The Project Gutenberg eBook #36640: Lectures on Elementary Mathematics

n^√3 A + m^√3 B = ^{−h − k}√⁻³ ,

2

which are real quantities. Consequently, if the root h is real, the two other roots also will be real in the irreducible case and they will be real in that case only.

But the invariable difficulty is, to demonstrate directly that

.³ f + g^√−1 + .³ f − g^√−1,

which we have supposed equal to h, is always a real quantity whatever be the values of f and g. In particular cases the demon- stration can be effected by the extraction of the cube root, when

that is possible. For example, if f = 2, g = 11, we shall find that the cube root of 2 + 11^√−1 will be 2 + ^√−1, and similarly that the cube root of 2 − 11^√−1 will be 2 − ^√−1, and the sum of

the radicals will be 4. An infinite number of examples of this

class may be constructed and it was through the consideration of such instances that Bombelli became convinced of the real- ity of the imaginary expression in the formula for the irreducible case. But forasmuch as the extraction of cube roots is in general possible only by means of series, we cannot arrive in this way at a general and direct demonstration of the proposition under consideration.

It is otherwise with square roots and with all roots of which the exponents are powers of 2. For example, if we have the

expression

.f + g^√−1 + .f − g^√−1,

composed of two imaginary radicals, its square will be

√

2f + 2 f ² + g²,

a quantity which is necessarily positive. Extracting the square root, so as to obtain the equivalent expression, we have

.2f + 2 f ² + g²,

√

for the real value of the imaginary quantity we started with. But if instead of the sum we had had the difference between the two proposed imaginary radicals we should then have obtained for its square the following expression

√

2f − 2 f ² + g²,

a quantity which is necessarily negative; and, taking the square root of the latter, we should have obtained the simple imaginary expression
. _√

2f − 2 f ² + g².

Further, if the quantity

.⁴ f + g^√−1 + .⁴ f − g^√−1

were given, we should have, by squaring, the form

.f + g^√−1 + .f − g^√−1 + 2 ⁴ f ² + g²

√

= .2f + 2 f ² + g² + 2 ⁴ f ² + g²,

√ √

a real and positive quantity. Extracting the square root of this expression we should obtain a real value for the original quantity;

and so on for all the other remaining even roots. But if we should attempt to apply the preceding method to cubic radicals we should be led again to equations of the third degree in the irreducible case.

For example, let

Cubing, we get

.³ f + g^√−1 + .³ f − g^√−1 = x.

2f + 3^√3 f ² + g² ..³ f + g^√−1 + .³ f − g^√−1Σ = x³;

that is

√

2f + 3x ³ f ² + g² = x³,

or, with the terms properly arranged,
√