or x + 2 = –5, and so x equals either 3 or –7. Alternatively, one can read the equation as |x – (–2)| = 5, interpreting it to mean that x is a point a distance of five
units from –2. Five units to the left means x is the point
–7; five units to the right means x is 3.
The notion of absolute value was not made explicit
until the mid-1800s. KARL WEIERSTRASS, in 1841, was
the first to suggest a notation for it—the two vertical
bars we use today. Matters are currently a little confusing, however, for mathematicians today also use this
notation for the length of a VECTOR and for the MODULUS 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 + a for all numbers a.
4. Inverse: For each number a there is another number,
denoted “–a,” so that: a + (–a) = 0 = (–a) + a.
5. Commutativity: For all numbers a and b we have:
a + b = b + a.
Having identified these five properties, mathematicians 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 satisfies 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. GROUP
THEORY is the study of all the results that follow from
these basic five axioms without reference to a particular mathematical system.
The study of RINGs and FIELDs considers mathematical systems that permit two fundamental operations (typically called addition and multiplication).
Allowing for the additional operation of scalar multiplication leads to a study of VECTOR SPACEs.
The theory of algebraic structures is highly developed. The study of vector spaces, for example, is so
extensive that the topic is regarded as a field of mathematics 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 c and corresponding
angles A, B, C opposite those sides, satisfies:
The angle C is acute only if cosC > 0, that is, only if a
2
+ b
2 > c
2
. Thus a triangle a, b, c is acute if, and only if,
the following three inequalities hold:
cosC
a b c
ab
=
+ −
2
2
2
2
acute angle 5
units from –2. Five units to the left means x is the point
–7; five units to the right means x is 3.
The notion of absolute value was not made explicit
until the mid-1800s. KARL WEIERSTRASS, in 1841, was
the first to suggest a notation for it—the two vertical
bars we use today. Matters are currently a little confusing, however, for mathematicians today also use this
notation for the length of a VECTOR and for the MODULUS 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 + a for all numbers a.
4. Inverse: For each number a there is another number,
denoted “–a,” so that: a + (–a) = 0 = (–a) + a.
5. Commutativity: For all numbers a and b we have:
a + b = b + a.
Having identified these five properties, mathematicians 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 satisfies 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. GROUP
THEORY is the study of all the results that follow from
these basic five axioms without reference to a particular mathematical system.
The study of RINGs and FIELDs considers mathematical systems that permit two fundamental operations (typically called addition and multiplication).
Allowing for the additional operation of scalar multiplication leads to a study of VECTOR SPACEs.
The theory of algebraic structures is highly developed. The study of vector spaces, for example, is so
extensive that the topic is regarded as a field of mathematics 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 c and corresponding
angles A, B, C opposite those sides, satisfies:
The angle C is acute only if cosC > 0, that is, only if a
2
+ b
2 > c
2
. Thus a triangle a, b, c is acute if, and only if,
the following three inequalities hold:
cosC
a b c
ab
=
+ −
2
2
2
2
acute angle 5