inverse element 279

The splitting game

application of E

ULER

’

S THEOREM

shows that any soccer

ball design that uses hexagons and pentagons, no mat-

ter how irregular those shapes may be, with three edges

meeting at each vertex, must include precisely 12 pen-

tagons. (The number of hexagons can vary.) The count

of pentagons is thus an invariant in soccer ball design.

Some invariants can be quite surprising. For exam-

ple, consider the following theoretical exercise:

Suppose 10 ft of length is added to a rope

that was just long enough to wrap snugly

around the equator of the Earth. Imagine that

the rope is again wrapped around the equator,

but this time—due to its extra length—it hov-

ers just above the ground. How high is the

gap between the ground and the suspended

rope?

To answer this question, let Rdenote the radius of the

Earth. The length of the extended rope is 2

π

R+ 10 =

feet, which corresponds to the circumference

of a circle of radius R+ feet. This shows that the

rope hovers ≈1.6 ft off the ground. Notice that the

answer, apart from being surprisingly large in value,

does not depend on the value of R. This means that the

size of the planet is immaterial. Thus if 10 ft of length

is added to a rope that fits snugly around the equator

of any planet—Mars, Jupiter, or a planet the size of a

pea—the extended rope will always hover ft off the

ground. This value is an invariant for the problem.

As another example, consider the following famous

“splitting game”:

Write the number 12 at the top of a page and

below it write a pair of positive whole numbers

that sum to 12, say, 7 and 5. On the side of the

paper record the product 7 ×5 = 35. Now

write below 7 a pair of numbers that sum to 7,

say, 3 and 4, and record the product 3 ×4 =

12. Continue in this manner, “splitting” each

number that appears in the diagram into two

and recording the product of the pair of num-

bers chosen. Do this until the number 1

appears 12 times. The following represents one

possible such splitting diagram:

Now sum all the products recorded. What

value is obtained?

Surprisingly, no matter which splitting diagram one

constructs, the sum of products is an invariant of the

game and will always have value 66. (The number 66

happens to be the 11th

TRIANGULAR NUMBER

. In gen-

eral, if one begins this game with a number N, then the

invariant that arises in the game is the (N–1)th trian-

gular number.)

inverse element An element of a set that, when com-

bined with another element produces the

IDENTITY ELE

-

MENT

of the set, is called an inverse element. More

precisely, if a set Scomes equipped with a

BINARY

OPERATION

“*” and an identity element e, then an

inverse for an element aof the set is another element b

such that a*b = b*a = e.

For example, for the set of numbers under the

operation of addition, the inverse of any number ais its

negative –a. In this context, the identity element is zero

and we do indeed have:

a+ (–a) = (–a) + a= 0

Under the operation of multiplication, the identity ele-

ment is 1, and the inverse of any (nonzero) number ais

its reciprocal :

a×= ×a= 1

Each and every element of a

GROUP

is required to have

an inverse. The inverse of the identity element is itself.

1

–

a

1

–

a

1

–

a

5

–

π

5

–

π

5

–

π