Computer Math

CPS120: Data Representation

Representing Data

• The computer knows the type of data stored in a particular location from the context in which the data are being used; – i.e. individual bytes, a word, a longword, etc– 01100011 01100101 01000100 01000000

• Bytes: 99(10, 101 (10, 68 (10, 64(10

• Two byte words: 24,445 (10 and 17,472 (10

• Longword: 1,667,580,992 (10

2

Natural NumbersZero and any number obtained by repeatedly adding one to it.

Examples: 100, 0, 45645, 32

Negative NumbersA value less than 0, with a – sign

Examples: -24, -1, -45645, -32

Numbers

3

IntegersA natural number, a negative number, zero

Examples: 249, 0, - 45645, - 32

Rational NumbersAn integer or the quotient of two integers

Examples: -249, -1, 0, ¼ , - ½

Numbers (Cont’d)

4

How many ones are there in 642?

600 + 40 + 2 ?

Or is it

384 + 32 + 2 ? -- Octal

Or maybe…

1536 + 64 + 2 ? -- Hexadecimal

Natural Numbers

5

642 is 600 + 40 + 2 in BASE 10

The base of a number determines the number of digits and the value of digit positions

Natural Numbers

6

Continuing with our example…642 in base 10 positional notation is:

6 x 10² = 6 x 100 = 600

+ 4 x 10¹ = 4 x 10 = 40 + 2 x 10º = 2 x 1 = 2 = 642 in base 10

This number is in base 10

The power indicates the position of

the number

Positional Notation

7

 dn * Rn-1 + dn-1 * R

n-2 + ... + d2 * R + d1

As a formula:

642 is:  63 * 102 +  42 * 10 +  21

R is the base of the number

n is the number of digits in the number

d is the digit in the ith position

in the number

Positional Notation

68

What if 642 has the base of 13?

642 in base 13 is equivalent to 1068 in base 10

+ 6 x 13² = 6 x 169 = 1014 + 4 x 13¹ = 4 x 13 = 52 + 2 x 13º = 2 x 1 = 2

= 1068 in base 10

Positional Notation

Representing Real Numbers

• Real numbers have a whole part and a fractional part. For example 104.32, 0.999999, 357.0, and 3.14159 the digits represent values according to their position, and those position values are relative to the base.

• The positions to the right of the decimal point are the tenths position (10-1 or one tenth), the hundredths position (10-2 or one hundredth), etc.

Representing Real Numbers (Cont’d)

• In binary, the same rules apply but the base value is 2. Since we are not working in base 10, the decimal point is referred to as a radix point.

• The positions to the right of the radix point in binary are the halves position (2-1 or one half), the quarters position (2-2 or one quarter), etc.

Representing Real Numbers (Cont’d)

• A real value in base 10 can be defined by the following formula:

• The representation is called floating point because the number of digits is fixed but the radix point floats.

Representing Real Numbers (Cont’d)

• Likewise, a binary floating –point value is defined by the following formula:sign * mantissa * 2exp

Representing Real Numbers (Cont’d)

• Scientific notation is a term with which you may already be familiar, so we mention it here. Scientific notation is a form of floating-point representation in which the decimal point is kept to the right of the leftmost digit.

• For example, 12001.32708 would be written as 1.200132708E+4 in scientific notation.

Representing Text

• To represent a text document in digital form, we simply need to be able to represent every possible character that may appear.

• There are finite number of characters to represent. So the general approach for representing characters is to list them all and assign each a binary string.

• A character set is simply a list of characters and the codes used to represent each one. By agreeing to use a particular character set, computer manufacturers have made the processing of text data easier.

Alphanumeric Codes

• American Standard Code for Information Interchange (ASCII)– 7-bit code– Since the unit of storage is a bit, all ASCII codes are

represented by 8 bits, with a zero in the most significant digit– H e l l o W o r l d– 48 65 6C 6C 6F 20 57 6F 72 6C 64

• Extended Binary Coded Decimal Interchange Code (EBCDIC)

The ASCII Character Set

• ASCII stands for American Standard Code for Information Interchange. The ASCII character set originally used seven bits to represent each character, allowing for 128 unique characters.

• Later ASCII evolved so that all eight bits were used which allows for 256 characters.

The ASCII Character Set (Cont’d)

The ASCII Character Set (Cont’d)

• Note that the first 32 characters in the ASCII character chart do not have a simple character representation that you could print to the screen.

The Unicode Character Set

• The extended version of the ASCII character set is not enough for international use.

• The Unicode character set uses 16 bits per character. Therefore, the Unicode character set can represent 216, or over 65 thousand, characters.

• Unicode was designed to be a superset of ASCII. That is, the first 256 characters in the Unicode character set correspond exactly to the extended ASCII character set.

The Unicode Character Set (Cont’d)

A few characters in the Unicode character set