Tuesday, 12 June 2012

Pebble notation as intro to number theory

I've been emailing with a friend of mine who is sub-teaching right now. I've been trying to convince him that this style of presentation of maths to children with rudimentary mastery over the four basic operations of arithmetic is the best way to introduce higher mathematics to young children. Below is what I've shown him so far (I hope it makes sense). I think capturing the imagination of children requires us to re-think what we know of numbers, perhaps the answers lay not in the future but the distant past.

The notion of “number” according to ancient Greek pebble notation:

Triangular numbers
 

*

**   1+2=3;



*

**

*** 1+2+3=6;



*

**

***

**** 1+2+3+4=10;



*

**

***

****

***** 1+2+3+4+5=15…

Square numbers


**

**  22=4;



***

***

*** 32=9;



****

****

****

**** 42=16

by summing consecutive odd numbers, this pattern for “square” numbers emerges:

1+3=4;

1+3+5=9;

1+3+5+7=16

-what would the next number in this sequence be? (solve through 1+3+5+…

because (2n-1) = odd number, which sum when squared come to n2 for square numbers: 4, 9, 16…

Solving for the next odd number:

(2x1-1) = 1;

(2x2-1) = 3;

(2x3-1) = 5;

(2x4-1) = 7;

(2x5-1) = 9;

1+3+5+7+9=25 = (52) or, in pebble notation:

*****

*****

*****

*****

*****

Oblong numbers


***

*** = 6;



****

****

**** = 12;



*****

*****

*****

***** = 20…

these “oblong” numbers may also be expressed as:  3+3=6; 6+6=12; 10+10=20… or (sum of two triangular numbers) see by drawing a dividing line diagonally
 

**

* /    *

        ** 3+3=6;



***

**

*/    *

       **

     *** 6+6=12;



****

***

**

*/    *

       **

     ***

   **** 10+10=20…

These interconnected “interactions” between the different pebble configurations: ie, triangular numbers to oblong numbers; the summing of consecutive counting numbers to construct triangular numbers; 1+2+3+4+5=15; the summing of consecutive odd numbers to construct square numbers:1+3+5+ (2n-1) = odd number which is squared (n2); the summing of two like triangular numbers to construct “oblong” numbers: 3+3=6; 6+6=12; 10+10=20; 15+15=30… oblong numbers may also be expressed like this: (b* x a**) = 3x2=6; 4x3=12; 5x4=20; 6x5=30… (*b is horizontal; **a is vertical)

-all of these connections have to do with the quality, quantity and deeper patterns that result from arranging “numbers” following rules that are allowed by pebble arrangements and other “mapping” techniques.

-what happens when we sum all even numbers: 

2+4=6;

2+4+6=12;

2+4+6+8=20;

2+4+6+8+10=30;

2+4+6+8+10+12=42

2+4+6+8+10+12+14=56…

-can this number sequence be arranged in a regular pebble pattern?

Next time, we will look at equations, functions and how they work using these new ways of looking at numbers. Mathematics is less about calculating numbers than making symbolic statements about how things relate to each other, what patterns connect with which other ones, etc. These are expressed as equations and functions in Math.

It is also about how to define a number by its given properties: What are “even” numbers; what are “odd” numbers; what are “composite” numbers; and, what are numbers that are not expressible into whole number fractions by any other number than itself and one?

We will also look at how “scientific notation” of very large and very small numbers work. We will look at how these “numbers” and their “magnitudes” are used in measuring systems, like the metric system.
_______________________________________________________

Oblong numbers revisited

***

*** 3x2=6;



****

****

**** 4x3=12;



*****

*****

*****

***** 5x4=20;



******

******

******

******

****** 6x5=30;



*******

*******

*******

*******

*******

******* 7x6=42…

 f we add one more pebble to these arrangements, like this:
 

***

**** 6+1=7;



****

****

***** 12+1=13;



******

******

******

******

******* 30+1=31…

these are all examples of numbers that are “not expressible into whole number fractions” (a/b) or, by whole number “ratios” (a:b). We call these “prime” numbers because they cannot be arranged into regular configurations like oblong numbers or triangular numbers, other than a straight line. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43…

These numbers may be used to generate all other counting numbers because they are either prime numbers themselves or numbers that factor into (at least a pair) n number of primes: 2; 3; 22 or (2x2) =4; 5; 2x3=6; 7; 23 or 2x(2x2) =8; 32 or (3x3)=9; 5x2=10; 11; 3x(2x2)=12; 13; 7x2=14; 5x3=15; 24 or (2x2x2x2) =16; 17;  and so on, with prime numbers in bold. –show how number 18 may be expressed.

Prime numbers can be found by using a technique called, the Sieve of Eratosthenes’ by arranging the counting number sequence like this:

1          2          3          4          5          6          7          8          9          10

11       12       13       14       15       16       17       18       19       20

21       22       23       24       25       26       27       28       29       30

31       32       33       34       35       36       37       38       39       40

41       42       43       44       45       46       47       48       49       50

51       52       53       54       55       56       57       58       59       60

61       62       63       64       65       66       67       68       69       70

71       72       73       74       75       76       77       78       79       80

81       82       83       84       85       86       87       88       89       90

91       92       93       94       95       96       97       98       99       100

since 2 is a prime number every multiple of it (or, even number here after) is eliminated with the sieve; since 3 is a prime number every multiple of it here after is eliminated; since 5 is a prime number every multiple of it here after is eliminated; so it is with 7; and so on.

Here is a special arrangement of counting numbers that seem to show a pattern for prime numbers:

1          2          3          4          5          6

7          8          9          10       11       12

13       14       15       16       17       18

19       20       21       22       23       24

25       26       27       28       29       30

31       32       33       34       35       36

37       38       39       40       41       42

43       44       45       46       47       48

49       50       51       52       53       54

55       56       57       58       59       60

61       62       63       64       65       66

67       68       69       70       71       72

73       74       75       76       77       78

79       80       81       82       83       84

85       86       87       88       89       90

91       92       93       94       95       96

97       98       99       100

every prime number greater than 3 (P > 3) occurs either in the first column or the fifth column. On a clock calculator with 6 numbers on the face, prime numbers greater 3 occur at 5 o’clock and 1 o’clock only. Twin primes, those that differ by two: (11 and 13; 17 and 19; 29 and 31; 41 and 43; and so on) occur at 5 o’clock for the first prime and its twin occurs at 1 o’clock.

Why this is so is still an unsettled question in maths.
________________________

Polynomials

The next thing we will examine here is what a polynomial is. A polynomial is an expression like the triangular generator that sums a sequence of numbers or multiplies numbers together, like oblong and, especially, square numbers. The difference between two consecutive numbers is constant and regular.

A polynomial may have a form like: ax + b. If a and b do not have the same whole-number divisors then its sum is a prime number: if we let a = 2 and b = 1, we get the basic polynomial, 2x + 1.

We let x = 1 so that (2x1) + 1 = 3;

when x = 2 the expression generates:

(2x2) + 1 = 5;

and so on.

The polynomial, 2x + 1, generates odd numbers: 3, 5, 7, 9, 11… as x increases up the sequence of numbers. This expression will generate prime numbers at about 39.6% for the first 1000 values of x. (eg, first two odd primes: 3 and 5 for x values 1 and 2 respectively). 2x + 1 is an example of a first degree polynomial.

A second-degree polynomial may take the form, ax2 + bx + c. The great Euler’s equation for generating prime numbers is in this form: x2 + x + 41 (ie, a second-degree polynomial).

For all values of x from 0 to 39, prime numbers from 41 to 1601 are generated:

x = 0:

0 + 0 + 41 = 41

x = 1:

1 + 1 + 41 = 43

x = 2:

4 + 2 + 41 = 47

x = 3:

9 + 3 + 41 = 53

x = 4:

16 + 4 + 41 = 61…

But things fall apart for this equation when x = 40:

1600 + 40 + 41 = 1681, which is factored into: 41x 41 = 1681.

A third-degree polynomial has the general form: ax3 + bx2 + cx + d. One productive third-degree polynomial, x3 + x2 – 349, which generates 411 prime numbers for the first 1000 values of x.

 One thing you’ll have noticed about polynomial equations in graphing them is that they grow quite fast, and they do not always begin at zero or one. Exponentiation, or a number multiplied by itself a number of times, is inherently geometric in growth.

Going back to the notion of triangular numbers, there is a deeper connection between them and an equation attributed to Gauss: ½ x (N + 1) x N.

Remember that polynomials have the property that “the difference between two consecutive numbers is constant and regular”; this fact allows for an equation like

½ x (N + 1) x N to work beautifully.

Using this equation, we can calculate the sum of any consecutive sequence whatsoever. Gauss, as an elementary student, is said to have used it to calculate the sum of the numbers from 1 to 100. Now, inputting 100 to the equation:

½ x (100 + 1) x 100

we get

½ x (101) x 100 = 50 x 101 x 100 = 5,050.

In examining this equation, we realize that the sum of the first (1) and last (100) terms equals 101. So do the second and penultimate numbers: 2 + 99 = 101; and the ones following, like 3 + 98 = 101; 4 + 97 = 101… this is where the (N + 1) happens: the last number N (100, in this case) + 1 (the first term). Then, we multiply by the last number (100) to 101, which we multiply again the result by half (50) – or, that which make up the 50 pairs of 101 between 1 and 100.

Triangular numbers may be generated like this: (n(n+1))/2.

n =1

(1(1+1)) /2 = (1x2)/2 = 2/2 = 1

n = 2

(2(2+1))/2 = (2x3)/2 = 6/2 = 3

n = 3

(3(3+1))/2 = (3x4)/2 = 12/2 = 6

n = 4

(4(4+1))/2 = (4x5)/2 = 20/2 = 10

n = 5

(5(5+1))/2 = (5x6)/2 = 30/2 = 15

The equation may be expressed in three different ways: (3(3+1))/2 = (3x4)/2 = 12/2

with (n(n+1))/2 being the “basic” expression as stated initially..

Given any equation, one may input numbers not necessarily just whole positive numbers: C = 2π says that the circumference of a circle equals πx2 radians. Rotating around a circle twice equals to πx4. The number, pi, is a transcendental number. It is not a rational number or is not a ratio of two whole numbers, p:q. Its decimal expansion never reveals what the next number in the sequence will be nor terminate and settle into a repeating sequence of numbers. In other words, its proper placement in the number line can ever only be approximated and never properly placed therein.

In the next instalment, we will look into the number or “real” line, what “different” types of numbers there are and what they mean.


Jay

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