Plausibility tests

20

G. D. Balk, M. B. Balk

Both the mathematics major and the schoolchild come across mathematical propositions, the truth of which he doubts. Otherwise, you need to “prove or disprove” any statement or choose the only correct one from several sentences of answers to a problem.

If the proof of a statement runs into cumbersome calculations, then it is often useful not to start with attempts at proof, but to make a separate report on whether this hypothesis is correct and whether it does not lead to clearly erroneous or inaccurate conclusions.

Let us dwell on some of the advantages of this kind of “check for legal completion” *).

This is what Bramagupta said…

Generalizing Heron’s well-known formula for the planes of a triangle, the Indian mathematician and astronomer Bramagupta (7th century AD) made a statement that can be formulated in the current notation as follows:

*) The article is based on the stepogram of the class of the 9th grade mathematics and mechanics club at school No. 7 in Smolensk (January 17, 1971).

the area S of any quadrilateral (Fig. 1) is expressed through its sides a, b, c, d according to the formula

$$S = \sqrt{(p-a)(p-b)(p-c)(p-d)}$$

here p, as usual, is the meter of the quadrilateral:

$$p = \frac{1}{2}(a+b+c+d).$$

Is Bramagupta’s statement true?

Before they rush to prove this statement (if it is true, then the proof will probably be cumbersome), it makes sense to use it to complement the truth. Let’s use the simplest method: testing for partial cases.

Consider the case when the given quadrilateral is a square; arbitrary rectangle; a rhombus other than a square.

In the first two cases, formula (1) leads to the correct result, in the third it leads to an error: according to the formula $S=a^2$, although it is clear that in this case $S<a^2$.

Therefore, Bramagupta’s statement is wrong.

Let us note in passing that this failure should not force us to abandon