Lecture 2: Digital Systems and Information Study Guide

Lecture 2 covers the foundational principles of digital systems, focusing on how information is represented through various number systems and mathematical frameworks.
The material is presented by the Computer Science Department at Florida Polytechnic University.

Positional notation is the method used to represent numbers where the value of a digit depends on its position relative to the radix point.
Key variables and components identifying positional notation include:
- $r$ : Represents the base or the radix of the number system (e.g., $r=10$ for decimal, $r=2$ for binary).
- $n$ : Indicates the number of digits located to the left of the decimal (or radix) point.
- $m$ : Indicates the number of digits located to the right of the decimal (or radix) point.
- $A$ : Represents a coefficient. This coefficient is a value that can be represented by one of the digits allowed within the specific base $r$ .

Understanding the powers of two ( $2^{n}$ ) is critical for binary system calculations and computer science architecture. The following table provides the values for $n$ from $0$ to $23$ :

$n$	$2^{n}$	$n$	$2^{n}$	$n$	$2^{n}$
$0$	$1$	$8$	$256$	$16$	$65,536$
$1$	$2$	$9$	$512$	$17$	$131,072$
$2$	$4$	$10$	$1,024$	$18$	$262,144$
$3$	$8$	$11$	$2,048$	$19$	$524,288$
$4$	$16$	$12$	$4,096$	$20$	$1,048,576$
$5$	$32$	$13$	$8,192$	$21$	$2,097,152$
$6$	$64$	$14$	$16,384$	$22$	$4,194,304$
$7$	$128$	$15$	$32,768$	$23$	$8,388,608$

This table illustrates the mapping between decimal (base 10) integers and their 4-bit binary (base 2) equivalents for values $0$ through $15$ :

Binary to Decimal: The process of converting a binary number into its decimal equivalent by summing the products of each bit and its corresponding power of $2$ .
Decimal to Binary (First Approach): Initial methodology for converting base 10 integers into base 2 representation.
Decimal to Binary (Second Approach): An alternative methodology, often involving successive division by $2$ and tracking the remainders to determine the binary string.

Decimal to Other Bases: Generalized procedures for converting decimal numbers into any target base $r$ .
Other Bases to Decimal: Using the polynomial expansion of positional notation to find the decimal value of a number in an arbitrary base $r$ .
Intersystem Conversions (Octal, Hexadecimal, and Binary):
- Octal and Hexadecimal to Binary: Techniques for direct conversion between these power-of-two bases without first converting to decimal.
- Binary to Octal and Hexadecimal: Grouping binary digits (bits) into sets of three (for octal) or four (for hexadecimal) to simplify representation.

Numbers with Radix Point to Decimal: Converting fractional components in non-decimal bases into their decimal fraction equivalents.
Decimal Fractions to Binary: Specialized procedures (typically successive multiplication) to convert a decimal fraction into a binary representation.
Decimal Fractions to Octal: Procedures for converting decimal fractional parts into the octal (base 8) number system.