Computer Data Representation

Data Representation

• This chapter covers data representation in computers, including number systems and character encoding.

Objectives

• Understanding binary, octal, and hexadecimal number systems.
• Converting numbers between decimal and other number systems.
• Executing binary arithmetic operations.
• Representing characters using ASCII and Unicode.

Number Systems

Decimal Number System (Base-10)

• Utilizes ten digits (0-9).
• Each position signifies a power of 10.
• Example: $524_{10} = 5 \times 10^2 + 2 \times 10^1 + 4 \times 10^0$

Binary Number System (Base-2)

• Employs two digits (0 and 1).
• Each position represents a power of 2.
• Example: $1011_2 = 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 11_{10}$

Octal Number System (Base-8)

• Uses eight digits (0-7).
• Each position signifies a power of 8.
• Example: $237_8 = 2 \times 8^2 + 3 \times 8^1 + 7 \times 8^0 = 159_{10}$

Hexadecimal Number System (Base-16)

• Employs sixteen digits (0-9 and A-F).
• Each position represents a power of 16.
• A=10, B=11, C=12, D=13, E=14, F=15
• Example: $1A3_{16} = 1 \times 16^2 + 10 \times 16^1 + 3 \times 16^0 = 419_{10}$

Number System Conversion

Decimal to Binary Conversion

• Repeatedly divide the decimal number by 2 and record the remainders.
• Example: $43_{10} = 101011_2$

Decimal to Octal Conversion

• Repeatedly divide the decimal number by 8 and record the remainders.
• Example: $159_{10} = 237_8$

Decimal to Hexadecimal Conversion

• Repeatedly divide the decimal number by 16 and record the remainders.
• Example: $419_{10} = 1A3_{16}$

Binary to Decimal Conversion

• Multiply each digit by the corresponding power of 2 and sum the results.
• Example: $1101_2 = 13_{10}$

Octal to Decimal Conversion

• Multiply each digit by the corresponding power of 8 and sum the results.
• Example: $365_8 = 245_{10}$

Hexadecimal to Decimal Conversion

• Multiply each digit by the corresponding power of 16 and sum the results.
• Example: $2B_{16} = 43_{10}$

Binary to Octal Conversion

• Group binary digits into sets of three (from right to left) and convert each group to its octal equivalent.
• Example: $110101_2 = 65_8$

Binary to Hexadecimal Conversion

• Group binary digits into sets of four (from right to left) and convert each group to its hexadecimal equivalent.
• Example: $111100001111_2 = F0F_{16}$

Octal to Binary Conversion

• Convert each octal digit to its 3-bit binary equivalent.
• Example: $47_8 = 100111_2$

Hexadecimal to Binary Conversion

• Convert each hexadecimal digit to its 4-bit binary equivalent.
• Example: $A9_{16} = 10101001_2$

Binary Arithmetic

Binary Addition

• Follows the same rules as decimal addition, but with base 2.
• 1 + 1 = 0, carry 1

Binary Subtraction

• Follows the same rules as decimal subtraction, but with base 2.

Character Encoding

ASCII (American Standard Code for Information Interchange)

• Uses 7 bits to represent characters (128 characters).
• Includes uppercase and lowercase letters, digits, punctuation, and control characters.

Unicode

• Uses variable number of bits(UTF-8, UTF-16, UTF-32) to represent characters from almost all writing systems.
• Offers a unique code point for each character, enabling multilingual support.

Solved Problems

• Number system conversions and arithmetic examples are provided for practice.

ChemSketch

Displaying Stereochemistry

• Wedged bonds signify bonds coming out of the plane, while dashed bonds indicate bonds going into the plane.
• Proper depiction of wedge and dashed bonds is crucial for representing stereochemistry accurately.

Naming Stereocenters: The Cahn-Ingold-Prelog (CIP) Priority Rules

• Assign priorities to atoms connected to the stereocenter based on atomic number.
• Resolve ties by moving to the next atom until the first point of difference is reached.
• Multiple-bonded atoms are treated as multiple single bonds.

Determining R/S Configuration

• Assign priorities using CIP rules.
• Orient the molecule with the lowest priority group away from you.
• Determine the configuration (R or S) based on the direction of the circle from priority group 1 to 2 to 3.

Fischer Projections

• Way of representing stereochemistry in organic molecules.
• Vertical lines represent bonds going into the plane, horizontal lines represent bonds coming out of the plane.

Optical Activity

• The ability of a chiral molecule to rotate the plane of plane-polarized light.
• Enantiomers rotate light in equal but opposite directions.
• Racemic mixtures are not optically active.
• Specific Rotation Formula: $[\alpha] = \frac{\alpha}{l \cdot c}$

Guía práctica para acelerar el tiempo de comercialización (Practical Guide to Accelerating Time-to-Market)

Introducción (Introduction)

• Time-to-Market (TTM) is the duration from an idea's conception to its availability for sale. Reducing TTM provides a competitive advantage and increases revenue.

Beneficios clave de un TTM más rápido (Key Benefits of Faster TTM)

• Competitive Advantage: Launching products quicker captures market share.
• Increased Revenue: Faster TTM means products generate revenue sooner.
• Improved Customer Satisfaction: Timely product launches meet customer needs.

Estrategias para acelerar el TTM (Strategies to Accelerate TTM)

Agilizar el proceso de desarrollo del producto (Streamline Product Development)

• Agile Methodology: Adopt agile for iterative, flexible development with quick adjustments based on feedback.
• Multifunctional Teams: Improve communication and reduce delays with teams from marketing, engineering, and sales.
• Standardization: Reduce complexity and accelerate development by standardizing product components and processes.

Mejorar la colaboración y la comunicación (Improve Collaboration and Communication)

• Collaboration Tools: Facilitate real-time communication and collaboration.
• Regular Meetings: Keep everyone informed and identify potential obstacles.
• Feedback Loops: Establish clear feedback loops with customers to validate product functions and make rapid improvements.

Aprovechar la tecnología (Leverage Technology)

• Automation: Automate repetitive tasks to free resources and reduce errors.
• Cloud Computing: Use cloud services for scalability, flexibility, and quicker access to resources.
• Prototyping Tools: Quickly visualize and test product ideas.

Centrarse en las pruebas y la garantía de calidad (Focus on Testing and Quality Assurance)

• Early Testing: Detect and correct problems early by incorporating testing at the beginning of development.
• Automated Testing: Ensure products meet quality standards and reduce testing time.
• Continuous Integration: Adopt continuous integration to frequently integrate and test code changes.

Optimizar la gestión de la cadena de suministro (Optimize Supply Chain Management)

• Collaboration with Suppliers: Ensure timely delivery of materials.
• Inventory Management: Implement efficient inventory management to avoid shortages and delays.
• Diversification: Mitigate risks and ensure consistent supply.

Métricas para medir el TTM (Metrics to Measure TTM)

Métricas clave (Key Metrics)

• Cycle Time: Measure the time required to complete each stage of the development process.
• Schedule Adherence: Track the percentage of projects completed on time.
• Development Cost: Monitor the product's development cost to stay within budget.

Herramientas para el seguimiento del TTM (Tools for TTM Tracking)

• Project Management Software: Track progress, manage resources, and monitor deadlines.
• Dashboards: Visualize TTM metrics and identify improvement areas.

Estudios de caso (Case Studies)

• Empresa X (Company X): Reduced TTM by 30% by adopting agile methodologies and implementing multifunctional collaboration tools.
• Empresa Y (Company Y): Accelerated product launches by automating testing and optimizing supply chain management, resulting in a 20% increase in revenue.

Conclusión (Conclusion)

• Accelerating time-to-market is essential for competitiveness. Companies can significantly reduce TTM and achieve benefits by streamlining processes, improving collaboration, leveraging technology, focusing on testing, and optimizing supply chain management.

Chemical Kinetics

Reaction Rates

• For $A + B \rightarrow C + D$, $Rate = -\frac{\Delta[A]}{\Delta t} = -\frac{\Delta[B]}{\Delta t} = \frac{\Delta[C]}{\Delta t} = \frac{\Delta[D]}{\Delta t}$
  • Rate is always positive and proportional to stoichiometry.
• For $2A + B \rightarrow 3C + 4D$, $Rate = -\frac{1}{2}\frac{\Delta[A]}{\Delta t} = -\frac{\Delta[B]}{\Delta t} = \frac{1}{3}\frac{\Delta[C]}{\Delta t} = \frac{1}{4}\frac{\Delta[D]}{\Delta t}$

Rate Law

• For $aA + bB \rightarrow cC + dD$, $Rate = k[A]^x[B]^y$
  • k: rate constant, x: order with respect to A, y: order with respect to B, x+y: overall order.
  • The rate law must be determined experimentally, it applies to the slowest step of the reaction mechanism.

Integrated Rate Laws

• Includes rate law, integrated rate law, linear plot, slope, and half-life for 0, 1, and 2 order reactions.

Collision Theory

• Molecules must collide with sufficient energy and correct orientation ($E_a$).

Arrhenius Equation

• $k = A e^{-E_a/RT}$ and its logarithmic form.
• Plot of ln(k) vs 1/T is linear with slope of $-E_a/R$

Reaction Mechanisms

• Elementary steps: series of simple steps that show the progress of a reaction at the molecular level
• Molecularity: number of molecules that participate in an elementary step
• Rate-determining step: slowest elementary step in a reaction mechanism

Catalysts

• Catalysts speed up a reaction, are not consumed, and lower the activation energy.
  • Homogeneous catalyst: same phase as the reactants
  • Heterogeneous catalyst: different phase as the reactants

Lecture 18: Gerschgorin Circle Theorem

Motivation

• A method for estimating the eigenvalues of a matrix.

Gerschgorin Circle Theorem

• For $A \in \mathbb{C}^{n \times n}$, the Gerschgorin discs are defined as:
  • $D_i = {z \in \mathbb{C} : |z - a_{ii}| \leq \sum_{j \neq i} |a_{ij}| }$
• Eigenvalues of A lie in the union of the Gerschgorin discs:
  • $\lambda(A) \subseteq \bigcup_{i=1}^n D_i$

Corollaries

• The same result holds for $A^T$ (or $A^*$), using column sums.
• An isolated Gerschgorin disc contains exactly one eigenvalue.
• A connected component of k Gerschgorin discs contains exactly k eigenvalues.

Diagonal Dominance

• Matrix $A$ is diagonally dominant if $|a_{ii}| > \sum_{j \neq i} |a_{ij}|$ for all i.
• If A is strictly diagonally dominant, then A is invertible.

Explore data representation in computers, covering number systems and character encoding. Understand binary, octal, and hexadecimal systems. Learn number conversions between decimal and other systems, and character representation using ASCII and Unicode.