Hello! We’re back again, now bringing
up the second topic to be discussed about. We picked the second topic from our
semester’s syllabus that is, Arithmetic for Computers (Number Systems and Operations).
From what we understand from the exact
definition of arithmetic, it is actually the oldest and most elementary branch
of mathematics that is used by almost every single studying person, for tasks
ranging from simple day-to-day counting to advanced science and business
calculations. It includes the study of quantity, especially as the result of
operations that combine numbers. In common usage, it refers to the simpler properties
when using the traditional operations of addition, subtraction, multiplication
and division with smaller values of numbers.
So basically, arithmetic is all about
numbers and operations involving numbers. As we always know, we humans use the
decimal number system to perform arithmetic operations including every numbers
to infinity. On the other hand, not for the computers, that uses the binary
system, containing only two digits that are 0 and 1. We need a way to convert
numbers from one system to another in order to allow computers to understand
us.
All the number systems we use practically
operate on some basic principles where each of it has a base as we will look at
decimal with base 10, binary with base 2 and hexadecimal owning base 16. The base denotes the number
of digits and the powers to be used. Base 10 has 10 digits that are numbered
from 0-9 and uses the powers of 10. However, base 2 has 2 digits that numbered
from 0-1 and uses powers of 2 whereas base 16 has 16 digits which numbered from
0-F (where the letters A-F serve as the digits following 9), and uses the powers
of 16.
Other than that, each of these systems
of number is a positional notation system meaning that the location of a digit
in the number determines its value. As an example, the value of 6 in the
decimal number 631 is not the same as the value of 6 in 361 or 136.
So in this explanation part, we'll be
using the decimal number system to show how the positional notation system
works. Assuming that we have the number 631, what this number means? Each digit
represents that digit itself amplifying the power of 10 represented by its
position. Don’t forget that the decimal point is supposed to be on the right of
the number if it is not explicitly shown. Starting at the decimal point and moving
to the left, the columns represent increasing powers of 10. Now we are showing
the steps.
102 101 100
6 3 1
The base of the numbers is 10 because
the number is initially in decimal form, and the exponent (the superscript)
tells what power of 10. The exponents increase by 1 with every move to the next
position to the left. Note that the exponents also decrease by 1 with each
column move to the right, so the first position to the right of the decimal
place tells how many 10-1's are in the number, and the second position to the
right of the decimal place tells how many 10-2's are in the number and so forth
for other bases.
Recall that any number to the zeroth
power is 1, and any number to the first power is that number itself. So the
column labeled 100 denotes how many 1's are in the number, the
column labeled 101 represents how many 10's are in the number, and the column
labeled 102 shows how many 100's are in the number as shown below;
100's column 10's column 1's column
6 3 1
Thus this number has 1 ones, 3 tens,
and 6 hundreds, where
1 x 100 = 1 x 1 = 1
3 x 101 = 3 x 10 = 30
6 x 102 = 6 x 100 = 600
So by summing up all those numbers, we
will get the number in base 10.
1 + 30 + 600 = 63110
But we basically do not put the
subscript for base 10 because base 10 numbers are the numbers we are always
using. On the other hand, we do for the other bases like 100011002,
112358 and AE4316. This is due to the using of these
bases that is not too wide excluding for those who are involved with computers.
Operations on Integers
There are some operations in dealing
with integers in number system. I’m sure every single studying person learn
basic operation involving numbers, right. What else could it be other than
addition, subtraction, multiplication, division and in this case we also had to
deal with overflow in those operations.
Let’s get started with the simplest
operation that is addition. If the yield is out of the range, there will be an
overflow. Simply, adding positive and negative operands will result in no
overflow. But adding two positive operands will result an overflow, but if only
the yield sign is equal to 1. In the other hand, the operation of summing up
two negative operands will result an overflow only if the result sign is 0.
Saem goes for subtraction operation of
integers. There will be an overflow when the value produced is out of the
range. There will be no overflow when the operation involves subtraction of two
positive or two negative operands. But if the result sign is 0 after
subtracting positive from negative operand, there will be an overflow. The overflow
will also exist when the result sign is 1 after subtracting negative from
positive operand.
Dealing with Overflow
Here are some ways to handle the
overflow cases. Somehow, some languages such as C. will just practically ignore
overflow. We can use MIPS addu, addui, subu instructions to deal with it. We can
also use other languages as Ada and Fortran. But using these languages require a
raising an exception. So it is quite difficult to be settled. We might also use
MIPS add, addi, sub instructions for other incentive. Plus, we can invoke
exception handler on overflow as one of the ways to deal with it. Next step is
just by saving the PC in exception program counter (EPC) register.
As multimedia-majoring students, we
absolutely need to know these, that arithmetic is such a basic thing in
multimedia. It is the foundation that makes the media up, compiling those media
becoming multimedia, such a perfect, attentive presentation medium to be used
by us, the users. In this prospect, graphics and media processing operates on
vectors of 8-bit and 16-bit data.
Next, let’s move on to multiplication
operation of integers. As an example taken from our notes,
The upper operand is called
multiplicand while the lower one is known as the multiplier. The length of
product is the sum of operand lengths.
The next one is division operation. Note
that if the divisor is less than or equal to its dividend bits, it will be 1
bit in quotient, and then subtracting the two operands. Otherwise, the other 0
bit in quotient, we will just bring them down next to dividend bit.
Floating Point
Floating points is the way of representation
for non-integral numbers including smallest and largest numbers. As if in
scientific notation;
–2.34 × 1056
+0.002 × 10–4
+987.02 × 109
Floating point standard is defined by “IEEE
Std 754-1985”. It is developed in response to divergence of representations
which is a portability issues for scientific code. IEEE Std 754-1985 is now
almost universally adopted. Other than that, there are two ways of representations
the floating point standards those are single precision (32-bit) and double
precision (64-bit).
Okay! Now we are finished with our short brief explanation about arithmetic involved in computers. We do hope that you understand and the information we have prepared will be able to help you in improving your understanding towards arithmetic Number System. A lot of thanks for spending your precious time to visit our page!
^_^
