a

–

b

1

––

x2+ 1

442 reciprocal

1

–

x

digit if necessary, and proceeds to the next column digits

one place to the left to repeat this process again. It is not

clear, then, how one should proceed when asked to add

two infinite decimal expansions. (What is the rightmost

digit with which to begin?)

In the mid-1800s, as mathematicians attempted to

establish a sound theoretical footing for

CALCULUS

,

scholars came to realize the necessity of a rigorous

understanding of what is meant by a “real number.” In

particular, in order to validate the three key theorems

on which calculus rests (namely, the

EXTREME

-

VALUE

THEOREM

, the

INTERMEDIATE

-

VALUE THEOREM

, and the

MEAN

-

VALUE THEOREM

), it was essential that the real-

number system be proved “complete,” in the sense that

no numbers are “missing” from the system. (For exam-

ple, the square root of 2 is missing from the set of

rational numbers).

French mathematician A

UGUSTIN

-L

OUIS

C

AUCHY

(1789–1857) proposed a definition of a real number

using Cauchy sequences. (This approach is based on

the idea that every irrational number is the

LIMIT

of a

sequence of rational numbers.) Although it is easy to

perform arithmetic operations on Cauchy sequences,

the completeness of the real-number system created is

not immediately clear. In 1872, German scholar J

ULIUS

W

ILHELM

R

ICHARD

D

EDEKIND

(1831–1916) proposed

the notion of a D

EDEKIND CUT

as a means of defining

the real numbers. This approach has the advantage of

making the completeness of the real-number system

readily apparent, but the disadvantage of making the

arithmetic manipulations of the real numbers less natu-

ral. Nonetheless, both approaches are valid and are

used today as the means for defining the real numbers.

The work of G

EORG

C

ANTOR

(1845–1918), with his

famous

DIAGONAL ARGUMENT

, shows that the

CARDI

-

NALITY

of the reals is greater than that of the rationals.

It is not surprising then that a more sophisticated

approach to understand the real-number system is

needed than that for understanding the rationals.

Today the set of real numbers is denoted Rand is

depicted geometrically as the set of points on a

NUMBER

LINE

. The real numbers constitute a mathematical

FIELD

.

See also

NUMBER

;

ORDER PROPERTIES

.

reciprocal (multiplicative inverse) The number 1

divided by a quantity is called the reciprocal of that

quantity. For example, the reciprocal of 2 is 1/2, and

the reciprocal of x2+ 1 is . The product of a

quantity and its reciprocal is always 1.

The reciprocal of any

FRACTION

is the corre-

sponding fraction with numerator and denominator

interchanged: . The reciprocal of a

COMPLEX

NUMBER

x+ iy is simplified by multiplying the numera-

tor and denominator each by the conjugate of the com-

plex number:

This is the same process as

RATIONALIZING THE

DENOMINATOR

of a fractional quantity.

An equation that is unchanged (that is, has the

same set of solutions) if the variables in that equation

are replaced by their reciprocals is called a reciprocal

equation. For example, x2– 3x+ 1 = 0 is a reciprocal

equation, since replacing xby yields

+ 1 = 0 which simplifies to 1 – 3x+ x2= 0.

The reciprocal series of a given

SERIES

is the series

whose terms are the reciprocals of the terms of the

given series. For example, any

HARMONIC SERIES

is the

reciprocal series of an

ARITHMETIC SERIES

.

See also

INVERSE ELEMENT

.

Recorde, Robert (ca. 1510–1558) Welsh Algebra

Born in Tenby, Wales, ca. 1510 (his exact birth date is

not known), mathematician Robert Recorde is remem-

bered for his instrumental work in establishing general

mathematics education in England as well as for intro-

ducing

ALGEBRA

to that country. Recorde wrote a num-

ber of influential elementary textbooks, all written in

English and all using clear and simple terminology. He

is also remembered for inventing the symbol “=” to

denote equals.

Recorde received a medical degree from Cambridge

University in 1545 and practiced medicine in London

for several years before being appointed as the general

surveyor of mines and monies in Ireland by King

Edward VI. It is worth noting that Recorde produced

the first coin in England to have the date written with