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 trigonometry, along with applications to both plane and spherical 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 Halley’s comet. (Halley was the first to compute the length
of the orbit of the comet 210 years later.) Regiomontanus also devised a mathematical method for determining longitudinal position on the EARTH 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 relationship 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 (x 1 , y 1 ), (x 2 , y 2 ), …, (x n , y n ), one
can attempt to find a formula y = f(x) that “best fits”
the data. The LEAST SQUARES METHOD seeks to minimize 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) = ax
2 + bx + c, and suspected 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 people of the world. Then the “mothered by” correspondence M that associates to each person his or her
biological mother is a relation. For the set of real numbers, the “greater than” relation G declares a to be
related to b if a is greater than b, and the “circle” relation C sets x as related to y if the two numbers satisfy
the equation x
2 + y
2 = 1.
If two members a and b of a set are related by a
relation R, then we write aRb. For instance, if Brian’s
mother is Joan, then “Brian M Joan” (that is, Brian is
“mothered by” Joan). We also have that 3G2, since 3 is
greater than 2, and 1C 0, since the point (1,0) lies on
the unit circle. (Note: the “greater than” relation G is
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 mothering relation M we have (Brian, Joan), but not (Joan,
Brian). Mathematicians generally prefer the orderedpair 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 discussion 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 corresponds 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 C forms 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 C also
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 = x
2
,
then this relation is a function: if xPy 1 and xPy 2 , then it
must be the case that y 1 = y 2 . (We have y 1 = y 2 = x
2
.)
relative error If the ERROR or uncertainty in a measurement 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 approximation 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 building should be written 62 ± 0.5 ft, and the relative error
of this measurement is 0.5/62 ≈ 0.008.
See also PERCENTAGE ERROR.
y f x
i
i
i
n
−
(
)
=
∑
( )
2
1
relative error 445
appropriate for a circle of a different radius. His
work also outlined all the basic theory of trigonometry, along with applications to both plane and spherical 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 Halley’s comet. (Halley was the first to compute the length
of the orbit of the comet 210 years later.) Regiomontanus also devised a mathematical method for determining longitudinal position on the EARTH 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 relationship 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 (x 1 , y 1 ), (x 2 , y 2 ), …, (x n , y n ), one
can attempt to find a formula y = f(x) that “best fits”
the data. The LEAST SQUARES METHOD seeks to minimize 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) = ax
2 + bx + c, and suspected 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 people of the world. Then the “mothered by” correspondence M that associates to each person his or her
biological mother is a relation. For the set of real numbers, the “greater than” relation G declares a to be
related to b if a is greater than b, and the “circle” relation C sets x as related to y if the two numbers satisfy
the equation x
2 + y
2 = 1.
If two members a and b of a set are related by a
relation R, then we write aRb. For instance, if Brian’s
mother is Joan, then “Brian M Joan” (that is, Brian is
“mothered by” Joan). We also have that 3G2, since 3 is
greater than 2, and 1C 0, since the point (1,0) lies on
the unit circle. (Note: the “greater than” relation G is
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 mothering relation M we have (Brian, Joan), but not (Joan,
Brian). Mathematicians generally prefer the orderedpair 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 discussion 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 corresponds 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 C forms 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 C also
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 = x
2
,
then this relation is a function: if xPy 1 and xPy 2 , then it
must be the case that y 1 = y 2 . (We have y 1 = y 2 = x
2
.)
relative error If the ERROR or uncertainty in a measurement 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 approximation 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 building should be written 62 ± 0.5 ft, and the relative error
of this measurement is 0.5/62 ≈ 0.008.
See also PERCENTAGE ERROR.
y f x
i
i
i
n
−
(
)
=
∑
( )
2
1
relative error 445