IRRATIONALS AND IMAGINARY NUMBERS 123

It is not possible to give any short explanation of what the
square root of minus 1 may be said to represent, although it
occurs frequently in calculations involved in wireless and
electricity theories. A full justification of its uses may be
found in advanced mathematical treatises, but it is of interest
to note that its practical use is made possible by the fact,
when it does appear in equations, it is usually cancelled out
before the final answer is reached.

The number ¢ has peculiar properties of its own, the most
obvious of these being that successive powers of it are
repetitive:

i =V —1 5=V —1
12=—1 8= —1
?=—V —1 sy
tt=41 8=41

This produces the relationship:
1%=—1

The numbers 7, ¢ and 7 may be related to each other by the
remarkable equation:

=—1I

whilst the two numbers ¢ and = are featured in Stirling’s
Theorem, which states that:
V/(27n) .e"n" =n! approx.

The introduction of imaginary numbers can have a re-
markable effect upon real numbers. If any integral number is
split into two integral parts and these two parts are multiplied
together, the highest quotient that can result is the square of
half the original number. If the original number be x, then

2
the limit for possible quotients is———<g> ‘

The number 14, for example, may be split into pairs of