about the real nature of the pebbles. Anyone who thought, on the basis of these findings, that he could pick up a pebble of 145 grams at the first try would be in for a serious disappointment. Indeed, it might well happen that however long he searched he would not find a single pebble weighing exactly 145 grams.”

Quite simple, quite brilliant.

types of number

24 12 2009

Reading the story of e, final chapter, and several ideas spring to mind. Relating to the continuity of the number line, certain numbers being unusual, eg √2 or π.

1. Pythagorean obsession with musical intervals and ratios of string length… I would not be surprised to find out that the circularity of the cochlea has something to do with musical note interpretation. That is, it’s not the sound frequency that we apply musical sense to, but our sensation, the modulation of nerve signals derived from the physical massing of cells around cochlea.

2. The rational numbers are enough to describe most phenomena because

… the accuracy of any measurement is inherently limited by the accuracy of our measuring device

This parallels the quantum world where our instruments influence the thing measured. That is, e and π are irrational in that they do not have an absolute rational description m/n, they deny this measuring. We can only get as accurate as the tools we use. We settle on a number that is sufficient for our purpose. It is always an approximation.

3. √2 is irrational in that it can not be written m/n, it is not resolve nicely. And yet we can manipulate it. We can construct it easily with a unit square. We may not be able to specify it, but we can point at it. We can refer to it with the symbols √2, and we can point to it if we make a unit square and point at the length of the diagonal. But with our number system, we can not apply a discrete number to it. We can only ask that the mind look at a thing, or carry the consequence of an imaginary process, the square root of 2.

4. The author refers to irrational numbers, written in decimal form as non-terminating and non-repeating, as holes in the number continuum. I guess I can understand this in terms of there not being a n/m or a fixed pattern, and hence it is un-numbered. I take it a step further and think of them as being asymptotes.

5. Transcendental numbers are those that can be expressed as a solution to an algebraic equation. Hence, √2 is algebraic even though it is irrational. If it is not algebraic, it must be irrational; but some irrational numbers can be algebraic eg √2. A transcendental number can be expressed in terms of an infinite series of fractions. π and e are transcendental.

What follows is a little more basic, regarding primes.

6. The fundamental theorem of arithmetic is that any number above 1 can be factored into primes in one and only one way. No problem with this, though the number of primes in a given composite is interesting. eg 4 is 2×2, 8 is 2x2x2, so the shape of a number is given by the number of primes used to derive it. 12 is also a volumetric number since it is 2x2x3. Hmmm… 1 is 1x1x1x… 2 is 2x1x1x1…. 3 is 3x1x1x1… 4 is 2x2x1x1…. 5 is 5x1x1x1… 6 is 3x2x1x1….

7. Primes seem to arrange themselves into p and p+2 eg 3 and 5 or 11 and 13 or 17 and 19. Why do these prime twins exist? Primes can not follow one another, only 2 and 3 the only successive prime numbers. Surely this has to do with the number 2?

8. The Goldbach Conjecture: every even number greater than or equal to 4 can be written as a sum of two primes. It is unsolved. No counter-example has been provided, but no proof either. hmmm 0 as 0+0+0… 1 as 1+0+0+0… 2 as 1+1+0+0+0… 3 as 2+1+0+0+0… or 1+1+1+0+0+0…  4 as 3+1+0+0+0… or 2+2+0+0+0… or 2+1+1+0+0+0… or 1+1+1+1+0+0+0… etc I know I am including 0 and 1 in these calculations, but it makes some kind of sense since 0 is the identity of addition as 1 is the identity of multiplication. There are multiple ways of writing a prime sum, eg 10 can be written 5+5 or 3+7.

9 Lambert’s Conjecture: for large x, the number of primes below a certain number x ~ x/ln x, where ln is the natural log base e, otherwise known as the Prime Number Theorem. This is surprising to mathematicians, that the average distribution of prime numbers can be described using logarithmic function; that is, primes as the domain of integers and e to the domain of limits and continuity. But it should not be, since the number of higher dimensional shaped number increases as x gets larger. That is, 4 is the first square number, 8 is the first cube before we even get our second square number 9, and even by 16 we have our first hypercube number well before we get our third square number 25 and second cube number 27. Every order of 10 we increase x by, the number of powered shaped numbers will increase, won’t it? Of the number of primes within 100 million, which is 5,761,455, how many of them are power-derived? This might contribute the accuracy that this 5.75% happens to be close to 0.0543 which is 1/ln 100 million.

This all seems to be related to addition, the higher function of multiplication, and then powers. And because of the nature of large numbers, and the different ways of thinking of them as sums or products, the statistical behaviour of large primes seems reasonable enough. Also, primes are effectively the numbers that are not repetitions of previous numbers, ie not multiples. The reason why a formula is not possible is because it is the absence of pattern, rather than the presence of one; the primes are what is left. Still, e and π turn up as interesting numbers and not through exclusion of other numbers solutions in a set.

can it be simpler?

23 11 2009

I watched a TED video. The chap looked a bit geeky, and didn’t seem too accustomed to people listening to him. Krank-like, I guess. However, his name is David Deutsch and he’s highly respected in the multi-universe physics world. Anyway, he offered a solution to why science has worked since the 16th century (namely because there is low variation), and this definitely rings true. (In my language: not surprising, really, since science is rooted in objective reality, independent of our minds, so there is less variation in terms of interpretation as we hone in on a explanation that matches actuality.) Of course, his concern is with the multiple interpretations of quantuum physics within science itself, as well as the plethora of new-age assigned interpretations…

So, what about XQ and the subjective side of maths? Reasonable amount of intuitive guessing going on here: notions of negative and the mind’s filitering, or addition and category systems. These are speculative. Clearly.

My only answer to this, as I think about it now, is that the amount of variation increases the more complex the maths we deal with. It is like smoke from a stick of incense. Especially when we consider iterative equations, or Riemann Spheres. Seems to me, then, that the simplest thing is where we might meet with more subjective alignment. That is, as close the burning ember as possible. And in maths, that’s counting, isn’t it?

When we count, what exactly are we counting? It looks like we are counting things, however, if the things are fabrications of our mind, how we cloth actuality, then is this a self-referential exercise? We are counting things that are fabrications of our mind, and if mind is process, then at some level we are counting processes, we are counting in time. The real abstraction, our side of maths, is that we are counting in time, and not any thing out there. This is demonstrated in the way we learn our multiplication tables, through repetition of counting steps. (Extract from One Two Many, from the XQ booklets.)

So, is this something which strikes you as true? Or, are there other interpretations, apart from the obvious that we are counting things. And this is the root to our understanding why music holds its endless fascination for us…

Somehow the mind compiles a soundscape of the environment in terms of events. Subjectively speaking, we are particularly attuned to patterns of sound in time, from simple rhythms and melodies in music to the near chaotic jumble of words. Music elicits movement from our bodies, from the tapping of feet, the nodding of head, or the gyration of our hips. Certain melodies grab our passions, others lift us to sublime heights. It is as if we are the instrument, we resonate to the frequencies. We can also invest music with our memories, with rich associations, with specific events. We may link the music to the performance, the violent extraction of sound from the instrument or its gentle teasing, or it may be completely divorced from production. Our primary attention can be led to follow a particular continuity of sound, focussing on the melody for example, enabling us to predict what it to happen. Complex interactions within the soundscape can be simultaneously appreciated, thus deriving our deep affinity to music. (Extract from Haphazard Sensory Observations, from XQ Conditional).

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