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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.

 




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