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          ARRoGANT                CoURiERS      WiTH     ESSaYS

Grade Level:       Type of Work           Subject/Topic is on:
 [ ]6-8                 [ ]Class Notes    [Essay on Computers in   ]
 [ ]9-10                [ ]Cliff Notes    [Math                    ]
 [x]11-12               [x]Essay/Report   [                        ]
 [ ]College             [ ]Misc           [                        ]

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                       Computer's in Math

     Ever since the first computer was developed in the early 1900's the
computer has been using math to solve most of it's problems. The Arithmetic
and Logical unit helps the computer solve some of these problems. All type
of math can be solved on computer's which it uses.

                        Binary Arithmetic

     A computer understands two states: on and off, high and low, and so
on. Complex instructions can be written as a combination of these two
states. To represent these two conditions mathematically, we can use the
digits 1 and 0. Some simple mathematical operations, such as addition and
subtraction, as well as the two's complement subtraction procedure used by
most computer's.

                Evaluating an Algebraic Function

     It is frequently necessary to evaluate an expression, such as the one
below, for several values of x.

                       y= 6x4+4x3-5x2+6x+4

     First to start with developing the power's of x to perform the
necessary multiplications by the coefficients, and finally produce the sum.
The following steps are the way the computer "thinks" when it is
calculating the equation.

     1.Select x
     2.Multiply x by x and store x2
     3.Multiply x2 by x and store x3
     4.Multiply x3 by x and store x4
     5.Multiply x by 6 and store 6x
     6.Multiply stored x2 by 5 and store 5x2
     7.Multiply stored x3 by 4 and store 4x3
     8.Multiply stored x4 by 6 and store 6x4
     9.Add 6x4
     10.Add 4x3
     11.Subtract 5x3
     12.Add 6x
     13.Add 4


                      Binary Coded Decimal

     One of the most convenient conversions of decimal to binary coded
decimal's is used today in present day computer's. BCD(Binary Coded
Decimal) is a combination of binary and decimal; that is each separate
decimal digit is represented in binary form. For example the chart below
represents the Binary and Decimal conversions.



                     Decimal         Binary

                         0              0
                         1              1
                         2             10
                         3             11
                         4             100
                         5             101
                         6             110
                         7             111 
                         8             1000
                         9             1001
                        10             1010


     BCD uses one of the above binary representations for each decimal
digit of a given numeral. Each decimal digit is handled separately.

     For example, the decimal 28 in binary is as follows:

                        (28)10 = (11100)2
     The arrangement in BCD is as follows:

                          2          8
                       0010       1000

     Each decimal digit is represented by a four-place binary     
     number.


                     Direct Binary Addition

     In binary arithmetic if one adds 1 and 1 the answer is 10. The answer
is not the decimal 10. It is one zero. There are only two binary digits in
the binary system. Therefore when one adds 1 and 1, one gets the 0 and a
carry of 1 to give 10. Similarly, in the decimal system, 5 + 5 is equal to
zero and a carry of 1. Here is an example of binary addition:


                         column 4 3 2 1
                                 0 1 1 1
                               + 0 1 1 1
                                 1 1 1 0


     I n column 1, 1+1=0 and a carry of 1. Column 2 now contains 1+1+1.
This addition, 1+1=0 carry 1 and 0+1=1, is entered in the sum. Column 3 now
also contains 1+1+1, which gives a carry of 1 to column 4. The answer to
the next problem is found similarly.

                         1 0 0 1 1 0 1 1
                       + 0 0 1 1 1 1 1 1
                          1 1 0 1 1 0 1 0


                    Direct Binary Subtraction

     Although binary numbers may be subtracted directly from each other, it
is easier from a computer design standpoint to use another method of
subtraction called two's complement subtraction. This will be illustrated
next. However direct binary subtraction will be discussed.

     Direct Binary Subtraction is similar to decimal subtraction, except
that when a borrow occurs, it complements the value of the number. Also
that the value of the number of one depends on the column it is situated.
The values increase according to the power series of 2: that is 20, 21,23,
and so on, in columns 1, 2, 3 and so on. Hence, if you borrow from column 3
you are borrowing a decimal 4. ex column 3 2 1 1 1 0 - 1 0 1 0 0 1

     In the example a borrow had to be made from column 2, which
changed its value to 0 while putting decimal 2 (or binary 11) in
column 1. Therefore, after the borrow the subtraction in column 1
involved 2-1=1; in column 2 we had 0-0=0; and in column 3 we had
1-1=0.

     If the next column contains a 0 instead of a 1 , then we
must proceed to the next column until we find one with 1 from
which we can borrow.

     ex
                                
                              1 0 0 0
                            - 0 1 0 1

     After the borrow from column 4,

                           0 1 1 (11)
                          - 0 1 0  1
                            0 0 1  1

     Notice that a borrow from column 4 yields an 8(23). Changing
column 3 to a 1 uses a 4, and column 2 uses a 2, thus leaving 2
of the 8 we borrowed to put in column 1.
     ex
                         0 1 1 0 0 0 1 0
                        - 0 0 0 1 0 1 1 1


     After the first borrow:

                       0 1 1 0 0 0 0 (11)
                     - 0 0 0 1 0 1 1  1



After the second borrow (from column 6):


                      0 1 0 1 1 1 (11) (11)
                    - 0 0 0 1 0 1  1    1  
                      0 1 0 0 1 0  1    1


     These operations are stored in the computer's memory then
performed in the computer's Arithmetic/Logic Unit in the CPU.

                         Approximations

     In computer's, it is very important to consider the error
that may occur in the result of a calculation when numbers which
approximate other numbers are used. This is important to the use
of computer's  because of computers are usually very long and
involve long numbers.

                            Division

     It is possible to divide one number from another by
successively subtracting the divisor from the dividend and
counting number of the subtractions necessary to reduce the
remainder to a number smaller than the divisor.

     For example, to divide 24 by 6:

                  Number of                Is remainder smaller
                  subtractions             than divisor?


    24
   - 6                1                         No
    18   
   - 6                2                         No
    12
   - 6                3                         No
     6                
   - 6                4                         Yes 
     0  

This shows how the computer "thinks" when it is calculating a
problem using the division operation.

     Here is another example when there is a remainder.
     For example to divide 27 by 5:
                   Number of              Is remainder smaller
                   Subtractions           than divisor?      
   
    27
   - 5                 1                        No
    22
   - 5                 2                        No
    17
   - 5                 3                        No
    12
   - 5                 4                        No
     7
   - 5                 5                        Yes
     2

 Therefore 27 = 5, with a remainder of 2.

These two diagrams show the flow of thinking for the operation of
division in a calculation.


               Evaluating Trigonometric Relations

     For many problems in mathematics, the relationships between
the sides of a right triangle are important, and this, of course,
may suggest a general definition of trigonometry. hat is,if a
computer is available, how trigonometric functions can be done by
hand. It is interesting to consider some of the features of this
field from a computer-oriented point of view.


     It is not necessary to consider the last three functions in
the same sense as the first three because, if any one of the
first three one can get, the last three one can get by the
reciprocal of the first three.

     Reference to the triangle above shows that:

                            tan A = a
                                    b

and that tan A is related to sin A and cos A by the following:

                     sin A = a/c = a = tan A
                     cos A   b/c   b

     Something similar is shown below using the Pythagorean
     Theorem:
                          a2 + b2 = c2

and dividing by c2:

                          a2 + b2 = c2
                          c2 + c2 = c2.


                  Applications of Computer Math

     Computer Math is used in various ways in the mathematics and
scientific field. Many scientists use the computer math to
calculate the equations and using formulas, there by making
calculating on computer much faster. For mathematicians computer
math can help mathematicians solve long and tedious problems,
quickly and efficiently.

     The introduction of computer's into the world's technology
has drastically increased the amount of knowledge helped by the
computer's. The different aspects of using computer math are
virtually limitless.
