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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.
For example, Zhi-Wei Sun [14] proved that each natural number can be written as a sum of four squares with some requirements on the choice of these four numbers. One may also wonder whether it is necessary to use the entire set of square integers to write each natural as the sum of four squares.
The sign of the square root needs to be chosen properly—note that if 2 π is added to θ, the quantities inside the square roots are unchanged, but the left-hand-sides of the equations change sign. Therefore, the correct sign to use depends on the value of θ.
Comment: The proof of Euler's four-square identity is by simple algebraic evaluation. Quaternions derive from the four-square identity, which can be written as the product of two inner products of 4-dimensional vectors, yielding again an inner product of 4-dimensional vectors: (a·a)(b·b) = (a×b)·(a×b).
The natural product is associative and commutative and distributes over the natural sum. The natural product is always greater or equal to the usual product, but it may be strictly greater. For example, the natural product of ω and 2 is ω · 2 (the usual product), but this is also the natural product of 2 and ω. Under natural addition, the ...
We prove associativity by first fixing natural numbers a and b and applying induction on the natural number c. For the base case c = 0, (a + b) + 0 = a + b = a + (b + 0) Each equation follows by definition [A1]; the first with a + b, the second with b. Now, for the induction. We assume the induction hypothesis, namely we assume that for some ...
with this last subject to the condition that the roots (defined below) , sum to a non-zero value: +. The E α {\displaystyle E_{\alpha }} are sometimes called ladder operators , as they have this property of raising/lowering the value of β {\displaystyle \beta } .
Small Eisenstein primes. Those on the green axes are associate to a natural prime of the form 3n + 2. All others have an absolute value equal to 3 or square root of a natural prime of the form 3n + 1. Eisenstein primes in a larger range. If x and y are Eisenstein integers, we say that x divides y if there is some Eisenstein integer z such that ...