Problem 19 - Discrete Mathematics with Applications

Question

Let \(A=132,33, \ldots, 1261 .\) Let \(f : A \rightarrow\) ASCII defined by \(f(n)=\) character with ordinal number \(n .\) Find \(f(n)\) for each value of \(n .\) $$ 90 $$

Step-by-Step Solution

Answer

The ASCII character corresponding to ordinal number 90 is 'Z'.

Step 1: Understand what is the ASCII value

ASCII (American Standard Code for Information Interchange) is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Each character (letter, digit, or any symbol) has an assigned numerical value (ordinal number) called the ASCII value.

Step 2: Find the ASCII character for the given ordinal number

The given ordinal number is 90. To find the corresponding ASCII character, we can look up an ASCII table and see which character has the ASCII value of 90. You can also use programming languages or online tools to find the character.

Step 3: Convert the ASCII value to the corresponding character

In the ASCII table, the character with the ASCII value of 90 is 'Z'. So, \(f(90) = 'Z'\). Hence, the ASCII character for the given ordinal number 90 is 'Z'.

Key Concepts

Character Encoding

Character encoding is a method used to convert characters in text into a format that can be easily processed and understood by computers. It acts like a bridge, translating human text into bits and bytes that a computer can handle. Without character encoding, our computers wouldn’t be able to recognize the text we input or store it efficiently.
Different character encodings exist for various purposes, such as UTF-8, ISO-8859-1, and the classic ASCII. Each encoding system has its own set of rules dictating how characters are represented.

• ASCII is often considered the foundational character encoding scheme, especially for English text.
• Character encoding ensures that text looks the same everywhere, be it on a screen or in a document.

Ordinal Number

An ordinal number is a concept in mathematics that represents position or order in a set. In the context of ASCII and character encoding, an ordinal number refers to the numerical position assigned to a specific character within a character set.

• Each character in an encoding system like ASCII is associated with a unique ordinal number.
• For example, in ASCII, the capital letter 'A' has the ordinal number 65.

These numbers are crucial because they allow computers to look up and manipulate characters efficiently. By knowing a character's ordinal number, a computer can understand which symbol it corresponds to and how it should be processed or displayed.

ASCII Table

The ASCII table is a standardized list mapping characters to their corresponding ordinal numbers in the ASCII encoding system. This table consists of 128 unique entries, representing symbols, digits, uppercase, and lowercase letters.

• The ASCII value ranges from 0 to 127.
• Control characters (like newline or tab) are also part of the ASCII table.

For example, the letter 'Z' is found at position 90 in the ASCII table, which means it has an ASCII value of 90. Whenever you need to determine the ASCII character for a specific number, or vice-versa, the ASCII table is the go-to reference. Many online tools and programming functions can also automate this lookup for convenience.

Text Representation

Text representation in computers is fundamentally about converting human-readable text into a computer-friendly format. Through character encoding schemes like ASCII, text representation ensures that all characters—whether letters, numbers, or symbols—are stored as specific sequences of bits.

• It ensures consistency across different devices and platforms.
• Stored data can be efficiently retrieved and displayed.

For instance, when you press a key on your keyboard, the corresponding character is turned into its numeric representation (like 'A' to 65 in ASCII) and then processed by your computer in binary form. Text representation is an underlying process, making sure what you write is saved accurately and appears the same on any screen.