a+ b

––

2

mean 333

a2+ b2

––

a+ b

1

–

an

1

–

a2

1

–

a1

1

–

2

the Tzolkin followed a 260-day year (which happens to

be the number of days between the two days of the

year in which the Sun is directly overhead in the

Yucatán Peninsula). The second calendar, the Haab, the

agricultural calendar, was the usual 365-day year,

divided into 18 months of 20 days, and a short 5-day

month called the Wayeb. The two calendars coincided

every 18,980 days (the lowest common multiple of 260

and 365) that is, every 52 years, yielding a period that

constituted a Mayan “century.” The Maya were also

fully aware that the most visible planet in the heavens,

Venus, returned to its exact same position every 584

days, a number that happens to divide 2 ×18,980.

That Venus returns to its same position precisely at the

passing of exactly two “centuries,” that is every 104

years, was of profound religious significance to the

Maya people.

It seems that the Maya had no standard methods for

multiplying or dividing large numbers, and they seemed

never to have developed the concept of a fraction. Yet

despite the cumbersome nature of their notational sys-

tem, the Maya performed some exceptionally accurate

astronomical computations. For instance, they correctly

calculated the exact length of a year to be 365.242 days,

and the length of the lunar month to be 29.5302 days.

(These results were not presented in terms of decimals,

of course. Records show, for example, that the Maya

computed that 149 lunar months span exactly 4,400

days.) The Maya made their astronomical observations

using tools no more sophisticated than a pair of sticks

tied together at a right angle through which to observe

the planets.

mean A mean of two numbers aand bis a number m

between aand bthat, in some sense, represents the

“middle” of the two numbers. The most common mea-

sure of “mean” is the average or arithmetic mean given

by m= . It represents the location on the number

line half way between positions aand b. Alternatively,

one could consider the geometric mean given by m=

√

–

ab. This represents the side-length of a square whose

area is the same as that of an a×brectangle.

In the fourth century

B

.

C

.

E

., members of the later

Pythagorean school identified 10 means, now called the

neo-Pythagorean means. For instance, the first two are

the arithmetic and geometric means. The third mean,

given by , is called the harmonic mean, and

the fourth, m= , is the counterharmonic mean.

All “means” have the property that if the numbers a

and bare each multiplied by k, then mis also multi-

plied by k.

The notion of mean can be extended to that of

more than two numbers. Given nnumbers a1,a2,…,an

we set:

For example, the arithmetic mean of 3, 8, and 9 is

20/3 = 6 2/3, and their geometric mean is 3

√

––

3·8·9 = 6. It

is a theorem of algebra that, for any set of positive

numbers a1,a2,…,an, the arithmetic mean is always

greater than or equal to the geometric mean:

This is called the arithmetic–geometric-mean inequal-

ity. (For the case with just two numbers aand b, the

statement (a+ b) ≥ √

–

ab is equivalent to the patently

true statement (a– b)2≥0.) By applying this inequality

to the numbers , ,…, , the harmonic-geometric

inequality follows:

In statistics, the arithmetic mean µof a set of

observations a1,a2,…,anis called a sample mean. The