recurrence relation 443

H

INDU

-A

RABIC NUMERALS

(the numbers we use today)

rather than R

OMAN NUMERALS

.

Maintaining an interest in mathematics throughout

his life, Recorde wrote a series of instructional texts in

mathematics. His first piece, The Grounde of Artes,

published in 1543, discussed the advantages of the

Hindu-Arabic numerals and general arithmetic tech-

niques useful for commerce. He later published an

extended version of this work with the same name in

1552, which also explored the theory of

WHOLE NUM

-

BERS

and

RATIONAL NUMBERS

. Around this time Recorde

also published Pathwaie to Knowledge, an abridged ver-

sion of Euclid’s famous work, T

HE

E

LEMENTS

.

Recorde is perhaps best remembered for his 1557

piece The Whetstone of Witte, considered the first sig-

nificant text on the topic of algebra written in English.

Its title is a pun not understood today. Early algebraists

used the Latin word cosa for “thing,” meaning the

unknown variable in an equation. This word closely

resembles the Latin word cos for “whetstone,” a stone

for sharpening tools. Thus Recorde’s title is referring to

the art of algebra as a device for sharpening one’s wit.

In this famous text, Recorde explains the theory of

equations and the arithmetic of square roots. He solves

QUADRATIC

equations (but dismisses negative solu-

tions). He notes, in particular, that for an equation of

the form x2= ax – b, the term aequals the sum of the

two roots of the equation, and the term btheir prod-

uct. (We see this today by noting that (x– r1)(x– r2) =

x2– (r1+ r2)x+ r1r2.) It is in this piece that Recorde

introduces the symbol “=” for equality, noting that the

use of two parallel line segments is apt “bicause noe 2

thynges can be moare equalle.”

Recorde died in London, England, in 1558. The

exact date of death is not known.

rectangle A

QUADRILATERAL

with all four angles

being right angles is called a rectangle. A

SQUARE

is

considered a rectangle.

The

AREA

of a rectangle is the product of its length

and its breadth, and its

PERIMETER

is double the sum of

these two quantities:

area = length ×breadth

perimeter = 2 ×length + 2 ×breadth

There are only two rectangles with integer side-

lengths whose areas and perimeters have the same

numerical value: the 3 ×6 rectangle of area/perimeter

18, and 4 ×4 square of area/perimeter 16.

The two

DIAGONAL

s of a rectangle are equal in

length and bisect each other, and the rectangle is the

only quadrilateral with this property. Carpenters mea-

sure diagonal lengths to check that their work is rect-

angular. One can also use this observation to draw

surprisingly accurate rectangles on the ground using

only two pieces of rope equal in length and a piece of

sidewalk chalk: tie the two ropes together at their mid-

points and pull them taut to form the two bisecting

diagonals of a rectangle. Mark the endpoints of the

ropes, these are the corners of the rectangle, and draw

its sides using a taut rope as a guide.

See also

GOLDEN RECTANGLE

.

recurrence relation (recursive relation, reduction for-

mula) An equation that allows one to calculate the

successive values of a

FUNCTION

or a

SEQUENCE

once an

initial set of values is given is called a recurrence rela-

tion. For example, the F

IBONACCI NUMBERS

Fnare com-

pletely specified by the recurrence relation Fn= Fn–1 +

Fn–2 for n> 2 once we are told that F1= 1 and F2= 1.

The number of moves required to solve the

TOWER OF

H

ANOI PUZZLE

with Ndiscs, T(N),is given by the

recurrence relation T(N+ 1) = 2T(N) + 1 with T(1) = 1.

The recurrence relation an+1 = an+ 3 with a1= 7 defines

the arithmetic sequence 7, 10, 13, 16,…

In

INTEGRAL CALCULUS

, the method of

INTEGRATION

BY PARTS

is often used to establish important reduction

formulae. For example, setting u= xnand v′= exin the

integral below establishes the relation ∫xnexdx = xnex–

n∫xn–1exdx. Repeated application of this formula allows

one to eventually evaluate the given integral.

Other reduction formulae include:

See also

DYNAMICAL SYSTEM

;

LOGISTIC GROWTH

;

RECURSIVE DEFINITION

.