Fractions

Definition

Part of a whole.A number written with the bottom part (the denominator) telling you how many parts the whole is divided into, and the top part (the numerator) telling how many you have.

     Here  3 –numerator , 7 –denominator.

Proper and improper common fractions

A common fraction is said to be a proper fraction if the absolute value of the numerator is less than the absolute value of the denominator—that is, if the absolute value of the entire fraction is less than 1. A vulgar fraction is said to be an improper fraction if the absolute value of the numerator is greater than or equal to the absolute value of the denominator.

 Mixed numbers

A mixed numeral (often called a mixed number, also called a mixed fraction) is the sum of a whole number and a proper fraction. This sum is implied without the use of any visible operator such as “+”. For example, in referring to two entire cakes and three quarters of another cake, the whole and fractional parts of the number are written next to each other:  

An improper fraction is another way to write a whole plus a part. A mixed number can be converted to an improper fraction as follows:

1. Write the mixed number     as a sum      .

2.  Convert the whole number to an improper fraction with the same denominator as the fractional part,  .

3.  Add the fractions. The resulting sum is the improper fraction. In the example,   .

Similarly, an improper fraction can be converted to a mixed number as follows:

  1. Divide the numerator by the denominator. In the example, , divide 11 by 4.  11 ÷ 4 = 2 with remainder 3.
  2. The quotient (without the remainder) becomes the whole number part of the mixed number. The remainder becomes the numerator of the fractional part. In the example, 2 is the whole number part and 3 is the numerator of the fractional part.
  3. The new denominator is the same as the denominator of the improper fraction. In the example, they are both 4.   Thus .

    \[ \frac{11}{4}\: = 2\frac{3}{4}\ \]

At first, we sample f(x) in the N (N is odd) equidistant points around x^*:

    \[ f_k = f(x_k),\: x_k = x^*+kh,\: k=-\frac{N-1}{2},\dots,\frac{N-1}{2} \]

where h is some step.
Then we interpolate points \{(x_k,f_k)\} by polynomial

(1)   \begin{equation*} P_{N-1}(x)=\sum_{j=0}^{N-1}{a_jx^j} \end{equation*}

Its coefficients \{a_j\} are found as a solution of system of linear equations:

(2)   \begin{equation*} \left\{ P_{N-1}(x_k) = f_k\right\},\quad k=-\frac{N-1}{2},\dots,\frac{N-1}{2} \end{equation*}

Here are references to existing equations: (1), (2).
Here is reference to non-existing equation (??).

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