numbers. In a written report to the academy, Riemann
presented his work on the famous zeta function and
his method of extending the scope of the function to
include complex numbers as inputs—another masterpiece. This work completely changed the direction of
mathematical research in number theory for the century that followed. Riemann had managed to connect
the notions of geometry and space to complex functions and, now, to the study of numbers. This significant achievement provided mathematicians the means
to translate insights and advances in one disparate
field into results and discoveries in another.
Riemann suffered from ill health most of his life
and died of tuberculosis at the age of 39 in Selasca,
Italy, on July 20, 1866.
right angle An ANGLE equal to one-quarter of a complete revolution is called a right angle. Such an angle
has measure 90° or
radians. The corner of a square
is a right angle.
One could say that a right angle is the “correct”
angle to use in architecture and the construction of
buildings. Egyptian architects of 1500 B.C.E. were aware
that a 3–4–5 triangle contains a right angle (the converse
of PYTHAGORAS’S THEOREM shows this) and used knotted ropes 3 + 4 + 5 = 12 units long to quickly construct
these triangles at a building site. A right angle is also the
angle made if one were to make a perfect right turn.
See also DEGREE MEASURE; EGYPTIAN MATHEMATICS; RADIAN MEASURE.
right-handed/left-handed system In three-dimensional space one identifies the location of points by
making reference to a set of three mutually perpendicular number lines, usually called the x-, y-, and z-axes,
intersecting at a point called the origin. There are two
possible ways to orient the axes.
An xyz-coordinate system is called right-handed if,
taking the right hand, the positive x-axis points in the
direction of the thumb, the positive y-axis in the direction of the index finger, and the positive z-axis in the
direction of the (bent) middle finger. An xyz-coordinate
system is called left-handed if it follows the directions
of these fingers of the left hand instead.
Reversing the direction of any one of the axes, or
switching the labels of any two axes, changes the orientation of the coordinate system. Mathematicians have
settled on the convention of preferring right-handed
systems over left-handed ones.
More generally, three vectors a, b, and c, in that
order, in three-dimensional space form a right-handed
system if pointing the thumb of the right hand in the
direction of a, and the index finger in the direction of b
has vector c lying on the side of the palm of the hand
(this is the direction the middle finger would need to
curl to point in direction c). Alternatively, the three vectors form a right-handed system if the TRIPLE VECTOR
PRODUCT a· (b × c) is a positive number.
In two-dimensional space, coordinate axes can
again be oriented one of two ways. A set of xy-coordinate axes is said to be positively oriented if a counterclockwise rotation is required to turn the positive
x-axis onto the positive y-axis (through the smallest
angle possible) and negatively oriented if instead clockwise motion is needed.
See also CARTESIAN COORDINATES; CROSS PRODUCT.
ring Motivated by the question of what makes arithmetic work the way it does, mathematicians have identified seven key principles satisfied by the operations of
addition and multiplication. Today, mathematicians
call any mathematical system satisfying these basic
axioms a ring.
Precisely, a ring is a set R together with two methods
for combining elements of R to produce new elements of
R, usually called addition “+” and multiplication “*,”
satisfying the following rules:
1. Commutative Law for Addition: For all elements a
and b of R, we have: a + b = b + a.
2. Associative Law of Addition: For all elements a, b,
and c of R, we have: a + (b + c) = (a + b) + c.
3. Existence of a Zero: The set R has an element 0
with the property that a + 0 = 0 + a = a for all elements a in R.
4. Existence of Additive Inverses: For each element a
in R there is an element, denoted –a, such that a +
(–a) = (–a) + a = 0.
5. Associative Law of Multiplication: For all elements
a, b, and c of R we have: a*(b*c) = (a*b)*c.
6. Existence of a 1: There is an element 1 of R with
the property that a*1 = 1*a = a for all a in R.
7. Distributive Laws: For all elements a, b, and c of R
we have: a*(b + c) = a*b + a*a and (b + c)*a = b*a
+ c*a.
π
–
2
448 right angle
presented his work on the famous zeta function and
his method of extending the scope of the function to
include complex numbers as inputs—another masterpiece. This work completely changed the direction of
mathematical research in number theory for the century that followed. Riemann had managed to connect
the notions of geometry and space to complex functions and, now, to the study of numbers. This significant achievement provided mathematicians the means
to translate insights and advances in one disparate
field into results and discoveries in another.
Riemann suffered from ill health most of his life
and died of tuberculosis at the age of 39 in Selasca,
Italy, on July 20, 1866.
right angle An ANGLE equal to one-quarter of a complete revolution is called a right angle. Such an angle
has measure 90° or
radians. The corner of a square
is a right angle.
One could say that a right angle is the “correct”
angle to use in architecture and the construction of
buildings. Egyptian architects of 1500 B.C.E. were aware
that a 3–4–5 triangle contains a right angle (the converse
of PYTHAGORAS’S THEOREM shows this) and used knotted ropes 3 + 4 + 5 = 12 units long to quickly construct
these triangles at a building site. A right angle is also the
angle made if one were to make a perfect right turn.
See also DEGREE MEASURE; EGYPTIAN MATHEMATICS; RADIAN MEASURE.
right-handed/left-handed system In three-dimensional space one identifies the location of points by
making reference to a set of three mutually perpendicular number lines, usually called the x-, y-, and z-axes,
intersecting at a point called the origin. There are two
possible ways to orient the axes.
An xyz-coordinate system is called right-handed if,
taking the right hand, the positive x-axis points in the
direction of the thumb, the positive y-axis in the direction of the index finger, and the positive z-axis in the
direction of the (bent) middle finger. An xyz-coordinate
system is called left-handed if it follows the directions
of these fingers of the left hand instead.
Reversing the direction of any one of the axes, or
switching the labels of any two axes, changes the orientation of the coordinate system. Mathematicians have
settled on the convention of preferring right-handed
systems over left-handed ones.
More generally, three vectors a, b, and c, in that
order, in three-dimensional space form a right-handed
system if pointing the thumb of the right hand in the
direction of a, and the index finger in the direction of b
has vector c lying on the side of the palm of the hand
(this is the direction the middle finger would need to
curl to point in direction c). Alternatively, the three vectors form a right-handed system if the TRIPLE VECTOR
PRODUCT a· (b × c) is a positive number.
In two-dimensional space, coordinate axes can
again be oriented one of two ways. A set of xy-coordinate axes is said to be positively oriented if a counterclockwise rotation is required to turn the positive
x-axis onto the positive y-axis (through the smallest
angle possible) and negatively oriented if instead clockwise motion is needed.
See also CARTESIAN COORDINATES; CROSS PRODUCT.
ring Motivated by the question of what makes arithmetic work the way it does, mathematicians have identified seven key principles satisfied by the operations of
addition and multiplication. Today, mathematicians
call any mathematical system satisfying these basic
axioms a ring.
Precisely, a ring is a set R together with two methods
for combining elements of R to produce new elements of
R, usually called addition “+” and multiplication “*,”
satisfying the following rules:
1. Commutative Law for Addition: For all elements a
and b of R, we have: a + b = b + a.
2. Associative Law of Addition: For all elements a, b,
and c of R, we have: a + (b + c) = (a + b) + c.
3. Existence of a Zero: The set R has an element 0
with the property that a + 0 = 0 + a = a for all elements a in R.
4. Existence of Additive Inverses: For each element a
in R there is an element, denoted –a, such that a +
(–a) = (–a) + a = 0.
5. Associative Law of Multiplication: For all elements
a, b, and c of R we have: a*(b*c) = (a*b)*c.
6. Existence of a 1: There is an element 1 of R with
the property that a*1 = 1*a = a for all a in R.
7. Distributive Laws: For all elements a, b, and c of R
we have: a*(b + c) = a*b + a*a and (b + c)*a = b*a
+ c*a.
π
–
2
448 right angle