absolute convergence 3

1

calculators. It is a useful teaching device to introduce

young children to the notion of place-value and to the

operations of basic arithmetic.

See also

BASE OF A NUMBER SYSTEM

; N

APIER

’

S

BONES

.

Abel, Niels Henrik (1802–1829) Norwegian Algebra

Born on August 5, 1802, Niels Abel might have been

one of the great mathematicians of the 19th century had

he not died of tuberculosis at age 26. He is remembered,

and honored, in mathematics for putting an end to the

three-century-long search for a

SOLUTION BY RADICALS

of the quintic equation. His theoretical work in the top-

ics of

GROUP THEORY

and

ALGEBRA

paved the way for

continued significant research in these areas.

Abel’s short life was dominated by poverty, chiefly

due to the severe economic hardships his homeland of

Norway endured after the Napoleonic wars, exacer-

bated by difficult family circumstances. A schoolteacher,

thankfully, recognized Abel’s talent for mathematics as a

young student and introduced him to the works of

L

EONHARD

E

ULER

, J

OSEPH

-L

OUIS

L

AGRANGE

, and other

great mathematicians. He also helped raise money to

have Abel attend university and continue his studies.

Abel entered the University of Christiania in the city of

Christiania (present-day Oslo), Norway, in 1821.

During his final year of study, Abel began working

on the solution of quintic equations (fifth-degree poly-

nomial equations) by radicals. Although scholars for a

long time knew general formulae for solving for

QUADRATIC

, cubic, and

QUARTIC EQUATION

s using noth-

ing more than basic arithmetical operations on the

COEFFICIENT

s that appear in the equation, no one had

yet found a similar formula for solving quintics. In

1822 Abel believed he had produced one. He shared the

details of his method with the Danish mathematician

Ferdinand Degen in hopes of having the work published

by the Royal Society of Copenhagen. Degen had trouble

following the logic behind Abel’s approach and asked

for a numerical illustration of his method. While trying

to produce a numerical example, Abel found an error in

his paper that eventually led him to understand the rea-

son why general solutions to fifth- and higher-degree

equations are impossible. Abel published this phenome-

nal discovery in 1825 in a self-published pamphlet

“Mémoire sur les équations algébriques où on démontre

l’impossibilité de la résolution de l’équation générale du

cinquième degré” (Memoir on the impossibility of alge-

braic solutions to the general equations of the fifth

degree), which he later presented as a series of seven

papers in the newly established Journal for Pure and

Applied Mathematics (commonly known as Crelle’s

Journal for its German founder August Leopold Crelle).

At first, reaction to this work was slow, but as the repu-

tation of the journal grew, more and more scholars took

note of the paper, and news of Abel’s accomplishment

began to spread across Europe. A few years later Abel

was honored with a professorship at the University of

Berlin. Unfortunately, Abel had contracted tuberculosis

by this time, and he died on April 6, 1829, a few days

before receiving the letter of notification.

In 1830 the Paris Academy awarded Abel, posthu-

mously, the Grand Prix for his outstanding work.

Although Abel did not write in terms of the modern-day

concepts of group theory, mathematicians call groups sat-

isfying the

COMMUTATIVE PROPERTY

“Abelian groups” in

his honor. In 2002, on the bicentenary of his birth, the

Norwegian Academy of Science and Letters created a

new mathematics prize, the Abel Prize, similar to the

Nobel Prize, to be awarded annually.

Research in the field of commutative algebra con-

tinues today using the approach developed by Abel

during his short life. His influence on the development

of

ABSTRACT ALGEBRA

is truly significant.

absolute convergence A

SERIES

containing

positive and negative terms is said to converge abso-

lutely if the corresponding series with all terms

made positive, , converges. For example, the series

converges absolutely because

the corresponding series

converges. (See

CONVERGENT SERIES

.) The “absolute

convergence test” reads:

If converges, then the original series

also converges.

It can be proved as follows: