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In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form + + to the form + for some values of and . [1] In terms of a new quantity x − h {\displaystyle x-h} , this expression is a quadratic polynomial with no linear term.
The stepped reckoner was based on a gear mechanism that Leibniz invented and that is now called the Leibniz wheel. It is unclear how many different variants of the calculator were made. Some sources, such as the drawing to the right, show a 12-digit version. [5] This section describes the surviving 16-digit prototype in Hanover. Leibniz wheel
He presented a method of completing the square to solve quadratic equations, sometimes called Śrīdhara's method or the Hindu method. Because the quadratic formula can be derived by completing the square for a generic quadratic equation with symbolic coefficients, it is called Śrīdharācārya's formula in some places.
In the position shown, the counting wheel meshes with three of the nine teeth of the Leibniz wheel. A Leibniz wheel or stepped drum is a cylinder with a set of teeth of incremental lengths which, when coupled to a counting wheel, can be used in the calculating engine of a class of mechanical calculators.
One side of the square is labeled with the sexagesimal number 30. The diagonal of the square is labeled with two sexagesimal numbers. The first of these two, 1;24,51,10 represents the number 305470/216000 ≈ 1.414213, a numerical approximation of the square root of two that is off by less than one part in two million.
Burroughs founded the American Arithmometer Company in 1886. After his death, in 1904 partner John Boyer renamed the business the Burroughs Adding Machine Company. He was awarded the Franklin Institute's John Scott Legacy Medal shortly before his death. [1]
The Arithmometer, invented in 1820 as a four-operation mechanical calculator, was released to production in 1851 as an adding machine and became the first commercially successful unit; forty years later, by 1890, about 2,500 arithmometers had been sold [16] plus a few hundreds more from two arithmometer clone makers (Burkhardt, Germany, 1878 ...
A typical use of this is the completing the square method for getting the quadratic formula. Another example is the factorization of x 4 + 1. {\displaystyle x^{4}+1.} If one introduces the non-real square root of –1 , commonly denoted i , then one has a difference of squares x 4 + 1 = ( x 2 + i ) ( x 2 − i ) . {\displaystyle x^{4}+1=(x^{2 ...