260 Incan mathematics

If the limits exist, then the improper integrals are said to

converge. (Thus, for example, is a convergent

improper integral.) If the limit does not exist, then the

improper integral is said to diverge. (One can check, for

example, that diverges.) The integral test used

in the study of

CONVERGENT SERIES

makes use of

improper integrals of this type.

It is possible to define even more-general improper

integrals by considering integrands that take the value

infinity at several places within the range of integration,

possibly over an infinite range of integration. One inter-

prets such improper integrals as the sum of several limits.

Incan mathematics The Incan Empire ruled the region

of Peru and its surroundings before the Spanish conquest

of the continent in 1532. It was an extraordinarily

sophisticated civilization. The Inca people had highly

developed systems of agriculture, transportation, road

construction, textile production, and administration, yet,

surprisingly, had no form of writing. Consequently, with

absolutely no cultural records or documents available,

very little is known of the mathematics of these people.

It is known that the Inca recorded numerical infor-

mation with a device known as a quipu, which consisted

of a collection of strings all tied to a common central

object, usually another rope. Administrators simply tied

knots in the strings to record numbers, with each string

representing a different count. Knots were grouped in

clusters to represent quantities of powers of 10. Thus,

for example, the number 283 was represented (back-

ward) by a cluster of three knots placed near the free end

of a string (three units), a space, a cluster of eight knots

in the center of the string (eight 10s), another space, and

finally two knots at the top of the string (two 100s). The

Incas, in effect, were following the

DECIMAL REPRESEN

-

TATION

system we use today. A large space indicated a

digit of zero (to distinguish 208 from 28, for instance).

Strings of different colors were used to signify

what the knots on that string were counting. For

example, white strings may have represented the

counts of cattle, while green strings may have repre-

sented the levels of punishment to be executed for

committing a certain crime.

Each town in the Incan Empire had its own quipca-

mayoc, a “keeper of the knots,” who recorded census

information about the population, the produce, and

weapons of the town. Quipues recording this informa-

tion were sent every year, by relay runners, to the capi-

tal of the empire, Cuzco.

The extent to which the Incan people manipulated

numbers and performed calculations is unclear. A

Spanish priest by the name of José de Acosta, who

lived among the Incas from 1571 to 1586, describes in

his book, Historia natural y moral de las Indias, resi-

dents shuffling kernels of corn on the ground as though

manipulating the beads of an

ABACUS

. Obviously some

kind of numerical computation was being conducted,

but Acosta did not understand the method and was

unable to explain what exactly was being done.

The Inca thought of numbers in very concrete terms.

The concept of “two-ness,” for instance, did not exist

for the Inca, and the count of 2 applied to very specific

things. This is evidenced by the fact that the Inca had

several different words for two, which were used to dis-

tinguish different contexts—the two objects that make a

pair, the 2 that came from one object divided in half, and

the one object and its specific partner that make a pair,

for instance. There was no single all-encompassing word

for the state of having two parts.

It is likely that the Inca people kept astronomical

records, a topic of interest to most ancient cultures, but

there is no direct evidence of this.

incircle A circle that lies inside a triangle and touches

all three sides is called the incircle of that triangle. The

center of that circle is called the incenter of the triangle,

and its radius is called the inradius of the triangle.

It is relatively straightforward to see that for any

triangle there is precisely one circle in the triangle that

touches all three sides. (Erase one side of the triangle

and note that there are infinitely many circles that fit

into the wedge formed by the two remaining sides. In

redrawing the third side, observe that precisely one of

those circles lies tangent to that additional edge.)

The incenter of a triangle is a point that lies the

same distance from each of the three sides of the trian-

gle. Note that any point that is

EQUIDISTANT

from just

two intersecting lines lies on the angle bisector of those

lines (that is, on a line that cuts the angle between the

two given lines in half). Consequently, we have that the

incenter of any triangle lies on each of the angle bisec-

tors of the vertices of the triangle. This establishes: