acute angle 5

or x+ 2 = –5, and so xequals either 3 or –7. Alterna-

tively, one can read the equation as |x– (–2)|= 5, inter-

preting it to mean that xis a point a distance of five

units from –2. Five units to the left means xis the point

–7; five units to the right means xis 3.

The notion of absolute value was not made explicit

until the mid-1800s. K

ARL

W

EIERSTRASS

, in 1841, was

the first to suggest a notation for it—the two vertical

bars we use today. Matters are currently a little confus-

ing, however, for mathematicians today also use this

notation for the length of a

VECTOR

and for the

MODU

-

LUS

of a

COMPLEX NUMBER

.

abstract algebra Research in pure mathematics is

motivated by one fundamental question: what makes

mathematics work the way it does? For example, to a

mathematician, the question, “What is 263 ×178 (or

equivalently, 178 ×263)?” is of little interest. A far

more important question would be, “Why should the

answers to 263 ×178 and 178 ×263 be the same?”

The topic of abstract algebra attempts to identify

the key features that make

ALGEBRA

and

ARITHMETIC

work the way they do. For example, mathematicians

have shown that the operation of

ADDITION

satisfies

five basic principles, and that all other results about the

nature of addition follow from these.

1. Closure: The sum of two numbers is again a number.

2. Associativity: For all numbers a, b, and c, we have:

(a+ b) + c= a+ (b+ c).

3. Zero element: There is a number, denoted “0,” so

that: a+ 0 = a= 0 + afor all numbers a.

4. Inverse: For each number athere is another number,

denoted “–a,” so that: a+ (–a) = 0 = (–a) + a.

5. Commutativity: For all numbers aand bwe have:

a+ b= b+ a.

Having identified these five properties, mathemati-

cians search for other mathematical systems that may

satisfy the same five relations. Any fact that is known

about addition will consequently hold true in the new

system as well. This is a powerful approach to matters.

It avoids having to re-prove

THEOREMS

and facts about

a new system if one can recognize it as a familiar one in

disguise. For example,

MULTIPLICATION

essentially sat-

isfies the same five

AXIOMS

as above, and so for any

fact about addition, there is a corresponding fact about

multiplication. The set of symmetries of a geometric

figure also satisfy these five axioms, and so too all

known results about addition immediately transfer to

interesting statements about geometry. Any system that

satisfies these basic five axioms is called an “Abelian

group,” or just a

GROUP

if the fifth axiom fails. G

ROUP

THEORY

is the study of all the results that follow from

these basic five axioms without reference to a particu-

lar mathematical system.

The study of

RING

s and

FIELD

s considers mathe-

matical systems that permit two fundamental opera-

tions (typically called addition and multiplication).

Allowing for the additional operation of scalar multi-

plication leads to a study of

VECTOR SPACE

s.

The theory of algebraic structures is highly devel-

oped. The study of vector spaces, for example, is so

extensive that the topic is regarded as a field of math-

ematics in its own right and is called

LINEAR ALGEBRA

.

acceleration See

VELOCITY

.

actuarial science The statistical study of life

expectancy, sickness, retirement, and accident matters

is called actuarial science. Experts in the field are called

actuaries and are employed by insurance companies

and pension funds to calculate risks and relate them to

the premiums to be charged. British mathematician and

astronomer, Edmund Halley (1656–1742) was the first

to properly analyze annuities and is today considered

the founder of the field.

See also

LIFE TABLES

.

acute angle An

ANGLE

between zero and 90°is called

an acute angle. An acute-angled triangle is one whose

angles are all acute. According to the

LAW OF COSINES

, a

triangle with side-lengths a, b, and cand corresponding

angles A, B, C opposite those sides, satisfies:

The angle Cis acute only if cosC> 0, that is, only if a2

+ b2> c2. Thus a triangle a, b, c is acute if, and only if,

the following three inequalities hold: