1

At the bottom of page 5 is the phrase "changes k to k ⊕ (1j+1)2". Isn't 1 to any power still 1 even in binary? I'm thinking this must be a typo. I sent an email to Dr. Knuth to report this, but I don't expect to hear back for months. In the meantime, I'm trying to figure out what this is supposed to be.

SSteve
  • 10,550
  • 5
  • 46
  • 72
  • Voting to close as "not programming related." Hehe, I kid. – Robert S. Mar 11 '09 at 18:16
  • For those of us who don't have a copy in front of us, to figure out what it is supposed to be we need a little more info – Adrian Archer Mar 11 '09 at 18:26
  • I didn't think Dr. Knuth /used/ email... – Arcane Mar 11 '09 at 18:26
  • Adrian: I'm presuming that only people who have the book will answer the question. – SSteve Mar 11 '09 at 23:19
  • 2
    Arcane: Yes, Dr. Knuth stopped using email on 1/1/90. But there's an address for reporting errors in taocp. His secretary prints the emails and gives them to him. http://www-cs-faculty.stanford.edu/~knuth/email.html – SSteve Mar 11 '09 at 23:21

2 Answers2

4

This can be resolved by using the convention that (...)2 represents a bitwise representation. (1j+1)2 then consists solely of j+1 ones, rather than referring to an exponentiation. You can see this convention explained more explicitly in TAOCP Volume 4 Fascicle 1 at page 8, for example:

If x is almost any nonzero 2-adic integer, we can write its bits in the form

x = (g01a10b)2

in other words, x consists of some arbitrary (but infinite) binary string g, followed by a 0, which is followed by a+1 ones and followed by b zeros, for some a >= 0 and b >= 0.

[I have substituted the symbol alpha by g to save encoding problems]

Going back to your original query; k ⊕(1j+1)2 is equated with k ⊕ (2j+1 - 1) implying that (1j+1)2 = (2j+1 - 1): this holds because the left-hand side is the integer whose significant bits are j+1 (contiguous) ones; the right-hand side is an exponentiation. For example, with j =3:

(14)2 = (1111)2 = (24 - 1)

Hope that helps.

Kyle Hale
  • 7,912
  • 1
  • 37
  • 58
mas
  • 1,107
  • 1
  • 11
  • 18
0

A list of known typos can be found on the errata page:

http://www-cs-faculty.stanford.edu/~knuth/taocp.html

Your reported typo is not there. If it really is a typo, you might be eligible for a cash reward from Knuth himself.

neesh
  • 5,167
  • 6
  • 29
  • 32
  • 2
    He doesn't actually do the checks any more, because it exposes checking account etc. now you get a certificate. Hardly anyone cashed the checks anyway: http://www-cs-faculty.stanford.edu/~knuth/news08.html – Charlie Martin Mar 11 '09 at 20:20