Lapses in Mathematical Reasoning by V. M. Bradis

Author:V. M. Bradis [V. M. Bradis, V. L. Minkovskii and A. K. Kharcheva]
Language: eng
Format: epub
Publisher: Dover Publications
Published: 2016-11-14T16:00:00+00:00

and namely

is equal to one of the values of the square root of the number zz′, namely to the number:

Now the question arises whether this latter number is the “first” value of the root of zz′, or whether this “first” value is the second value of the root of zz′, designated by us by the letter υ.

From the conditions and , we conclude that and . In other words, the point representing the number u may lie either on the positive portion of the x-axis or above the x-axis , or on the negative portion of the x-axis , or below the x-axis . Thus the product of the “first” values of the roots of z and z′ may be equal in some cases to the “first” value of the root of zz′, and in other cases to its “second” value.

By considering instead of the values of half the sum of the arguments , the value of the whole sum α + α′, and collecting in pairs the four cases considered above, we arrive at the following two cases:

Case I. If α + α′ < 360°, then either , or ; the number u is represented either by a point of the positive portion of the x-axis, or by a point of the upper half-plane; u is equal to the “first” value of the root of zz′.

Case II. If α + α′ > 360°, then either , or , the number u is represented either by a point of the negative portion of the x-axis, or by a point of the lower half-plane; u is equal to the “second” value of the root of zz′.

We now obtain the final conclusion: the product of the “first” values of square roots of complex numbers z and z′ is equal to the “first” value of the square root of the product of these numbers if, and only if, the sum of the arguments of these numbers is less than 360° (it is assumed that the arguments are selected within the limits 0–360°).

Noting that the “second” value of the square root is equal to its “first” value, taken with the opposite sign, we can, by making use of the last proposition, easily construct the following “table of multiplication” for square roots of complex numbers (the “first” values of the square root we shall denote by the plus sign in front of the square root symbol, and the “second” value by the minus sign). Just as before, we assume that the arguments α and α′ of the numbers z and z′ are taken between 0° and 360°.

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