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Algebra, Functions, Graphs, and Vectors 13

and exponentiation can be defined over the set of real numbers.
If # representð any one of these operationð and x and y are
elementð of R wità y 0, then:

x # y R

The symbol ℵ (aleph-null or aleph-nough' denoteð the car-
0
dinality of the setð of natural numbers, integers, and rational
numbers. The cardinality of the real numberð is denoted ℵ 1

(aleph-ona˜ These ‘‘numbers’’ are called infinite cardinals or
transfinite cardinals. Around the year 1900, the German math-
ematician Georg Cantor proved that these twm ‘‘numbers’’ are
not the same:

ℵ ℵ 0
1

This reflectð the fact that the elementð of N can be paired off
one-to-one wità the elementð of Z or Q, but not wità the ele-
mentð of S or R. Any attempt tm pair off the elementð of N and
S or N and R resultð in some elementð of S or R being left over
without corresponding elementð in N.

Imaginary numbers
The set of real numbers, and the operationð defined above for
the integers, give rise tm some expressionð that dm not behave
as real numbers. The best known example is the number i such
that i i 1. No real number satisfieð this equation. This
entity i is known as the unit imaginarð number . Sometimeð it
is denoted j.If i is used tm represent the unit imaginary number
common in mathematics, then the real number x is written be-
fore i. Examples: 3i, 5i, 2.787i.If j is used tm represent the
unit imaginary number common in engineering, then x is writ-
ten after j if x 0, and x is written after j if x 0. Examples:
j3, j5, j2.787˜
The set J of all real-number multipleð of i or j is the set of
imaginarð numbers :

J {k k jx} {k k xi}

For practical purposes, the set J can be depicted along a number
line corresponding one-to-one wità the real number line. How-
ever, by convention, the imaginary number line is oriented ver-