Rational Mathematics

by G. P. Jelliss

Power Series

A power series (or polynomial) is a sum of multiples of powers of a variable.

Theprem: Power Series Theorem: Any expression formed by addition and multiplication of numbers and a single variable (say n or x) can be reduced to the form of a power series. Proof is by application of the commutative, associative and distributive laws. The exponent of the highest power of the variable in the expression is called its degree.

An expression of degree 1 is often termed linear, an expression of degree 2 quadratic (this misleading term comes from the Latin for 'square' not 'four'), and one of degree 3 is cubic (these three terms come from geometrical applications). An expression of degree 4 is quartic, one of degree 5 is quintic, and similar Latinate terms exist for higher powers.

In positional numeration the value of a digit depends on its position. A digit moved one step to the left represents n times its previous value. For example a five-digit expression abcde in base n represents the number: (((((a.n) + b).n + c).n + d).n + e which when multiplied out becomes: a.(n^4) + b.(n^3) + c.(n^2) + d.n + e, the sum of a series of multiples of powers of n. In any positional number system the compound symbol 10 represents the base n, 100 represents the square of the base, 1000 its cube, and 1 followed by k noughts its kth power. A number with m zeros at the end can be written in the form k.(10)^m. Some 'scientific' calculators show this as kEm, where the E stands for 'exponent'. The number of digits needed to express a number N in base n is k if n^(k1) ≤ N < n^k. The number k is the 'order' of N relative to n.

Exercise: Prove that if g(i) are numbers >1 then every number ≥1 can be expressed uniquely in the form a(0) + a(1)g(1) + a(2)g(1)g(2) + ... + a(k)g(1)...g(k) where 0≤a(i)<g(i+1). [Baker] This is effectively a monetary or other multi-unit system with more than one conversion factor.

Forms

An expression that determines for each number n a number f(n) is called a formula. Numbers expressible by a formula f(n) are said to be of the form f(n). A formula f(n) determines a number sequence (f(0), f(1), f(2), ...).

Various sets of numbers can be described by the geometrical formations in which they can be represented, and relationships between them can be visualised as patterns of dots or squares or other shapes. The same results can be presented arithmetically as the summation of simple series.

Forms of degree 1

The sequence 0, 1, 2, 3, ..., u is that of the identity function, denoted by n itself (or by such formulas as 1.n or 0+n). The formula 2.n determines the sequence of even numbers 0, 2, 4, 6, 8, 10, ..., while 2.n + 1 determines the odd numbers 1, 3, 5, 7, 9, 11, ... The even and odd numbers together make up the complete set of numbers.

If m > 0 then for any number n there exist unique numbers q and r (quotient and remainder) such that n = q.m + r where r < m. We have n = (n|m).m + (nr). Each number m thus classifies the nonzero numbers into m first degree forms (or linear forms): m.n, m.n + 1, m.n + 2, ... m.n + (m1).

Forms of degree 2

By summing the successive numbers from 1 we get the triangular numbers 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, ... The nth number in this sequence is given by n.(n+1)|2. This division by 2 is always possible since the product of two successive numbers is even, because one of them is even and the other odd.

Beginning the summation at 2 we get 2, 5, 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, ... which are 1 less than a triangular number. Beginning at 3 we get 3, 7, 12, 18, 25, 33, 42, 52, 63, 75, 88, 102, ... which are 3 less. Beginning at 4 we get 4, 9, 15, 22, 30, 39, 49, 60, 72, 85, 99, ... which are 6 less. Generally: beginning at r we get a series which is equal to the triangular numbers less the (r1)th triangular number: n.(n+1)|2 r.(r1)|2. Unlike the first degree forms these sets of numbers are not mutually exclusive.

The sum of the first n odd numbers is the nth square number, n.n = n^2 = 1 + 3 + 5 + ... + (2n1). The series proceeds: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, ... The nth square is the sum of two successive triangular numbers, the nth and (n1)th: n^2 = (n^2 n)|2 + (n^2 + n)|2. The square of the nth odd number is one more than 8 times the nth triangular number: (2n + 1)^2 = 4.(n^2) + 4.n + 1 = 8.[(n^2 + n)|2] + 1. In other words odd squares are of the form 8.m + 1. Even squares are of the form 8.m or 8.m + 4. Numbers of the forms 8.m + 2, 3, 5, 6, or 7 are never squares.

The sum of all the odd numbers up to the nth odd square 1 + 3 + 5 + ... + (2n1)^2 = (2.n.(n1)+1)^2 is thus the sum of two squares: (2.n1)^2 + (2.n.(n1))^2.

Examples of this are
3^2 + 4^2 = 5^2
5^2 + 12^2 = 13^2
7^2 + 24^2 = 25^2
9^2 + 40^2 = 41^2
11^2 + 60^2 = 61^2

The numbers 2.n.(n1) and 2.n.(n1)+1 are consecutive numbers whose sum is a square (2.n1)^2, and the difference of their squares is the same square.

N^2 1 = (N1).(N+1). i.e. One less than a square is the product of two successive odd numbers or two sucessive even numbers. N^2 = (N1).(N+1) + 1, i.e. the square of a number is one more than the product of the numbers one above and one below. More generally we have N^2 − X^2 = (NX).(N+X). This is often useful to simplify calculations.

The sum of the first n even numbers is a near-square (or oblong) number: 0 + 2 + 4 + 6 + ... + 2.n = n.(n+1) = n^2 + n, i.e. a product of two successive numbers. The series runs: 0, 2, 6, 12, 20, 30, ... The nth near-square is twice the nth triangular number. The nth near-square is n^2 + n and also (n+1)^2 (n+1).

The near-square numbers plus one (i.e. twice the triangular numbers plus one) form the sequence 1, 3, 7, 13, 21, 31, ... which I call intercube numbers, i.e. odd numbers of the form n.(n1) + 1 = n^2 n + 1.

Numbers of the form 6.m+1 where m is a triangular number are called hexagonal numbers, since they can be pictorially represented as hexagonal arrays. The nth hexagonal number is thus 6.[n.(n+1)|2] + 1 = 3.n^2 + 3.n + 1. The series runs: 1, 7, 19, 37, 61, 91, ... Its differences are the successive multiples of 6: namely 6, 12, 18, 24, 30, ...

The number 5 is the first number expressible as the sum of two different squares 5 = 1 + 4 = 1^2 + 2^2. The next numbers with this property are: 10, 13, 17, 20, 25, ... They are pythagorean numbers.

Forms of degree 3

Summing the first n triangular numbers gives the tetrahedral (or triangular pyramid) numbers: 1 + 3 + 6 + .. + n.(n+1)|2. The series runs: 1, 4, 10, 20, 35, 56, 84, 120, ...

The sums of the series of square numbers are called pyramidal numbers, since they give the numbers of cannonballs in a square pyramidal arrangement. The series runs: 1, 5, 14, 30, 55, 91, 140, ....

The nth pyramidal number answers the puzzle of counting the number of squares of all sizes (i.e sets of four points at the corners of a geometrical square) in a square array, n by n. There are (n+1 r)^2 of size r by r, from which we deduce there are n^2 + (n1)^2 + (n2)^2 + ... + 3^2 + 2^2 + 1^2, i.e. the reverse of the previous summation.

The sums of the squares of n+1 consecutive numbers of which the greatest is 2.n.(n+1), is equal to the sum of the squares of the next n numbers.

This series of results runs:
3 + 4 = 5
10 + 11 + 12 = 13 + 14
21 + 22 + 23 + 24 = 25 + 26 + 27

The sum of the n odd numbers commencing with the nth intercube number is the nth cube number n.n.n = n^3. The series of cubes runs: 0, 1, 8, 27, 64, 125, 216, ... Thus 1 = 1^3, 3 + 5 = 2^3, 7 + 9 + 11 = 3^3, 13 + 15 + 17 + 19 = 4^3, ...

The sum of the first n hexagonal numbers is the nth cube. Thus 1 = 1^3, 1 + 7 = 2^3, 1 + 7 + 19 = 3^3, 1 + 7 + 19 + 37 = 4^3, ... And conversely the difference of two successive cubes is a hexagonal number: (n+1)^3 n^3 = 3.n^2 + 3.n + 1 = 3.n.(n+1) + 1.

All cubes are of the forms 7.m or 7.m 1.

N^3 = (N2).(N+1)^2 + 3.N + 2.

Some sums of cubes: 3^3 + 4^3 + 5^3 = 6^3, 1^3 + 6^3 + 8^3 = 9^3.

Forms of degree 4 or more

The sum of the first n cubes equals the square of the nth triangular number, i.e. the square of the sum of the first n numbers. That is 1^3 + 2^3 + 3^3 + ... + n^3 = (1 + 2 + 3 + ... + n)^2 = (n^2).(n+1)^2|4 = (n^4 + 2.n^3 + n^2)|4. A cube number is the difference of the squares of two successive triangular numbers: n^3 = [(n^2 + n)|2]^2 − [(n^2 − n)|2]^2.

Differences and sums of powers: (xHy) is a divisor of (x^n H y^n) for any number n. If a≤|b then x^a H y^a ≤| x^b H y^b.

Any power number of the form n^x can be built up from n consecutive odd numbers, irrespective of the value of x. Where n is odd the central number in the odd series for n^x is equivalent to n^(x1). For example 5^4 = 121 + 123 + 125 + 127 + 129 = 625, where 125 = 5^3. Where n is even the central pair of numbers in the series are n^(x1) 1. For example 4^4 = 61 + 63 + 65 + 67 = 256, where 63 and 64 are 4^3 1.

Any power number can be derived by repeated summing beginning from a constant value. Or, working the other way, by repeatedly calculating differences of successive terms we eventually reach a constant value. The fourth powers n^4 form the series: 0, 1, 16, 81, 256, 625, 1296, ... The first differences of the fourth powers are : 1, 15, 65, 175, 369, 671, ... The second differences are: 14, 50, 110, 194, 302, ... The third differences are: 36, 60, 84, 108, ... The fourth differences are 24, constant.

The natural numbers have the constant difference 1, the squares have the constant second difference 2, the cubes have the constant third difference 6, the fourth powers have the constant fourth difference 24, and so on. The sequence 1, 2, 6, 24, ... is that of the factorial numbers.