Basic Math and Pre-Algebra For Dummies (108 page)

You can turn any real number into a complex number by just adding 0
i
(which equals 0):

• 3 = 3 + 0
  i
   –12 = –12 + 0
  i
   3.14 = 3.14 + 0
  i
  

These examples show you that the real numbers are just a part of the larger set of complex numbers.

Like the rational numbers and real numbers (check out the sections earlier in this chapter), the set of complex numbers is closed under the Big Four operations. In other words, if you add, subtract, multiply, or divide any two complex numbers, the result is always another complex number.

The
transfinite numbers
are a set of numbers representing different levels of infinity. Consider this for a moment: The counting numbers (1, 2, 3, ...) go on forever, so they're infinite. But there are
more
real numbers than counting numbers.

In fact, the real numbers are
infinitely more infinite
than the counting numbers. Mathematician Georg Cantor proved this fact. He also proved that, for
every level of infinity, you can find another level that's even higher. He called these ever-increasing levels of infinity
transfinite,
because they transcend, or go beyond, what you think of as infinite.

The lowest transfinite number is ℵ
₀
(aleph null), which equals the number of elements in the set of counting numbers ({1, 2, 3, 4, 5, ...}). Because the counting numbers are infinite, the familiar symbol for infinity (∞) and ℵ
₀
mean the same thing.

The next transfinite number is ℵ
₁
(aleph one), which equals the number of elements in the set of real numbers. This is a higher order of infinity than ∞.

The sets of integers, rational, and algebraic numbers all have ℵ
₀
elements. And the sets of irrational, transcendental, imaginary, and complex numbers all have ℵ
₁
elements.

Higher levels of infinity exist, too. Here's the set of transfinite numbers:

• {ℵ
  ₀
  , ℵ
  ₁
  , ℵ
  ₂
  , ℵ
  ₃
  , ...}
  

The ellipsis tells you that the sequence of transfinite numbers goes on forever — in other words, that it's infinite. As you can see, on the surface, the transfinite numbers look similar to the counting numbers (in the first section of this chapter). That is, the set of transfinite numbers has ℵ
₀
elements.

Mark Zegarelli
is a math and test prep teacher, as well as the author of eight
For Dummies
(Wiley) books, including
SAT Math For Dummies,
ACT Math For Dummies,
and
Calculus II For Dummies
. He holds degrees in both English and math from Rutgers University and lives in Long Branch, New Jersey, and San Francisco, California.

I dedicate this book to the memory of my mother, Sally Ann Zegarelli (Joan Bernice Hanley).

Writing this second edition of Basic Math & Pre-Algebra For Dummies was an entirely enjoyable experience, thanks to the support and guidance of Lindsay Lefevere of John Wiley & Sons, Inc., editors Tracy Barr and Krista Hansing, and technical reviewer Mike McAsey. And many thanks to my assistant, Chris Mark, for his unfailing diligence and enthusiasm.

And thanks to the great folks at Borderlands Café, on Valencia Street in San Francisco, for creating a quiet and friendly place to work.

Finally, a shout out to my awesome nephews, Jake and Ben.

Publisher's Acknowledgments

Executive Editor:
Lindsay Sandman Lefevere

Project Editor:
Tracy L. Barr

Copy Editor:
Krista Hansing

Technical Editor:
Michael McAsey

Project Coordinator:
Sheree Montgomery

Cover Image:
©Jiri Moucka illustrations/Alamy