relative error 445

discussed the means to convert these values to ones

appropriate for a circle of a different radius. His

work also outlined all the basic theory of trigonome-

try, along with applications to both plane and spheri-

cal geometry. Regiomontanus’s work was extremely

influential.

Regiomontanus was an able astronomer. In 1472

he made detailed observations of a comet sufficiently

accurate enough for scholars to later identify it as Hal-

ley’s comet. (Halley was the first to compute the length

of the orbit of the comet 210 years later.) Regiomon-

tanus also devised a mathematical method for deter-

mining longitudinal position on the E

ARTH

by making

use of the location of the Moon in the sky. He died in

Rome, Italy, on July 8, 1476.

regression If a

SCATTER DIAGRAM

indicates a rela-

tionship between two variables in a scientific study,

then any attempt to quantify that relationship is

called regression. For example, if an experiment

yields data points (x1, y1), (x2, y2), …, (xn, yn), one

can attempt to find a formula y= f(x) that “best fits”

the data. The

LEAST SQUARES METHOD

seeks to mini-

mize the sum of squares:

using techniques of

CALCULUS

. This approach works

particularly well if one suspects a linear relationship

f(x) = ax + b, but it can be used too for possible

quadratic relationships f(x) = ax2+ bx + c, and sus-

pected relationships following other simple formulae.

See also

CORRELATION COEFFICIENT

.

relation (relationship) Any pairing between elements

of one set with some elements of another, or elements

of one set with other elements of the same set, is called

a relation. For example, consider the set of all the peo-

ple of the world. Then the “mothered by” correspon-

dence Mthat associates to each person his or her

biological mother is a relation. For the set of real num-

bers, the “greater than” relation Gdeclares ato be

related to bif ais greater than b, and the “circle” rela-

tion Csets xas related to yif the two numbers satisfy

the equation x2+ y2= 1.

If two members aand bof a set are related by a

relation R, then we write aRb. For instance, if Brian’s

mother is Joan, then “Brian MJoan” (that is, Brian is

“mothered by” Joan). We also have that 3G2, since 3 is

greater than 2, and 1C0, since the point (1,0) lies on

the unit circle. (Note: the “greater than” relation Gis

usually denoted “>”.)

If the relation being discussed is understood, it is

also convenient to write two elements that are related

simply as an ordered pair. For instance, in the mother-

ing relation Mwe have (Brian, Joan), but not (Joan,

Brian). Mathematicians generally prefer the ordered-

pair presentation of a relation and in fact will define a

relation to simply be any collection of ordered pairs.

This has the advantage that if the elements under dis-

cussion are numbers, then one can graph the points that

satisfy a given relation. For example, under the “greater

than” relation G, a pair (a,b) satisfying this relation cor-

responds to a point in the plane above the diagonal line

y= x. The set of all pairs (x,y) that belongs to the circle

relation Cforms a circle in the xy-plane.

A relation is called a

FUNCTION

if no element in the

first set (first coordinate) of the relation is matched

with more than one element of the second set (second

coordinate). For example, as Brian, as with every

human being, can be matched with only one biological

mother, the “mothered by” relationship is a function.

Since 3 > 2 and 3 > 1, for example, the “greater than”

relation is not a function. The “circle” relation Calso

fails to be a function since, for instance, both (0,1) and

(0, –1) belong to the relation.(We have 0C1 and 0C–1.)

If we consider, instead, the “parabola” relation P

defined as the set of all points (x, y) that satisfy y= x2,

then this relation is a function: if xPy1and xPy2, then it

must be the case that y1= y2. (We have y1= y2= x2.)

relative error If the

ERROR

or uncertainty in a mea-

surement of a quantity is expressed as a ratio to the

total measurement, then the error is called a relative

error. For example, when 1.2 is used as an approxima-

tion for the quantity 1.16, then the relative error is

0.04/1.16 ≈0.034. If, in measuring the height of a

building as 62 ft, engineers used a device that measures

to the nearest half a foot, then the height of the build-

ing should be written 62 ±0.5 ft, and the relative error

of this measurement is 0.5/62 ≈0.008.

See also

PERCENTAGE ERROR

.