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Casus irreducibilis occurs when none of the roots are rational and when all three roots are distinct and real; the case of three distinct real roots occurs if and only if q 2 / 4 + p 3 / 27 < 0, in which case Cardano's formula involves first taking the square root of a negative number, which is imaginary, and then taking the ...
We are not taking the square root of any negative values here, since both and are necessarily positive. But we have lost the solution x = − 2. {\displaystyle x=-2.} The reason is that x {\displaystyle x} is actually not in general the positive square root of x 2 . {\displaystyle x^{2}.}
Square roots of negative numbers are called imaginary because in early-modern mathematics, only what are now called real numbers, obtainable by physical measurements or basic arithmetic, were considered to be numbers at all – even negative numbers were treated with skepticism – so the square root of a negative number was previously considered undefined or nonsensical.
The solution in radicals (without trigonometric functions) of a general cubic equation, when all three of its roots are real numbers, contains the square roots of negative numbers, a situation that cannot be rectified by factoring aided by the rational root test, if the cubic is irreducible; this is the so-called casus irreducibilis ...
Laguerre's method may even converge to a complex root of the polynomial, because the radicand of the square root may be of a negative number, in the formula for the correction, , given above – manageable so long as complex numbers can be conveniently accommodated for the calculation. This may be considered an advantage or a liability ...
The imaginary unit's core property is that i 2 = −1. The term "imaginary" was coined because there is no number having a negative square. There are in fact two complex square roots of −1, namely i and −i, just as there are two complex square roots of every other real number (except zero, which has one double square root).
A method analogous to piece-wise linear approximation but using only arithmetic instead of algebraic equations, uses the multiplication tables in reverse: the square root of a number between 1 and 100 is between 1 and 10, so if we know 25 is a perfect square (5 × 5), and 36 is a perfect square (6 × 6), then the square root of a number greater than or equal to 25 but less than 36, begins with ...
In the case of three real roots, the square root expression is an imaginary number; here any real root is expressed by defining the first cube root to be any specific complex cube root of the complex radicand, and by defining the second cube root to be the complex conjugate of the first one.