Error involving countable numbers in Glencoe Algebra 2 (2014)

Errors in textbooks happened when Pebbles Flintstone and Bamm-Bamm Rubble attended Flintstone Elementary, and they still happen on occasion today. But even with that historical perspective, this howler is a doozy.

This was sent to me by a former student of mine. It appears in the chapter study guide for Section 2.1 of Glencoe’s Algebra 2 textbook (published in 2014), presumably as an enrichment activity for students learning about the definitions of “one to one functions” and “onto functions.”

countable infinityJust how bad is this mistake?

  • The above “proof” is only a blatant assertion, without any justification, either formal or informal, for why the authors think that the statement is false.
  • The ordering of the rational numbers in the way listed above is actually reasonably close to the listing that actually does produce the one-to-one correspondence between \mathbb{Q} and \mathbb{Z}.
  • Just above Example 2 was Example 1, which gives the correct proof that there’s a one-to-one correspondence between \mathbb{Z} and \mathbb{N}. If the authors had double-checked this proof in any reputable book, they should have also been able to double-check that their Example 2 was completely false.

countable infinity 2

green line

The full chapter study guide can be found here (it’s on the last page):

Reactions can be found here:

Reference to this can be seen on page 10 of the teacher’s manual here:


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