The Rock

Introduction

This page is about an extended math adventure for a young person who knows just a bit about numbers. Her name is "B".

The Rock is triangular and solid and densely mathematical! The image is intended to sound like a foundation, although of course it is not a foundation in the way that arithmetic is a foundation. The Rock is in fact often called Pascal's Triangle, it contains many patterns, and it is not square, so it feels like an interesting basis for exploring that is alternative to arithmetic. Monsieur Pascal did a good job writing it up, so there is how he gets his name stuck to it, though it was around before him. Wikipedia has a good article on it that I have not yet read... so I'll look up more after I run through the little bit I do know.

Kids like to find patterns so mathematics as a study of patterns seems a fun match; this is my empirical generalization based on a sample study size of 2 (myself and B). So more properly I will say that B notices patterns and seems to enjoy pointing them out to me. She also has an affinity for things that go on and on to infinity; and here we have both; so off we go.

Starting off with a^b

I said to B, who understands some arithmetic including multiplication and who calculates on her fingers mostly: "One day some ones got together in a line." I like to personalize numbers particularly because people have motivations. 'The ones stood in a line and asked "what if we all multiply ourselves together, what do we get?"' Pause.

A note on 'Pause': This is for B to work out an answer. She came up with the correct answer in this case, which is 1. When she does not get to the correct answer I always assume I have asked the question poorly; so I drop it and come back to it later from a new angle. "Pause" means we got the right answer and moved on.

A note on 'Cutesy': The ones story is a bit cute the way I've written it; but in practice there's a lot more dialog in this discussion with B than I represent here. We may put the 1s in a bar, they may be drinking and talking pirate talk (swearing) and fighting and so on. So it doesn't have to stay cute or on-topic. The main thing is to get B engaged, which is generally pretty easy, whether or not it is cute.

My first goal in the flow here is to get B comfortable with calculating things from new notation, and in particular I want her to be very happy with the notation a^b. Raising numbers to powers is crucial to The Rock at the outset, particularly because we need powers of two. The problem with powers is that a number raised to the zero power looks like it ought to equal zero. 'See the little number up top? That is how many times you write the big number in a row, and you multiply them together.' So 2⁰ should obviously but incorrectly equal zero. I've been reading about Euler who did a lot with infinite strings of things. Riffing on that I'll create a different definition for exponents and get in some infinity thinking as well.

So I tell B "Some more ones came into the room, and they lined up. What did it multiply out to?" Pause. "And then an infinity number of ones came into the room..." (I draw this using ...) "and what did that multiply out to?" Pause.

I don't remember having any trouble with infinity as an evolving concept. It just got bigger and bigger as I got older; now the problem I have with infinity is how to work with it correctly. For example I feel considerable discomfort shall we say when thinking about the sum of reciprocals of primes being unbounded.

That is, choose any number n; then there is some other number m such that the first m terms of that sum are greater than n. No matter how big you make n there is a corresponding m; but boy let me tell you m is a gosh darn big number for even small values of n.

Anyway to convert the point I now have to permit a finite number of twos into the room, and now what is the new product, say three twos and the infinite number of ones all multiplied together? And then once we have that we can let a different number of twos in the room, or some threes, or whatever; but the ones are always there. Then I say "Ok we have a way to write how many twos there are in the room with all the ones." Which is the introduction of exponential notation, and it works great when we get to the payoff question: "What is 2⁰?" Pause.

Making a catalog of powers of two

The next thing to do is write down some powers of two so they are ready to be noticed at any moment.

1
2
4
8
16
32
64
128
256
512
1024
2048
4096
8293
16384
32768
65536 (the first one I have to calculate)
131072
262144

and so on

Defining even and odd numbers

Here the key point is that no matter how big the number is you just need to see its last digit to know whether it is even or odd.

Making the rules for summing even and odd numbers

Here B is quite happy making an immutable law based on a single example; so we might want to check with a couple.

The Rock version one: Odd and Even

Here we need a large piece of graph paper and two colored pens. We draw the diagonals as all-odd and go from there.

Diagonal patterns

I'll number diagonals from zero.

The zeroth diagonal is all odds.

The next (one'th) diagonal is even / odd.

The next (second) is even / even / odd / odd.

Then skip one to the fourth: Even even even even odd odd odd odd.

So diagonals numbered as powers of two are one set of patterns. The ones in between the powers of two follow patterns as well, but more complex.

Here is (to me) an interesting thought: The horizontal rows are each a finite number of numbers, and there are an infinite number of them. Meanwhile the diagonals are an infinite number of numbers, and there are an infinite number of them. And they are the same numbers. So .

At upper left are visible the rules for adding evens and odds.

Incidentally this raises the question: What is more common in The Rock: Even numbers or odd numbers? Is it an asymptotic ratio or does it keep changing as you build more rows?

Flipped triangles and boundary rows

Note row 2ⁿ is solid odd.

It must add up to a power of two; which is even.

But the only way to do that is to add up pairs of odds so this row must always have an even number of numbers in it. This is an extension of the core rule: If you add up a bunch of them and the result is even then you added up an even number of odds. In other words, odd times even is even, odd times odd is odd. And even times even is odd; so this is not symmetrical the way it was for addition. An even result is more likely in a multiplication product.

• even + even = even
• even + odd = odd
• odd + even = odd
• odd + odd = even

• even x even = even
• even x odd = even
• odd x even = even
• odd x odd = odd

Fractal pattern of the Serpienski gasket

This pattern can be generated two ways that I know of: Through a set of four rules for cellular automata (which is what The Rock is doing) and through a random process described here. This is intended to eventually migrate to the Toy Universe page of this Main Sequence wiki.

Row sums

Up to this point the patterns all involve even/odd-ness; so now I turn to the actual numbers. After playing around with the odd/even version of the rock we begin again, this time writing ones along the left and right edge diagonals, and then adding to fill the interior of The Rock. We verify that the odd even pattern is true and discuss how our general rule for addition of odds and evens has played out correctly in specific practice of using actual numbers and actual addition. I would try and talk about abstraction in relation to specific cases with B here, just to plant that idea. It leads to proofs which don't require looking for a counter-example in an infinite number of cases.

So now we have The Rock filled in with numbers; and ask "what do the rows add up to?" We start adding up rows to find the sums seem to be powers of two. This would be a good time for B to already know that 2⁰ = 1.

More advanced: On the An Infinite Quantity Of Math page I asserted without proof that if you arbitrarily slap alternating +1 and -1 factors across each row the sum across all rows is 0. Unfortunately on that page I did not prove that the sum across rows of The Rock (without the alternating signs) is 2ⁿ. This is left as a hanging exercise. I tried doing it this evening but I think I'm too tired. Another nice thing to get out of the definition of a combination expression is that while it is defined as a fraction it comes out to be an integer, always.

Oh one path to a proof might be to expand (a + b)ⁿ to arrive at a definition of the coefficients of the expansion, which are then set as equal to row elements. Then you evaluate for a = 1 and b = 1 and you're done.

Addendum: Need to write it up but that bit above is part of it, in fact the last step. The first bit is define a combination in the usual way and show that the elements of the Rock are combinations. This gives a nice recursion relation that applies from one row to the next. Then have to prove the binomial theorem which is mostly a matter of induction on sort of thing. Then you have the binomials and the Rock and the combinations all tied together and you substitute a = 1 and b = 1 and you get the 2ⁿ rule. Pick up that combinations are always integers along the way.

The pine cone

Diagonal sums, done in a slightly tricky manner, are the Fibonacci numbers. Also shown briefly here and here.

More advanced: I wrote some remarks on generalizing Fibonacci ratios (of successive pairs of terms) here.

Side Remark 1: This is not arithmetic

Last I checked schools teach kids to count and add and subtract and multiply and divide, and eventually do fractions and decimals. This takes what? about six years? but I suspect it takes that long for reasons that have nothing to do with the arithmetic itself. Granted it is a special case but Erdos had arithmetic down at age three so it is possible to learn arithmetic before age eleven. I wouldn't expect everyone to do so; rather my thesis is that stretching the arithmetic out over six or more years is an institutionalized impediment to learning mathematics. There is a math stigma in this culture because we spend six years instilling students with concrete numerical thinking: Every problem is solved by a mechanical calculation to arrive at a definite number which is somehow "the point" of the exercise. As a result students have little or no capacity for abstraction; and then of course it is often daunting when they arrive at mathematical abstractions, for example in algebra... so kids are mystified and frustrated in many cases.

I'm going to regard arithmetic as a useful toolbox which kids like B will pick up in school. Eventually. But mathematics is too important for me to wait around for B to start learning when she is 12, so we start mathematics today. You do not need to know long division to do combinatorics or trigonometry or calculus. Furthermore a more interesting question is how to start a kid thinking abstractly when her way of thinking is concrete and definite. Well, is it? No, actually no it's not. She thinks abstractly all the time when she is playing, for example when she is singing. I think the challenge is just to bring her abstract thinking to the table when we talk about mathematics.

Incidentally, to be fair to the school curriculum, based on what I see of B's math they have broken down arithmetic / numbers into constituent concepts and they do have the kids work with those abstractions. Numbers have 'partners' for example, two smaller numbers that added together equal the starting number. I don't want to disparage the hard work that has gone into the system we have today, and I don't want to climb on the critical bandwagon. I just think we can have some fun here outside of school.

Side Remark 2: Towards Abstract Thinking

In the philosophy of teaching and learning my approach is fairly easy to state and difficult to do. I try and look at a large chunk of the subject, pick the point at which it all became difficult and too much to handle for me, and start teaching towards that idea from day 1. I figure the sooner we get to it, the more the idea will appear to be a natural part of the process, and we can hope to make it invisibly "easy" to understand. When I taught a course in C programming in 1993 the first thing we did was pointers. As I did number games with B one of the first things we did was "numbers below zero". One of the next things we'll do will be double- or complicated-numbers.

One idea is to create simple mysteries that eventually lead to results that may be abstract or definite. An example might be an n mystery:

• One day n got up and put on a number T-shirt. "Today I will do everything based on my T-shirt" she thought.
• n then put on a jacket so nobody would know her number. (It was hidden under the jacket.)
• This arrives -- eventually -- at n + 7 = 12.

So the only point of all this story-telling is to have a bit more basis for linear equations and what it means to solve for n. That's maybe the spiral at the beginning of the yellow brick road of algebra.

Side Remark 3: A simple formalism

I mentioned complicated numbers above. These are pairs of regular numbers, in order, like (2, 3). This is not the same as (3, 2). Those are different. Adding these numbers is straightforward, as is plotting them. Multiplying them will be a bit of a journey, but it will be interesting to see the progression to polar form and how easy that will be to understand.

I was thinking that even more abstract would be to consider pairs of numbers and a symbol indicating a relationship. So a comma means "turn left". A slash means "break apart". A plus means + and a x means x and a - means "+ the number below zero". So we end up with a b where there could be any number of symbols in the box meaning different things. So you end up with a family of operations on two numbers. Don't know if this accomplishes anything but it's worth thinking about a bit maybe.

Normal distribution

Somewhere on this wiki I wrote that the binomial distribution is similar to the normal distribution. This is not really something to throw at B in these terms; but as long as we have those rows of The Rock just sitting there we might as well make some graphs out of them. I could see doing this in Excel after a few by-hand cases. This needs to connect back to dice that are labelled in some unusual way (like all sides = 1, and then all sides = 1 except one side = 2, then all 1 except one six) to generate an empirical view of the CLT.

Other questions

• Do multiples of 10 (say) pick out an interesting pattern?
• Do the centers of alternate rows (even rows or odd rows) converge in ratio to something? (In the way that a Fibonacci sequence ratio converges)
• Look at overplotting rows, normalizing to the row peak value vertically and normalize to the width of the row horizontally: Does this illustrate the narrowing of a peak as seen in thermodynamics?

• From W-pedia the entry in the triangle is equal to the number of paths to that point if one is allowed to traverse row-to-row using only diagonal adjacency.
• The self-similarity of the four embedded triangles clearly expands to any size (power of two) one could wish; but it stops in the other direction at the level of quantization of the rock. There is no triangle smaller than the "2" in the third row. Is it possible to build a fraction triangle that goes back the other way to very small numbers? Is there a rock balanced upside-down atop the rock with negative numbers? Is there a rock emanating sideways with complex numbers in it? A quick look shows making one arm (1, 0) and the other (0, 1) produces a sparse symmetrical first-quadrant rock. Pedagogically it would be a simple (harmless) introduction to "a new sort of number" of the form (a, b) and how to add them.