299 lines
11 KiB
Plaintext
299 lines
11 KiB
Plaintext
ÜÜÜÜÜÜÜÜÜÜÜÜÜ ÜÜÜ ÜÜÜÜ
|
|
ÜÛÛÛÛÛÛÛÛßÛßßßßßÛÛÜ ÜÜßßßßÜÜÜÜ ÜÛÜ ÜÛÛÛÛÛÛÛÛÜÜÜÜÜÛßß ßÛÛ
|
|
ßÛÛÛÛÛÛÛÛÛÛÛÛÛÛÜ ßÛÛ ÜÛÛÛÜÛÛÜÜÜ ßÛÛÛÛÜ ßÛÛÛÛÛÛÛÜÛÛÜÜÜÛÛÝ Ûß
|
|
ßßßÛÛÛÛÛÛÛÛÛÛÜ ÞÝ ÛÛÛÛÛÛÛÛÛÛÛßßÛÜÞÛÛÛ ÛÛÛÛÛÜ ßßÛÛÛÞß
|
|
Mo.iMP ÜÛÛÜ ßÛÛÛÛÛÛÛÝÛ ÞÛÛÛÛÛÛÛÛÛ ÞÛÛÛÛ ÞÛÛÛÛÛÝ ßÛß
|
|
ÜÛÛÛÛÛÛÛ ÛÛÛÛÛÛÛÛÝ ÞÛÛÛÛÛÛÛÛÝ ÛÛÛ ÛÛÛÛÛÛ
|
|
ÜÛÛÛÛÛÛÛÝ ÞÛÛÛÛÛÛÛÛ ÞÛÛÛÛÛÛÛÛ ß ÞÛÛÛÛÛÛÜ ÜÛ
|
|
ÜÛÛÛÛÛÛÛÝ ÛÛÛÛÛÛÛÛ ÛÛÛÛÛÛÛÛÝ ÞÞÛÛÛÛÛÛÛÛÛß
|
|
ÜÛßÛÛÛÛÛÛ ÜÜ ÛÛÛÛÛÛÛÛÝ ÛÛÞÛÛÛÛÛÝ ÞÛÛÛÛÛÛßß
|
|
ÜÛßÛÛÛÛÛÛÜÛÛÛÛÜÞÛÛÛÛÛÛÛÛ ÞÛ ßÛÛÛÛÛ Ü ÛÝÛÛÛÛÛ Ü
|
|
ÜÛ ÞÛÛÛÛÛÛÛÛÛÛß ÛÛÛÛÛÛÛÛÛ ßÛÜ ßÛÛÛÜÜ ÜÜÛÛÛß ÞÛ ÞÛÛÛÝ ÜÜÛÛ
|
|
ÛÛ ÛÛÛÛÛÛÛÛß ÛÛÛÛÛÛÛÛÛÛÜ ßÛÜ ßßÛÛÛÛÛÛÛÛÛß ÜÜÜß ÛÛÛÛÜÜÜÜÜÜÜÛÛÛÛÛß
|
|
ßÛÜ ÜÛÛÛß ßÛÛÛÛÛÛÛÛÛÛÜ ßßÜÜ ßßÜÛÛßß ßÛÛÜ ßßßÛßÛÛÛÛÛÛÛßß
|
|
ßßßßß ßßÛÛß ßßßßß ßßßßßßßßßßßßß
|
|
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 [ ]
|
|
|
|
Dizzed: 10/94 # of Words:1383 School: ? State: ?
|
|
ÄÄÄÄÄÄÄÄÄ>ÄÄÄÄÄÄÄÄÄ>ÄÄÄÄÄÄÄÄÄ>Chop Here>ÄÄÄÄÄÄÄÄÄ>ÄÄÄÄÄÄÄÄÄ>ÄÄÄÄÄÄÄÄÄ>ÄÄÄÄÄÄÄÄÄ
|
|
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.
|