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  2. Casus irreducibilis - Wikipedia

    en.wikipedia.org/wiki/Casus_irreducibilis

    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 ...

  3. Mathematical constant - Wikipedia

    en.wikipedia.org/wiki/Mathematical_constant

    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).

  4. Complex number - Wikipedia

    en.wikipedia.org/wiki/Complex_number

    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 ...

  5. Imaginary unit - Wikipedia

    en.wikipedia.org/wiki/Imaginary_unit

    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.

  6. Quadratic irrational number - Wikipedia

    en.wikipedia.org/wiki/Quadratic_irrational_number

    The square root of 2 was the first such number to be proved irrational. Theodorus of Cyrene proved the irrationality of the square roots of non-square natural numbers up to 17, but stopped there, probably because the algebra he used could not be applied to the square root of numbers greater than 17. Euclid's Elements Book 10 is dedicated to ...

  7. Proof of impossibility - Wikipedia

    en.wikipedia.org/wiki/Proof_of_impossibility

    It shows that the square root of 2 cannot be expressed as the ratio of two integers. The proof bifurcated "the numbers" into two non-overlapping collections—the rational numbers and the irrational numbers. There is a famous passage in Plato's Theaetetus in which it is stated that Theodorus (Plato's teacher) proved the irrationality of

  8. Nested radical - Wikipedia

    en.wikipedia.org/wiki/Nested_radical

    In the case of two nested square roots, the following theorem completely solves the problem of denesting. [2]If a and c are rational numbers and c is not the square of a rational number, there are two rational numbers x and y such that + = if and only if is the square of a rational number d.

  9. Irrationality measure - Wikipedia

    en.wikipedia.org/wiki/Irrationality_measure

    Rational numbers have irrationality exponent 1, while (as a consequence of Dirichlet's approximation theorem) every irrational number has irrationality exponent at least 2. On the other hand, an application of Borel-Cantelli lemma shows that almost all numbers, including all algebraic irrational numbers , have an irrationality exponent exactly ...