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Computers 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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