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### 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 and17,472 (10
- Longword: 1,667,580,992 (10

Numbers

Natural Numbers

Zero and any number obtained by repeatedly adding one to it.

Examples: 100, 0, 45645, 32

Negative Numbers

A value less than 0, with a – sign

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

2

Numbers (Cont’d)

Integers

A natural number, a negative number, zero

Examples: 249, 0, - 45645, - 32

Rational Numbers

An integer or the quotient of two integers

Examples: -249, -1, 0, ¼ , - ½

3

Natural Numbers

How many ones are there in 642?

600 + 40 + 2 ?

Or is it

384 + 32 + 2 ? -- Octal

Or maybe…

1536 + 64 + 2 ? -- Hexadecimal

4

Natural Numbers

642 is 600 + 40 + 2 in BASE 10

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

5

Positional Notation

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

The power indicates

the position of

the number

This number is in

base 10

6

Positional Notation

R is the base

of the number

As a formula:

dn * Rn-1 + dn-1 * Rn-2 + ... + d2 * R + d1

n is the number of

digits in the number

d is the digit in the

ith position

in the number

642 is: 63 * 102 + 42 * 10 +21

7

Positional Notation

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

6

8

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)

- 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

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