## Finally On Natural Analogarithms – student

Over the course of the year my fellow students and I have spent much of our spare time investigating the properties of the set of infinite dimensional vectors associated with the roots of rational numbers by way of the former's elements being the powers to which the latter's prime factors are raised, which we have dubbed -space.
We proceeded to define functions of such numbers by applying operations of linear algebra to their -space vectors; firstly with their magnitudes and secondly with their inner products. This time, I shall report upon our explorations of the last operation that we have taken into consideration; the products of matrices and vectors.

## Further Still On Natural Analogarithms – student

For several months now my fellow students and I have been exploring -space, being the set of infinite dimensional vectors whose elements are the powers of the prime factors of the roots of rational numbers, which we chanced upon whilst attempting to define a rational valued logarithmic function for such numbers.
We have seen how we might define functions of roots of rationals employing the magnitude of their associated -space vectors and that the iterative computation of such functions may yield cyclical sequences, although we conspicuously failed to figure a tidy mathematical rule governing their lengths.
The magnitude is not the only operation of linear algebra that we might bring to bear upon such roots, however, and we have lately busied ourselves investigating another.

## Further On Natural Analogarithms – student

My fellow students and I have of late been thinking upon an equivalence between the roots of rational numbers and an infinite dimensional rational vector space, which we have named -space, that we discovered whilst defining analogues of logarithms that were expressed purely in terms of rationals.
We were particularly intrigued by the possibility of defining functions of such numbers by applying linear algebra operations to their associated vectors, which we began with a brief consideration of that given by their magnitudes. We have subsequently spent some time further exploring its properties and it is upon our findings that I shall now report.

## On Natural Analogarithms – student

Last year my fellow students and I spent a goodly portion of our free time considering the similarities of the relationships between sequences and series and those between derivatives and integrals. During the course of our investigations we deduced a sequence form of the exponential function ex, which stands alone in satisfying the equations

D f = f
f(0) = 1

where D is the differential operator, producing the derivative of the function to which it is applied.
This set us to wondering whether or not we might endeavour to find a discrete analogue of its inverse, the natural logarithm ln x, albeit in the sense of being expressed in terms of integers rather than being defined by equations involving sequences and series.

## Finally On A Calculus Of Differences – student

My fellow students and I have spent much of our spare time this past year investigating the similarities between the calculus of functions and that of sequences, which we have defined for a sequence sn with the differential operator

Δ sn = sn - sn-1

and the integral operator
 n Δ-1 sn = Σ si i = 1
where Σ is the summation sign, adopting the convention that terms with non-positive indices equate to zero.

We have thus far discovered how to differentiate and integrate monomial sequences, found product and quotient rules for differentiation, a rule of integration by parts and figured solutions to some familiar-looking differential equations, all of which bear a striking resemblance to their counterparts for functions. To conclude our investigation, we decided to try to find an analogue of Taylor's theorem for sequences.

## Further Still On A Calculus Of Differences – student

For some time now my fellow students and I have been whiling away our spare time considering the similarities of the relationships between sequences and series and those between the derivatives and integrals of functions. Having defined differential and integral operators for a sequence sn with

Δ sn = sn - sn-1

and
 n Δ-1 sn = Σ si i = 1
where Σ is the summation sign, we found analogues for the product rule, the quotient rule and the rule of integration by parts, as well as formulae for the derivatives and integrals of monomial sequences, being those whose terms are non-negative integer powers of their indices, and higher order, or repeated, derivatives and integrals in general.

We have since spent some time considering how we might solve equations relating sequences to their derivatives, known as differential equations when involving functions, and it is upon our findings that I shall now report.