An Imaginary Tale Quotes

“The next, magnificent step would of course have been to write , but the Stereometria records it as , and so Heron missed being the earliest known scholar to have derived the square root of a negative number in a mathematical analysis of a physical problem.”
― Paul J. Nahin, An Imaginary Tale: The Story of i

“today they are usually called the Fresnel integrals. One does still see them also called the Euler integrals, however, and it was Euler who first evaluated them.”
― Paul J. Nahin, An Imaginary Tale: The Story of i

“partial sum is correct within an error less than the first term neglected.”
― Paul J. Nahin, An Imaginary Tale: The Story of i

“Even more astonishing, perhaps, is that 1π—a real number to a real power—has an infinity of distinct complex values, i.e., Only for n = 0, the principal value, is 1π real.”
― Paul J. Nahin, An Imaginary Tale: The Story of i

“Euler’s constant, which is γ = 0.577215664901532 …. After π and e, γ is perhaps the most important mathematical constant not appearing in elementary arithmetic.”
― Paul J. Nahin, An Imaginary Tale: The Story of i

“Finally, in 1748, Euler published the explicit formula in his book Introductio in Analysis Infinitorum.”
― Paul J. Nahin, An Imaginary Tale: The Story of i

“the differential arc length ds along a curve y = y(x) is The arc length from x = 0 to x = x̂, then, is simply”
― Paul J. Nahin, An Imaginary Tale: The Story of i

“In particular, the three roots to a cubic polynomial either must all be real, or there must be one real root and one conjugate pair.”
― Paul J. Nahin, An Imaginary Tale: The Story of i

“French mathematician Jacques Hadamard (1865–1963): “The shortest path between two truths in the real domain passes through the complex domain.” I”
― Paul J. Nahin, An Imaginary Tale: The Story of i

“the tangent function is periodic with period 180°,”
― Paul J. Nahin, An Imaginary Tale: The Story of i

“mean proportional, i.e., if b and c are two given positive numbers, then x is the mean proportional of b and c if it satisfies the statement “b is to x as x is to c.” Or, in algebra which”
― Paul J. Nahin, An Imaginary Tale: The Story of i

“the product of two sums of two squares of integers is always expressible, in two different ways, as the sum of two squares of integers.”
― Paul J. Nahin, An Imaginary Tale: The Story of i

“The breakthrough for came not from quadratic equations, but rather from cubics which clearly had real solutions but for which the Cardan formula produced formal answers with imaginary components.”
― Paul J. Nahin, An Imaginary Tale: The Story of i

“for the irreducible case with all three roots real, there is just one positive root; that is, the root given by the Cardan formula”
― Paul J. Nahin, An Imaginary Tale: The Story of i

“That is, he would have started over from the beginning to solve with, again, both p and q non-negative. This is totally unnecessary, however, as at no place in the solution to x3 + px = q did he ever actually use the non-negativity of p and q. That is, such assumptions have no importance, and were explicitly made simply because of an unwarranted aversion by early mathematicians to negative numbers. This”
― Paul J. Nahin, An Imaginary Tale: The Story of i

“The conclusion is that the original assumption of ordering leads us into contradiction, and so that assumption must be false.”
― Paul J. Nahin, An Imaginary Tale: The Story of i

“the quadrature problem is a measure of the greatest difficulty, since it was shown in 1882 to be impossible.”
― Paul J. Nahin, An Imaginary Tale: The Story of i

“The quadrature of a circle, the construction by straightedge and compass alone of the square equal in area to the circle,”
― Paul J. Nahin, An Imaginary Tale: The Story of i