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.”

- For starters, is most definitely a countable set, and so there is a one-to-one and onto correspondence between integers and rational numbers. See https://proofwiki.org/wiki/Rational_Numbers_are_Countably_Infinite or http://www.homeschoolmath.net/teaching/rational-numbers-countable.php for details.

- 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 and .

- Just above Example 2 was Example 1, which gives the correct proof that there’s a one-to-one correspondence between and . 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.

The full chapter study guide can be found here (it’s on the last page): http://nseuntj.weebly.com/uploads/1/8/2/0/18201983/2.1relations_and_functions.pdf

Reactions can be found here: https://www.reddit.com/r/math/comments/3k1qe6/this_is_in_a_high_school_math_textbook_in_texas/

Reference to this can be seen on page 10 of the teacher’s manual here: http://msastete.com/yahoo_site_admin1/assets/docs/Chpte2-1.25882808.pdf