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The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares.It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, [1] and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. [2]
However, the square of the distance (denoted d 2 or r 2), which has a paraboloid as its graph, is a smooth and analytic function. The dot product of a Euclidean vector with itself is equal to the square of its length: v⋅v = v 2. This is further generalised to quadratic forms in linear spaces via the inner product.
For example, the square of 65 can be calculated by n = 6 × (6 + 1) = 42 which makes the square equal to 4225. If the number is of the form m 0 where m represents the preceding digits, its square is n 00 where n = m 2 .
The sum of the reciprocals of the square numbers (the Basel problem) is the transcendental number π 2 / 6 , or ζ(2) where ζ is the Riemann zeta function. The sum of the reciprocals of the cubes of positive integers is called Apéry's constant ζ(3) , and equals approximately 1.2021 .
The roots of the quadratic function y = 1 / 2 x 2 − 3x + 5 / 2 are the places where the graph intersects the x-axis, the values x = 1 and x = 5. They can be found via the quadratic formula. In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation.
Another geometric proof proceeds as follows: We start with the figure shown in the first diagram below, a large square with a smaller square removed from it. The side of the entire square is a, and the side of the small removed square is b. The area of the shaded region is . A cut is made, splitting the region into two rectangular pieces, as ...
In 2035, the adjusted deficit equals 6.1% of GDP – significantly more than the 3.8% that deficits have averaged over the past 50 years. Congress has run a deficit every year since 2001.
The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2. It may be written in mathematics as 2 {\displaystyle {\sqrt {2}}} or 2 1 / 2 {\displaystyle 2^{1/2}} .