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The golden ratio's negative −φ and reciprocal φ−1 are the two roots of the quadratic polynomial x2 + x − 1. The golden ratio is also an algebraic number and even an algebraic integer. It has minimal polynomial. This quadratic polynomial has two roots, and. The golden ratio is also closely related to the polynomial.
An a × b rectangle can be packed with 1 × n strips if and only if n divides a or n divides b. [ 15 ] [ 16 ] de Bruijn's theorem : A box can be packed with a harmonic brick a × a b × a b c if the box has dimensions a p × a b q × a b c r for some natural numbers p , q , r (i.e., the box is a multiple of the brick.) [ 15 ]
The Rhind Mathematical Papyrus ( RMP; also designated as papyrus British Museum 10057, pBM 10058, and Brooklyn Museum 37.1784Ea-b) is one of the best known examples of ancient Egyptian mathematics. It is one of two well-known mathematical papyri, along with the Moscow Mathematical Papyrus. The Rhind Papyrus is the larger, but younger, of the two..
Calculus. In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions : The first terms of the series sum to approximately , where is the natural logarithm and is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it ...
The silver ratio is a Pisot–Vijayaraghavan number (PV number), as its conjugate 1 − √ 2 = −1 δS ≈ −0.41421 has absolute value less than 1. In fact it is the second smallest quadratic PV number after the golden ratio. This means the distance from δ n. S to the nearest integer is 1 δ n. S ≈ 0.41421n. Thus, the ...
It is also the densest possible packing of discs with this size ratio (ratio of 0.6375559772 with packing fraction (area density) of 0.910683). [8] There are also a range of problems which permit the sizes of the circles to be non-uniform.
3.4.2 Decay of Fourier transform. ... Euler's formula; Partial fractions; Changing order; ... The total area A 1 + A 2 is equal to the area of the bigger rectangle, x ...
In this document a two-part set of fractions was written in terms of Eye of Horus fractions which were fractions of the form 1 / 2 k and remainders expressed in terms of a unit called ro. The answers were checked by multiplying the initial divisor by the proposed solution and checking that the resulting answer was 1/2 + 1/4 + 1/8 + 1/16 ...