1. Data Representation

2. Topics Bit patterns Binary numbers Data type formats
Character representation
Integer representation
Floating point number representation

3. Data Representation
Data representation refers to the manner in which data is stored in the computer
There are several different formats for data storage
It is important for computer problem solvers to understand the basic formats

4. Why is it Important? As an example:
We will learn that since we have finite storage, it is possible to overflow a storage location by trying to store too large a number
Most programming languages provide multiple data types each providing different length storage for variables
It is up to the programmer to choose the data type with a length that won’t overflow
Knowing how numbers are represented in storage helps one to understand this

5. Bit Pattern
As you recall from an earlier presentation, data may take various forms: characters, numbers, graphical, etc.
All data is stored in the computer as a sequence of bits, that is, binary digits
This is a universal storage format for all data types, and it is called a bit pattern

6. Bits and Bytes
A bit is the smallest unit of data stored in a computer and it has a value of 0 or 1
It’s like a switch, on (1) or off (0)
In computers, bits are stored electronically in RAM and auxiliary storage devices by two-state digital circuits
The storage device itself doesn’t know what the bit pattern represents, but software (application software, operating system, and I/O device firmware) stores and interprets the pattern
That is, data is coded then stored and when retrieved it is decoded
A byte is a string of 8 bits and is called a character when the data is text

7. Binary Numbers
Each bit pattern is a binary number, that is, a number represented by 0’s and 1’s rather than 0, 1, 2, …, 9 as decimal numbers are
For example, bit patterns like 1010 and are also binary numbers
Binary numbers are based on powers of 2 rather than powers of 10 as decimal numbers are

8. Data Type Formats
As we have learned, fundamentally all data is stored as a bit pattern
But the different data types have different bit pattern formats
We want to learn the formats for:
Characters (for example, left, Lane, a, ?, \)
Integers (for example, 1, 453, -10, 0)
Floating point numbers (for example, , 1.2, , 0.009)

9. Character Representation
The American Standard Code for Information Interchange (ASCII) is the scheme used to assign a bit pattern to each of the characters
ASCII charts come in different flavors:
Some have 7 bit strings, some 8 or more
Some show the binary code for the various characters as well as the code represented in other number systems, e.g., decimal, hex, octal
For example, the letter A has the ASCII code:
in binary for a 7-bit chart
65 in decimal
41 in hex
101 in octal
Note: All four of these numbers represent the same value but using different number systems

10. Subset of ASCII Chart

11. ASCII Chart
Uppercase characters have different ASCII codes than lowercase
Uppercase characters come before lowercase
Numbers come before letters
The special characters are spread around
The numbers and upper and lowercase characters are in adjacent groupings, so that their codes increment by one

12. ASCII Chart
The only difference between the binary codes for the upper and lowercase characters is the sixth bit, that is, the decimal code for a lowercase character is 32 greater than the uppercase character’s decimal code
ASCII codes before decimal 32 are control characters (nonprintable) like bell, backspace, and carriage return
The final ASCII code in the 7-bit chart is the control character DEL with decimal code 127

13. Extended ASCII & Unicode
The eight-bit ASCII chart is sometimes called Extended ASCII
The seven-bit ASCII codes are the same in eight-bit chart except have a zero at the left
Some manufactures use the extra bit to create additional special characters, these however are nonstandard, e.g., using decimal 171 for ½, or 246 for ÷
Unicode is another scheme developed so that the many symbols in international languages may be represented. It also uses bit patterns. UTF-32 uses 32 bits.

14. Numeric Representation
ASCII codes are an inefficient method for representing numbers
For example, the number 1,024 using 8-bit ASCII would require four bytes or 32 bits of storage
Arithmetic operations on numbers represented in ASCII are very complicated
Representing the precision of a number, that is, the number of digits stored, may require large amounts of space when stored in ASCII
There are more efficient schemes for numbers

15. Integer Representation
An integer is a whole number, that is, a number without a decimal portion
Integers may be positive, negative, or zero
A plus-sign or minus-sign in front of the number is used to represent positive and negative numbers
The plus-sign is not required for positive numbers and zero
There are two categories of integer representation: unsigned and signed

16. Unsigned Integer
An unsigned integer is an integer without a sign, that is, a non-negative integer
They range from zero to infinity, but no computer can store all the integers in that range
So, a maximum unsigned integer is defined
This maximum is based on the number of bits used to store an integer
Let’s use 8 and 16-bit (1 and 2 bytes) storage locations in our examples
The length of storage is set by the data type the programmer specifies for a variable

17. Unsigned Integer
An unsigned integer is stored as its value when represented as a binary number
Leading zeros are added to fill out the storage location
For example, the decimal number 9 is represented as when stored in 1-byte because = 910
When stored in a 2-byte location, 9 would be represented as

18. Unsigned Integer
One may use the following table to work with binary numbers:
For example, given , what decimal number does it represent?
Add the non-negative powers of two, that is, = 9

19. Unsigned Integer
One may use the same table to go the other way, that is, given the decimal number 13, what is its binary representation?
Find the largest power of 2 that doesn’t exceed the number and place a 1 in that cell:
Subtract that power of 2 from the number and use this as the new number: 13 – 8 = 5

20. Unsigned Integer
Then continue in this way until the sum of the powers of two equals the number:
Now, 5 – 4 = 1, and so finally:
Note that = 13

21. Unsigned Integer Then fill in the remaining cells with zeros:
So, the unsigned integer representation of decimal 13 is when stored in 1-byte

22. Unsigned Integer
If one tries to store a number in a memory location that is not large enough we have what is called overflow
In this case, depending on the system, one may or may not receive an error message
So, one must not store a number that is larger than the maximum for a given length of storage
The maximum number storable in 1-byte is 255

23. Unsigned Integer
For example, if one tries to store 256 in 1-byte there is overflow because the largest value storable in 8 bits is 255 as one can see from the following table:
Note that = 255

24. Signed Integer
A sign-and-magnitude format is used to allow for positive and negative numbers (and zero)
The leading bit is designated as the sign bit: 0 for positive or zero, 1 for negative
The remaining bits represent the value
So, in 1-byte of storage the maximum number storable is not 255 as it was for the unsigned integer representation, but 127:
Note that = 127

25. Signed Positive Integer
To determine what the sign-and-magnitude representation of a positive decimal number is:
Convert the decimal number to binary
If needed add leading zeros to fill the storage location
For example, decimal 12 is represented in 1-byte as because = 12:

26. Signed Positive Integer
Going the other way, given a sign-and-magnitude representation for a positive number, one can interpret it as follows:
Leftmost bit will be 0 indicating positive
Convert the remaining bits to a decimal number
For example, is decimal 17:
Because = 17

27. Signed Negative Integer
For negative numbers, two’s complement format is used
Two’s complement is still a sign-and-magnitude format
In two’s complement, some of the magnitude bits are flipped from 0 to 1 or 1 to 0

28. Signed Negative Integer
To determine what the two’s complement representation of a negative decimal number is:
Ignore the sign and convert the decimal number to binary
If needed add leading zeros to fill the storage location
Leave all the rightmost 0’s and first 1 unchanged, but flip the remaining bits
Make the sign bit 1
For example, decimal -14 is represented in 1-byte as because (see next slide)

29. Signed Negative Integer
Convert 14 to binary ( = 14) and make leading bits zero:
Leave the rightmost 0’s and first 1 as is, but flip the remaining bits:
Make sign bit 1:

30. Signed Negative Integer
Going the other way, given a two’s complement representation for a negative number, one can interpret it as follows:
Leave the rightmost bits up to and including the first 1 unchanged, but flip the remaining bits
Convert the binary number to decimal
Put a minus-sign in front
For example, is decimal –22 because (see next slide)

31. Signed Negative Integer
Flip all but the rightmost 1 and any following 0’s:
Convert the binary number to decimal:
We get 22 because = 22
Put a minus-sign in front yielding -22

32. Signed Negative Integer
Two’s complement is the standard representation for negative integers in modern computers
This is because arithmetic operations are simple to implement when integers are stored this way (but this concept is beyond the scope of the course)
Although on the surface it seems complicated, at a deeper level it allows for simplicity of operations

33. Signed Negative Integer
An alternative but equivalent method for converting a negative number to its two’s complement representation is:
Ignore the sign and convert the decimal number to binary
If needed add leading zeros to fill the storage location
Flip all the bits
Add 1 to the result of the last step
Make the sign bit 1
Some people find this easier

34. Signed Negative Integer
For example, First, convert 14 to binary ( = 14) and make leading bits zero:
Flip all the bits:
Add 1:
Make the sign bit 1:

35. Floating Point Number Representation
Float point numbers are those that have a decimal portion (mathematicians call these real numbers)
Numbers like , , and
The method that is used allows for very large or very small numbers to be stored using the same format

36. Floating Point
The main idea in this format is that the decimal point is allowed to “float”
That is, there is an “actual” decimal location in the original number, and there is “stored” decimal location that is usually different
The original number is normalized by moving the decimal place so there is only one digit to the left

37. Floating Point
The basic idea can be seen from an example although this description glosses over many details
The number is normalized by moving its decimal point two places to the left to become , and this number is stored and is called the mantissa
Also, the fact that the decimal point was moved left by 2 is stored so that the original number may be reconstructed and this is called the exponent
The sign of the number is also stored (0 for positive or zero, 1 for negative)

38. Floating Point However, it is actually more complicated than that
The exponent and mantissa are actually stored in binary
And the value stored as the mantissa is only the fractional part of the binary number once the decimal point has been moved so that there is a binary 1 at the left, that is, is stored as and the leading 1 is assumed

39. Floating Point
The representation of numbers in floating point involves a couple procedures that are complicated and beyond the scope of the course
These are “repetitive multiplication of a decimal fraction by 2,” and the “excess system” for storing positive and negative numbers
So, we won’t be converting the numbers manually ourselves

40. Floating Point
However, the procedure used to store a number in floating point representation is:
Store a 0 (positive) or 1 (negative) in the sign field
Convert the integer part to binary
Convert the decimal part (fraction) to binary by using “repetitive multiplication by 2”
Combine the two binary numbers with a decimal point between
Move the decimal point so that there is a 1 bit at the left and store the remaining bits in the mantissa field
Store the number of places moved using the “excess system” in the exponent field

41. Floating Point
Computers store data in binary and in finite space, i.e., they are discrete, finite systems
However real numbers form a continuous, infinite system
Hence, computers can only approximate real numbers
The precision of a floating point number is how close the stored number is to the original number

42. Floating Point Small Basic Example:
Mathematically c should be 0 but what does the program display for c?
a = 2 / 3
b = 2 * (1 / 3)
c = a - b
TextWindow.WriteLine(c)

43. Floating Point
The more bits available for the mantissa field the more digits of the original number may be stored
Programming languages normally allow the programmer to define the precision by the data type chosen

44. Floating Point
Institute of Electrical and Electronics Engineers (IEEE) standards:
Single-Precision (4 bytes)
Double-Precision (8 bytes)

45. Floating Point Trade off:
Double precision numbers require more space and therefore programs using them may run slower
But operations using double precision numbers will be more precise

46. Summary Data are stored as bit patterns
A bit pattern is a binary number
There are various data type formats
Characters are represented in ASCII

47. Summary Integers are represented as either
Unsigned – stored as the binary number equivalent to the original
Signed Positive – stored using the sign-and-magnitude format where the magnitude is the binary equivalent
Signed Negative – stored using the sign-and-magnitude format where the magnitude is in the two’s complement format
Floating point numbers are represented using sign, exponent, and mantissa

48. Terminology Data representation Unsigned integer representation Bit
Byte
Bit pattern
Binary number
Character
Integer
Floating point number
ASCII
Control characters
Extended ASCII
Unicode
Unsigned integer representation
Overflow
Sign-and-magnitude representation
Sign
Two’s complement representation
Floating point representation
Normalize
Exponent
Mantissa
Precision
Single-precision
Double-precision

49. End