expressed in radians (for more about
angles, sines, cosines, and polar forms,
see “Geometry and Trigonometry”). This
is seen in equation form as: r(cos
i
sin ), in which r is the radius vector, is
the angle, and iis the imaginary number.Is there such a thing as a perfect
number?
Yes, there is such a thing as a perfect
number, but it is not what we think of as
true perfection. To mathematicians, per-
fect numbers are somewhat rare. They
are defined as a natural number (or posi-
tive integer) in which the sum of its posi-
tive divisors (or the bottom number in a
fraction that divides the number to equal
another whole number, and includes 1
but not the number itself) is the number
itself. For example, 6 is considered a per-
fect number because its divisors are 1, 2,
and 3—or 1 2 3 6. The next perfect numbers are 28 (1 2 4 7 14),
496; 8,128; 33,550,336; 8,589,869,056; 137,438,691,328; 2,305,843,008,139,952,128,
and so on. Larger and larger perfect numbers are still being discovered, especially with
the help of today’s faster and more memory-packed computers.What is meant by one-to-one correspondence?
A one-to-one correspondence means just what it implies: that the number of objects,
numbers, or whatever is the same as the set of other objects, numbers, or whatever.
(In set theory, the one-to-one correspondence means something different; for more
about set theory, see “Foundations of Mathematics.”)
Everyone has no doubt had contact with one-to-one correspondence without even
thinking about it. For example, there is a one-to-one correspondence of the number
10 to the number of fingers on both hands (ten). Counting a deck of cards is a one-to-
one correspondence—each number, from 1 to 52, representing a card in the deck.
When you compare two decks of cards, putting the cards side-by-side to equal 52 in
each deck can also be considered a one-to-one correspondence.
Not everything is counted in such a way. For example, when mathematicians want
to know the size of an unknown quantity, they put the unknown quantity in a one-to-
78 one correspondence with a known quantity.
Engineers, such as this man making calculations on
a CAD (computer aided design) of an electric motor
core, employ complex mathematics every day to per-
form their jobs. Taxi/Getty Images.