Elementary mathematics

Computing the compounding rate

A common problem is computing the rate of compounding,
given the initial amount, final amount, and number of periods.
We start with an initial amount a.
It increases for n periods;
in each period the amount then current
is multiplied by a constant factor r
to obtain the amount for the next period.
The final amount is b.
The formula that results from this situation
is the compounding or exponential growth formula

b = a × rⁿ .

It frequently happens that
we know
the initial amount a, the final amount b, the number of periods n,
and wish to determine
what multiplier r
would have yielded that final amount
when compounded as described.

To solve this problem we need only the following facts:

exp and log (aka ln) are inverse functions
exp ( x + y ) = (exp x) × (exp y).

From those two facts it easily follows that

log ( a × b ) = (log a) + (log b)
log ( b/a ) = (log b) - (log a)
and
log ( rⁿ ) = n × log r.

[Pedantic note 1:
So the exponential is a homomorphism, in fact an isomorphism,
from the (abelian group of) reals under addition and subtraction
to the (abelian group of) positive reals under multiplication and division,
while its inverse the logarithm is a homomorphism, in fact an isomorphism,
in the opposite direction.]

Now
to solve the problem of computing the rate of compounding as stated above,
we observe the following string of logically equivalent equalities:

b = a × rⁿ
b/a = rⁿ
log (b/a) = log ( rⁿ ) = n × log r
log r = ( log (b/a) ) / n
r = exp ( ( log (b/a) ) / n )

(The second line above is equivalent to r being the nth root of b/a, r = (b/a)^1/n,
but to actually compute the nth root you’re back to working with log and exp.)

For an application of the last formula,
see the calculation of the annualized rate of tuition increase at Harvard here.

Pedantic note 2:
The key part of the final formula for the rate of compounding,
exp ( log (??) / n ),
is an example of conjugation,
generalized from the theory of groups to the theory of groupoids;
the groupoid in which this conjugation takes place is the
groupoid of (abelian) groups and isomorphisms.

This conjugated function may be represented as a simple diagram
in the groupoid of (abelian) groups and isomorphisms:

log (R+, ×, 1) ------------------> (R, +, 0) | | ?/n | V (R+, ×, 1) <------------------ (R, +, 0) exp

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Miscellaneous Articles

2010-03-06-NYT-Bayley-Algebra-in-Wonderland

Algebra in Wonderland
By MELANIE BAYLEY
New York Times Op-Ed, 2010-03-06

Oxford, England

[1]
SINCE “Alice’s Adventures in Wonderland” was published, in 1865, scholars have noted how its characters are based on real people in the life of its author, Charles Dodgson, who wrote under the name Lewis Carroll. Alice is Alice Pleasance Liddell, the daughter of an Oxford dean; the Lory and Eaglet are Alice’s sisters Lorina and Edith; Dodgson himself, a stutterer, is the Dodo (“Do-Do-Dodgson”).

[2]
But Alice’s adventures with the Caterpillar, the Mad Hatter, the Cheshire Cat and so on have often been assumed to be based purely on wild imagination. Just fantastical tales for children — and, as such, ideal material for the fanciful movie director Tim Burton, whose “Alice in Wonderland” opened on Friday.

[3]
Yet Dodgson most likely had real models for the strange happenings in Wonderland, too. He was a tutor in mathematics at Christ Church, Oxford, and Alice’s search for a beautiful garden can be neatly interpreted as a mishmash of satire directed at the advances taking place in Dodgson’s field.

[4]
In the mid-19th century, mathematics was rapidly blossoming into what it is today: a finely honed language for describing the conceptual relations between things. Dodgson found the radical new math illogical and lacking in intellectual rigor. In “Alice,” he attacked some of the new ideas as nonsense — using a technique familiar from Euclid’s proofs, reductio ad absurdum, where the validity of an idea is tested by taking its premises to their logical extreme.

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