Few basic things to know
- Most Significant Bit (MSB) also called as Most Significant Digit (MSD). This is indeed the first digit (from left) in a binary number. More accurately the digit with the highest radix (base) value.
- Least Significant Bit (LSB) or Least Significant Digit (LSD). This is the digit of a binary number with the least radix value, probably the last digit.
- In binary number one digit is called as a bit.
- Computer memory is a place that store data and it has a fixed length of bits. Generally 8bit, 16bit, 32bit and 64bit in length.
Data Representation
- Computer uses a fixed number of bits to represent a single piece of data. Generally computers use 8bit, 16bit, 32bit and 64bit in size of binary numbers to represent different type and size of data.
- N-number of bit memory (location) can store 2n of different types of values.
- For an example 8bit memory can store 256 of different data values.
- 3bit memory can store 8 distinct values (patterns) 000, 001, 010, 011, 100, 101, 110, or 111
- When we interpret these values, we need to know exactly what represent with these values.
- If we know that these represent decimal numbers, then we know it is from 0-7 then it is a different value.
- So this interpretation of a binary pattern called data representation or encoding.
Integer Representation
Integers are whole numbers or fixed point numbers. The radix point is fixed after the LSB.
Example: 3774 is a fixed point number (integer)
2949.94 is a floating point number (real number)
qIn a
computer integers and floating point numbers are treated differently and
processed separately. Floating point numbers are processed in a separate
processor called floating point processor.
qComputers
use fixed number of bits to represent integers. Commonly use bit lengths are
8bit, 16bit, 32bit and 64bit.
qThere
are two types of integers
qUnsigned
integers: can represent zero and positive numbers only
qSigned
integers: can represent zero, positive and negative numbers.
q
qThere
are three schemas (ways) to represent signed integers.
qSign
Magnitude representation
q1’s
complement representation
q2’s
complement representation
qWe
learned about two different types of Integers;
qUnsigned Integers – that can represent
only zero and positive numbers. The value of an unsigned integer interpreted as
the “magnitude of underlying binary pattern or the normal way we convert binary
to decimal.
q
Example: 8bit binary number 0100
0010B (B = Binary)
0100 0010B = 2x21 + 2x26 = 4+64 = 68D (D = Decimal)
8bit = 0000 0000
16bit = 0000 0000 0000 0000
32bit = 0000
0000 0000 0000 0000
0000 0000 0000
64bit = 0000
0000 0000 0000 0000
0000 0000 0000 0000
0000 0000 0000 0000
0000 0000 0000
All above numbers represent zero in
different bit systems. The bit length of each system is different.
n-bit Unsigned Integers
qAn
n-bit pattern can represent 2n
distinct integers. This range from 0 to (2n)-1
When you are programming you need to
select the correct and appropriate bit size to store integers. If you want to
store small numbers less than 255, you can select 8bit number, selecting 32bit
length to store small numbers is a waste of resources.
Signed Integers
qSigned
integers can represent zero, positive and negative numbers.
qThe
MSB (Most Significant Bit) – Normally the first bit from left, represent the
sign of the integer. Zero (0) to represent positive (+) sign and One (1) to
represent negative (-) sign.
qThe
magnitude of the integer represent differently in different schemas.
qThere
are three schemas to represent signed integers.
1.Sign
Magnitude Representation
2.1’s
Complement Representation
3.2’s
Complement Representation
Example of Signed Integers:
+2344
is a positive integer -23434 is a
negative integer
n-bit Sign Magnitude Representation
qMost
Significant Bit (MSB) is the sign bit. 0 representing positive and 1
representing negative signs.
qThe
remaining (n-1) bits represent the magnitude / absolute value of the integer
interpreted according to its binary pattern.
Examples:
n=8 (8bit number) and binary number
is 01000001
Sign bit (MSB) is 0 à this
is a positive number
Absolute value 1000001 à 65
01000001B = +65D
B = Binary, D = Decimal
Examples:
n=8 (8bit number) and binary number
is 00000000
Sign bit (MSB) is 0 à this
is a positive number
Absolute value 0000000 à 0
00000000B = +0D
n=8 (8bit number) and binary number
is 10000000
Sign bit (MSB) is 1 à this
is a negative
number
Absolute value 0000000 à 0
10000000B = -0D
Problems (Drawbacks / Limitations)
of Sign Magnitude Representation:
qThere
are two representation for the number zero
qEven
the number zero need to have a sign, since it is not necessary nor accurate and
may leads to confusions.
qPositive
and negative numbers need to process separately.
n-bit 1’s Complement Representation
In this representation, the MSB is
the sign bit 0 represent positive and 1 represent negative numbers. The
remaining n-1 bits represent the magnitude (value) of the integer.
qIn
positive integers, the value of the integer is the binary pattern (value) of
the n-1 bit magnitude. Same as Sign Magnitude Representation.
qFor
negative integers, the absolute value is the n-1 binary pattern value of the
inverse magnitude (complement).
Example: 1 000 0001B (sign bit = 1,
negative integer)
Complement of 000 0001 is 111 1110
= 126D
1 000 0001B = -126D
In the 1’s Complement
Representation, there are two different representation for zero;
0000 0000B and 1111 1111B hence +0
and -0
Since the processing method of
positive and negative integers are different, computer need to process the
positive and negative numbers separately.
n-bit 2’s Complement Representation
In 2’s Complement Representation
also the MSB is the sign bit 0 represent positive and 1 represent negative
integers.
For positive integers, the value of
the integer equal to the magnitude of n-1 binary pattern
For negative integers, the absolute
value of the integer equal to the magnitude of the complement of n-1 binary
pattern plus one.
Example: 1 000 0001B (MSB = 1 is a
negative number)
Complement of 000 0001 = 111 1110
Let’s add 1 to the complement = 111
1111 = 127
1 000 0001B = -127D
Advantages of 2’s Complement
Representation
qThere
is only one representation for the number 0
qIn
binary addition and subtraction, both positive and negative numbers can treated
together (process together)
qSubtraction
can use the addition logic
qComputers
use 2’s Complement Representation for processing integers.
Addition, Subtraction, Overflow and Underflow
Addition of positive integers
Example: n=8, 65D + 5D = 70D
65D à 0100 0001B
5D à 0000 0101B
65D + 5D à 0100 0110B OK 70D within the range of -128 to
+127
Subtraction is the addition of
positive and negative integers
Example: n=8, 65D – 5D = 60D à 65D
+ (-5D)
65D à 0100 0001B
-5D à 1111 1011B
65D + (-5D)
à 0011 1100B à 60D
OK within
the range of -128 to +127
Overflow and underflow happens when
the result is not within the acceptable range
Example of Overflow: n=8, 127D + D
= 129D (not within -128 and +127, exceed maximum value)
127D à 0111 1111B
2D à 0000 0010B
127D + 2D à 1000 0001B à
-127D wrong answer
Example of underflow: n=8, -125D –
5D = -130D (not within -128 and +127, below the minimum value)
-125D à 1000 0011B
-5D à 1111 1011B
-125D + (-5D)
à 0111 1110B à
+126D wrong answer
Range of n-bit 2’s complement signed integers
An n-bit 2’s complement integer can
represent integers from -2(n-1) to
+2(n-1)-1. This can represent all the
integers within the range without missing any integers.
8bit from -128 to +127
16bit from -32,768 to +32,767
32bit from -2,147,483,648 to + 2,147,483,647
64bit from -9,223,372,036,854,775,888
to + 9,223,372,036,854,775,887
Decoding 2’s complement numbers
•Check
the sign bit first. If sign bit = 0, the number is positive and the absolute
value if the binary value of remaining n-1 integers.
•If
the sign bit = 1, the integer is negative. To get the absolute value;
•Invert remaining n-1 bits and add 1 or
•Scan from left until the first occurrence
of 1. Flip all values from that point to the left. Binary value of this pattern
gives the absolute value of the negative integer.
Example:
N=8 bit pattern = 1 100 0100 B
Negative number
Scan from right, until the first
occurrence of 1, flip all values from that point to left
011 1100 B
= 60 D
Represented negative value is -60 D
Big Endian vs. Little Endian
Modern computers store one byte of
memory in single memory location / address. 1 byte = 8 bits. Therefore in 32bit
integer stores in 4 memory locations.
Term “Endian” means the order of
storing bytes in memory. In Big Endian standard – the most significant byte
stores first in the lowest memory location. In Little Endian standard, the
least significant byte (LSB) stores in the lowest memory location.
Floating Point Number Representation
A floating point number (also
called as a real number) is a number with a decimal point. A floating point
number can represent a very large positive or negative number as well as a very
small negative or positive number. A real number has a range of -1.23x1088 to 1.25x1088 including
zero.
Scientifically a floating point
number represent as a number with a fraction and exponent of a certain radix. F x rE
Same floating point number can
represent in different ways. Consider 5.456D
We can represent this number in
following ways:
54.532 x 100
5.4532 x 101
0.54532 x 102 and so on….
Since there are many ways to
represent the same floating point number, we need to define a standard, so
every floating point number should be in this standard. To make it standard, we
can normalized the fraction part. In normalized form, there is only single none
zero digit represent before the radix point. In above example 5.4532 x 101 is the normalized form of 54.532
number.
In computers, there is an issue of
“Loss of Precision” in storing floating point numbers. An n-bit system can only
stores 2n distinct finite numbers. Even
within a small range 0.00 – 0.01, there are infinite number of decimal numbers.
But the number of bits in a computer system is limited, only a the approximate
number used, resulting in loss of accuracy.
Processing of a real number
(floating number arithmetic) is very much less efficient and slower than
processing of integers. In computers, real numbers uses fraction (F) and
exponent (E) with radix (r) of 2. Both F and E can be negative or positive. In
modern computing uses IEEE 754 standard to represent floating point numbers.
There are two representation schemes
32 bit single precision and
64 bit double precision
IEEE-754 32-bit Single-Precision Floating-Point Numbers
In 32-bit single precision floating
point number representation, decimal number represent in following structure:
•The
most significant bit is the sign bit. Zero for positive numbers and One for
negative numbers.
•Next
following 8bit represent the exponent (E)
•Remaining
23bit represent the fraction (F)
You may wonder why radix (r) is not
represent here. Because radix is always 2 for binary numbers (in computers).
IEEE-754 64-bit Double-Precision Floating-Point Numbers
64-bit double precision floating
point number representation is similar to 32-bit single precision floating
point number representation. Only difference is assignment of bit range for
exponent and fraction.
•The
most significant bit is the sign bit. Zero for positive numbers and One for
negative numbers.
•Next
following 11-bit represent the exponent (E)
•Remaining
52-bit represent the fraction (F)
Character Encoding
In computer memory, characters are
represented (encode) using character schemes called as “character set”,
“charset”, “character map” or “code page”. There are several standard schemes
such as ASCII, EBCDIC, UTF, etc.
7-bit
ASCII Code (aka
US-ASCII, ISO/IEC 646, ITU-T T.50)
•ASCII
(American Standard Code for Information Interchange) is one of the earlier
character coding schemes.
•ASCII
is originally a 7-bit code. It has been extended to 8-bit to better utilize the
8-bit computer memory organization. (The 8th-bit was originally used for parity
check in
the early computers).
This is called ASCII-8 Code.
EBCDIC Code
This is stands for “Extended Binary Code
Decimal Interchange Code”. This is a 8-bit code scheme without parity. With
this up to 256 characters can be coded.
Unicode (aka ISO/IEC 10646
Universal Character Set)
•Before
Unicode, no single character encoding scheme could represent characters in all
languages.
•Unicode
aims to provide a standard character encoding scheme, which is universal,
efficient, uniform and unambiguous. Unicode standard is maintained by a
non-profit organization called the Unicode Consortium (@ www.unicode.org).
Unicode is an ISO/IEC standard 10646.
•Unicode
is backward compatible with the 7-bit US-ASCII and 8-bit Latin-1 (ISO-8859-1).
That is, the first 128 characters are the same as US-ASCII; and the first 256
characters are the same as Latin-1.
•Unicode
originally uses 16 bits ,
which
can represent up to 65,536 characters.
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