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In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/ x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a / b is b / a. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one ...
The reciprocal of a proper fraction is improper, and the reciprocal of an improper fraction not equal to 1 (that is, numerator and denominator are not equal) is a proper fraction. When the numerator and denominator of a fraction are equal (for example, 7 / 7 ), its value is 1, and the fraction therefore is improper. Its reciprocal is ...
A unit fraction is a positive fraction with one as its numerator, 1/ n. It is the multiplicative inverse (reciprocal) ... For example, the matrix ...
The golden ratio φ and its negative reciprocal −φ −1 are the two roots of the quadratic polynomial x 2 − x − 1. The golden ratio's negative −φ and reciprocal φ −1 are the two roots of the quadratic polynomial x 2 + x − 1. The golden ratio is also an algebraic number and even an algebraic integer. It has minimal polynomial
In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio. The ratio is called coefficient of proportionality (or proportionality constant) and its reciprocal is known as constant of normalization (or normalizing constant).
A method analogous to piece-wise linear approximation but using only arithmetic instead of algebraic equations, uses the multiplication tables in reverse: the square root of a number between 1 and 100 is between 1 and 10, so if we know 25 is a perfect square (5 × 5), and 36 is a perfect square (6 × 6), then the square root of a number greater than or equal to 25 but less than 36, begins with ...
This, for cyclic fractions with long repetends, allows us to easily predict what the result of multiplying the fraction by any natural number n will be, as long as the repetend is known. A proper prime is a prime p which ends in the digit 1 in base 10 and whose reciprocal in base 10 has a repetend with length p − 1.
For example, setting c = d = 0 produces a diagonal complex matrix representation of complex numbers, and setting b = d = 0 produces a real matrix representation. The norm of a quaternion (the square root of the product with its conjugate, as with complex numbers) is the square root of the determinant of the corresponding matrix.
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