Internally, those devices use one of a variety of division algorithms.

The Lattice Method This algorithm is included mainly for its historical interest, and the fact that it provides fine practice with the multiplication facts and adding single-digit numbers.

It is not easy to explain exactly why it works, but it does have the reliability that all algorithms must have. It is also very efficient, no matter how many digits are in the factors, as indicated by the second example below.

The lattice method appeared in what is said to be the first printed arithmetic book, printed in Treviso, Italy, in It was in use long before that, with some historians tracing it to Hindu origins in India before This is a student favorite because of the direct relationship to known multiplication facts and its easy expandability to very large numbers.

Lattice Method Explanation First, make a table of the right size: Divide each cell of the resulting table in half by drawing a diagonal line. Write the digits for the first factor across the top one for each box.

Write the digits for the second factor down the right side one for each box. Multiply each digit across the top with the digit s down the right side, placing the tens digit of each product above the diagonal of the cell, and the ones digit below the diagonal.

For example, the upper left hand cell contains the answer 12, the result of multiplying the 3 first top digit by 4 first right-side digit.

The second box in the top row is the result of multiplying 5 second top digit by 4 first right-side digit. Start at the bottom right corner and add the digits upward along the diagonal.

Place the sum at the bottom of the diagonal outside the box carrying the tens digit to the next diagonal, if needed. Move left one column, and do the next diagonal. Continue along the bottom, then turn the corner and do the diagonals up the left side. The answer is read down the left side and across the bottom, left to right.

You check each estimate.Because G is an undirected graph, we know that the distance from x to y is the same as the distance from y to x, and so we can ignore the shaded cells in the above table for the purposes of computing diameter and average distance.

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Useful Computational Methods: The Long Division Algorithm.

As I write this, in A.D., students in the third or fourth grades in school are still being taught long division, although in the higher grades most students obtain quotients using calculators. At the end we are left with the gradient in the variables [dfdx,dfdy,dfdz], which tell us the sensitivity of the variables x,y,z on f!.This is the simplest example of backpropagation.

Going forward, we will want to use a more concise notation so that we donâ€™t have to keep writing the df part. That is, for example instead of dfdq we would simply write dq, and always assume that the gradient.

Division is one of the four basic operations of arithmetic, the others being addition, subtraction, and rutadeltambor.com division of two natural numbers is the process of calculating the number of times one number is contained within another one.: 7 For example, in the picture on the right, the 20 apples are divided into four groups of five .

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Long Division and Repeated Subtraction