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 master-
piece. This work completely changed the direction of
mathematical research in number theory for the cen-
tury that followed. Riemann had managed to connect
the notions of geometry and space to complex func-
tions and, now, to the study of numbers. This signifi-
cant 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 com-
plete 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 P
YTHAGORAS
’
S THEOREM
shows this) and used knot-
ted 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
; E
GYPTIAN MATHEMAT
-
ICS
;
RADIAN MEASURE
.
right-handed/left-handed system In three-dimen-
sional space one identifies the location of points by
making reference to a set of three mutually perpendicu-
lar 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 direc-
tion 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 orien-
tation 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 clying 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 vec-
tors 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-coordi-
nate axes is said to be positively oriented if a counter-
clockwise rotation is required to turn the positive
x-axis onto the positive y-axis (through the smallest
angle possible) and negatively oriented if instead clock-
wise motion is needed.
See also C
ARTESIAN COORDINATES
;
CROSS PRODUCT
.
ring Motivated by the question of what makes arith-
metic work the way it does, mathematicians have iden-
tified 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 Rtogether with two methods
for combining elements of Rto produce new elements of
R, usually called addition “+” and multiplication “*,”
satisfying the following rules:
1. Commutative Law for Addition: For all elements a
and bof R, we have: a+ b= b+ a.
2. Associative Law of Addition: For all elements a, b,
and cof R, we have: a+ (b+ c) = (a+ b) + c.
3. Existence of a Zero: The set Rhas an element 0
with the property that a+ 0 = 0 + a= afor all ele-
ments ain R.
4. Existence of Additive Inverses: For each element a
in Rthere is an element, denoted –a, such that a+
(–a) = (–a) + a= 0.
5. Associative Law of Multiplication: For all elements
a, b, and cof Rwe have: a*(b*c) = (a*b)*c.
6. Existence of a 1: There is an element 1 of Rwith
the property that a*1 = 1*a= afor all ain R.
7. Distributive Laws: For all elements a, b, and cof R
we have: a*(b+ c) = a*b + a*a and (b+ c)*a= b*a
+ c*a.
π
–
2
448 right angle