{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Continuum - ECE 3250 THE POWER OF THE CONTINUUM Fall 2008...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
ECE 3250 THE POWER OF THE CONTINUUM Fall 2008 My main goal is to demonstrate that R , the set of real numbers, is uncountably infinite. I’ll employ a classic proof technique known as Cantor’s diagonal argument. Invoking Cantor’s argument requires some other results about the real numbers that are interesting and useful in their own right, and I’ll elaborate on those as they arise. Before discussing decimal expansions of real numbers, which will play a crucial role in what follows, I’d like to remind you about an extraordinarily useful piece of machinery that you’ve learned about before. It’s the geometric series. We’ll be talking more formally later on about series (particularly power series), but for now consider this expression: X n =0 γ n . Here, γ is some given real or complex number. The official meaning of the series expression is lim N →∞ N - 1 X n =0 γ n . Of course, the limit’s existence is not guaranteed. Observe, however, that (1 - γ ) N - 1 X n =0 γ n = 1 - γ N , so that N - 1 X n =0 γ n = 1 - γ N 1 - γ . If | γ | < 1, the right-hand side converges as N → ∞ to 1 / (1 - γ ), and hence so does the left-hand side. The bottom line is that X n =0 γ n = 1 1 - γ if | γ | < 1 . I’d like now to address decimal expansions of real numbers. I’ll focus on the open unit interval (0 , 1) = { x R : 0 < x < 1 } . It turns out that any x (0 , 1) has at least one decimal expansion x = .a 1 a 2 a 3 a 4 a 5 . . . , where each a n is a natural number between 0 and 9, inclusive. The expansion means x = X n =1 a n 10 - n = lim N →∞ N X n =1 a n 10 - n . A decimal expansion of x is one way of representing x as the limit of a sequence of rational numbers. If you define q N = N X n =1 a n 10 - n , then each q N is rational and the sequence { q N } is a Cauchy sequence (check this for yourself) that converges to x as n → ∞ . Some decimal expansions terminate; for those expansions, there’s a smallest M > 0 such that a n = 0 for n > M . It happens that if x has a terminating decimal expansion, then x has at least one other decimal expansion. To wit, suppose x = .a 1 a 2 . . . a M is a terminating expansion for x with a M = 0 and a n = 0 for all n > M . Then a non- terminating expansion for x is x = .a 1 a 2 a 3 . . . a M - 1 ( a M - 1)999999 . . . , 1
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
2 where the 9’s go on forever. The M th decimal place has a M - 1 in it; in other words, you decrement the last nonzero decimal place in x ’s terminating expansion by 1, and you replace all the trailing zeroes in x ’s terminating expansion by 9’s. As an example, . 183746285647 = . 1837462856469999999999999999999999 . . . The geometric series makes this work. Re-write the second decimal expansion above as a 1 10 - 1 + a 2 10 - 2 + · · · + a M - 1 10 - ( M - 1) + a M 10 - M - 10 - M + X n = M +1 9 × 10 - n . Change the index in the last sum to m = n - ( M + 1) and you get 9 × 10 - ( M +1) X m =0 10 - m = 9 × 10 - ( M +1) 1 1 - 1 / 10 = 10 - M . So the sum of all the 9-terms in the expansion adds up to 10 - M , which cancels the - 10 - M in the expansion, which makes the expansion equal to a 1 10 - 1 + a 2 10 - 2 + · · · + a M - 1 10 - ( M - 1) + a M 10 - M , which is just our first (terminating) expansion for x . The bottom line is that you can expand such an x
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern