Collections of author's errors

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REVIEWS, BIBLIOGRAPHY

The ever-increasing desire of young people for knowledge, the broad prospects for the further development of higher education in the country, outlined by the XXIV Congress of the CPSU, place special demands on the quality of books published for young people, and, in particular, on the quality of aids for those entering universities. The editors have already drawn attention to the fact that among such manuals there are books containing significant shortcomings and even gross errors (see «Kvant» No. 11, 1970). Unfortunately, we are forced to return to this same issue once again.

In this issue we publish solutions to two manuals published in Kyiv and Leningrad.

Published in large numbers, these books were sold throughout the country. We would like to warn our readers against being too trusting in such books.

The editors of the magazine «Kvant» sent a letter to the Ministry of Higher and Secondary Special Education of the USSR about the need to establish reliable control over the publication of manuals for applicants to universities.

Collections of author’s errors

Recently, the Kyiv University Publishing House published two manuals on mathematics *), which, as stated in the abstract, are designed for students entering universities and can be useful for ministers and teachers of preparatory courses and for high school students. Familiarization with these manuals shows that they are written at an extremely low scientific and pedagogical level, and therefore can only lead to harm to the mathematical preparation of students.

Many key theoretical questions in school *) V. F. Kovaale, Practical classes in mathematics for university applicants. Arthmetic and algebra. Kyiv, Kyiv University Publishing House, 1970, 388 pages. 75,000 copies, price 88 kopecks.

V. F. Kovaale, Practical classes in mathematics for university applicants. Geometry and trionometry. Kyiv, Kyiv University Publishing House, 1971, 316 pp., print. 100,000 copies, price 74 kopecks.

For brevity, in future references these books will be referred to as I and II.

The author’s mathematics course fails, inaccurately or even with gross errors, and a significant number of the problems analyzed are solved incorrectly. A detailed list of comments and explanations of the shortcomings of these books could fill an entire magazine, and therefore we will give here only a few examples.

The definitions of the basic concepts of elementary mathematics given in books are not always accurate, and the explanations given are unclear and sometimes incorrect. So, the question of the arithmetic root is completely confused.

Is it possible to understand the exact meaning of, say, the following: “… $\sqrt{a^2}$ takes two meanings: $a$ and $-a$. But there is only one positive of these two numbers, and is it equal to $\sqrt{a^2}$ (I, p. 45)? It is not surprising that with this understanding of the sign of the square radical, the author makes numerous mistakes when solving problems: he proves that $\sqrt{7 - \sqrt{24}} = \sqrt{6} - 1$ (I, p. 170), and then reports that $\sqrt{7 - 4\sqrt{3}} = \pm (2 - \sqrt{3})$ (I, p. 347); $y = 1$ declares $\sqrt{y} = -1$ to be the root of the equation (I, p. 311); when calculating the values ​​of some trionometric functions through others, he uses formulas like $\cos \alpha = \sqrt{1 - \sin^2 \alpha}$ (II, pp. 149, 152, 164), etc.

The manuals pay attention only to purely formal calculations, without any analysis of the operations being carried out. It is because of this that when solving equations and systems of equations, incorrect results are often obtained: solutions are lost, the construction of the solution is not discarded, etc. (I, pp. 134-135, 202-203; II, pp. 214, 222-227, 312, etc.). The incorrect types of equations presented by the author are meaningless due to the lack of necessary analysis.

By way of illustration, we reproduce the “solution” of equation $|ax+b| = c$: “According to the definition of model $|ax+b| = c$”:

$$ax+b = c; \quad -(ax+b) = c.$$