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The problem of evil is generally formulated in two forms: the logical problem of evil and the evidential problem of evil. The logical form of the argument tries to show a logical impossibility in the coexistence of a god and evil, [ 2 ] [ 10 ] while the evidential form tries to show that given the evil in the world, it is improbable that there ...
The formula is a special case of the Euler–Boole summation formula for alternating series, providing yet another example of a convergence acceleration technique that can be applied to the Leibniz series. In 1992, Jonathan Borwein and Mark Limber used the first thousand Euler numbers to calculate π to 5,263 decimal places with the Leibniz ...
Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics". [2] When x = π, Euler's formula may be rewritten as e iπ + 1 = 0 or e iπ = −1, which is known as Euler's identity.
God could have created a world without the possibility of evil, but he willed to create the world in a "state of journeying" to its consummation (the time when evil will no longer exist). [75] God could have created beings without the possibility of committing sin, but he willed to create free beings, e.g., beings that have free-will and must ...
The entry of evil into the world is generally explained as consequence of original sin and its continued presence due to humans' misuse of free will and concupiscence. God's goodness and benevolence, according to the Augustinian theodicy, remain perfect and without responsibility for evil or suffering.
And Paul Nahin, a professor emeritus at the University of New Hampshire, who has written a book dedicated to Euler's formula and its applications in Fourier analysis, describes Euler's identity as being "of exquisite beauty". [8] Mathematics writer Constance Reid has opined that Euler's identity is "the most famous formula in all mathematics". [9]
For example, the square root of 2 is an irrational number, but it is not a transcendental number as it is a root of the polynomial equation x 2 − 2 = 0. The golden ratio (denoted φ {\displaystyle \varphi } or ϕ {\displaystyle \phi } ) is another irrational number that is not transcendental, as it is a root of the polynomial equation x 2 − ...
Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin(α + β) = sin α cos β + cos α sin β.