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.