Long division

Using simple algebraic long division, we can divide through term by term and rearrange terms to obtain... 

If the order of q(s) is higher, we need first carry out "long division" until we are left with a partial fraction "residue." Thus the coefficients ai are also called residues. We then expand this... 

We will skip the algebraic details. The simple idea is that we can do long division of a function of the form in Eq. (3-30) and match the terms to a Taylor s expansion of the exponential function. If we do, we ll find that the (1/1) Pade approximation is equivalent to a third order Taylor... 

TERMINATING DECIMAL a decimal that terminates. Eventually, when performing long division, the divisor divides evenly into one of the sub-dividends. Example 4.5, 23.6003. REPEATING DECIMAL a decimal whose fractional part follows a repeating pattern. The divisor never divides evenly into one of the sub-dividends, but a pattern emerges. Examples 8.99999. . ., 0.1 21 21 2. . . ., 4.567777... . ... 

To divide with decimal numbers, first change the problem to division by a whole number. It may be necessary to move the decimal point in the divisor (the number you are dividing by) to make it a whole number. Move the decimal in the dividend (the number you are dividing into) the same number of places, and copy the new decimal place holder straight up into the quotient (the answer to the division problem). Once the decimal point is placed, divide as you normally would with long division. 

Another way to convert a ratio in fractional form into a percent is to first change the ratio to a decimal by long division and then multiply by 100. Changing a fraction into a decimal was covered in Chapter 4 of this book. 

C. LONG DIVISION. The most interesting and most useful z-transform inversion technique is simple long division of the numerator by the denominator of The ease with which z transforms can be inverted by this technique is one of the reasons why z transforms are often used. 

Inversion of z transforms by long division is very easily accomplished numerically by a digital computer. The FORTRAN subroutine LONGD given in Table 18.2 performs this long division. The output variable X is calculated for NT sampling times, given the coefficients AQ, X 1), A(2), A M) of the numerator and the coefficients 6(1), 6(2),..., B N) of the denominator. 

Find the outputs X2, T,) of the two systems of Example 18.7 for a unit step input in m, . Use partial fractions expansion and long division. 

Long division shows that the manipulated variable changes at each sampling period, so rippling occurs. 

How do you get the 8into fractions automatically. 

This process results in the correct position of the decimal point in the quotient. The problem can now be solved by performing simple long division ... 

A long division in which most or all of the digits have been replaced by asterisks to form a cryptarithm. slide rule... 

We could translate this into terms of single molecules—by means of some very long division—because we know the number of molecules in each flask is the same, namely about 26,870,000,000,000,000,000,000. 

Example 28.6 Long Division for a More Complex Inversion... 

We can invert modified z-transforms through long division, taking care to divide separately terms involving m and those that do not include m. This allows us to find the value of a sampled-value function between sampling instants. For example, suppose that we have a signal with... 

VII.15 Find the inverse z-transform of the following expressions, using long division. 

Let us now proceed with the mechanics of two methods used to determine inverse z-transforms. The first is the partial-fractions expansion and the second is based on the long division of two polynomials. 

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