IT103 Algebra

Unit 1: Introduction to the course

R Batzinger

2025-07-31

Welcome to IT103 Algebra

  • How do you wish to be called?
  • Where are you from?
  • What is your experience with Math?
  • What is your goal?

YOUR INSTRUCTOR

  • Email: robert_b@payap.ac.th
  • Office: PC314
    (Office hrs by appointment)

Branches of Mathematics

Why study this course?

  • Prerequisite for advanced courses: provides essential skills needed for calculus, discrete math, machine learning, relational databases, cryptography, data science, algorithms, and machine learning.

  • Foundation for important functions: key to understanding algorithms, such as those for searching, sorting, and optimization. Machine learning and statistical functions algorithms depend onlinear algebra.

  • Exercise Problem-Solving Skills: Algebra is actually a process for applying rules to solve problems. This is crucial for thinking abstractly to tackle complex problems as well as for debugging and developing efficient code.

  • Skill used in Computer Graphics: Algebra is used in transformations, rotations, and scaling used in computer graphics, gaming, and data visualizations.

REQUIRED TEXTBOOK

REFERENCE BOOKS

Graphing Calculator

DESMOS Graphings Calculator

https://www.desmos.com/calculator

Topics to be covered

  1. Introduction to Prerequisites: Real Numbers: Algebra Essentials, Exponents and Scientific Notation, Radicals and Rational Exponents, Polynomials, Factoring Polynomials, Rational Expressions

  2. Introduction to Equations and Inequalities: The Rectangular Coordinate Systems and Graphs, Linear Equations in One Variable, Models and Applications, Complex Numbers, Quadratic Equations, Other Types of Equations, Linear Inequalities and Absolute Value Inequalities

  3. Introduction to Functions: Functions and Function Notation, Domain and Range, Rates of Change and Behavior of Graphs, Composition of Functions, Transformation of Functions, Absolute Value Functions, Inverse Functions

  4. Introduction to Linear Functions: Linear Functions, Modeling with Linear Functions, Fitting Linear Models to Data

  5. Introduction to Polynomial and Rational Functions: Quadratic Functions, Power Functions and Polynomial Functions, Graphs of Polynomial Functions, Dividing Polynomials, Zeros of Polynomial Functions, Rational Functions, Inverses and Radical Functions, Modeling Using Variation

  6. Introduction to Exponential and Logarithmic Functions: Exponential Functions, Graphs of Exponential Functions, Logarithmic Functions, Graphs of Logarithmic Functions, Logarithmic Properties, Exponential and Logarithmic Equations, Exponential and Logarithmic Models, Fitting Exponential Models to Data

  7. Introduction to Systems of Equations and Inequalities: Two Variables, Three Variables, Nonlinear Equations and Inequalities, Partial Fractions, Matrices and Matrix Operations, Solving Systems with Gaussian Elimination, Inverses, Cramer’s Rule

  8. Introduction to Analytic Geometry: The Ellipse, The Hyperbola, The Parabola, Rotation of Axes, Conic Sections in Polar Coordinates

  9. Introduction to Sequences, Probability and Counting Theory

Course Website

https://classroom.google.com/c/NzczOTcxNjM1NTcx?cjc=xzswzwc7

Course Assessment

\[\small\begin{matrix}\rm \hline {\bf Type} & {\bf Frequency} & {\bf Time} & {\bf Weight}\\ \hline \rm Class\ participation&\rm Daily & \rm In class& \rm 5\% \\ \rm Assignments & \rm Weekly &\rm Online&\rm 10\% \\ \rm Quizzes & \rm Biweekly &\rm Online &\rm 15\% \\ \rm Midterm & \rm 8\ Oct\ 2025 &\rm 09:00-11:00 & 30\%\\ \rm Finals &\rm 1\ Dec\ 2025 &\rm 13:00-16:00 & 40\%\\ \hline \end{matrix}\]

Course Grading

\[\bf\small\begin{matrix} Score\ Range & Grade\ Letter & Meaning\\ 80 - 100 & A & Excellent \\ 75 - 79 & B+ & Very\ good \\ 70 - 74 & B & Good \\ 65 - 69 & C+ & Above\ adequate\\ 60 - 64 & C & Adequate \\ 55 - 59 & D+ & Poor \\ 50 - 54 & D & Very\ Poor \\ \lt 50 & F & Fail\\ \end{matrix}\]

Algebra

The use of mathematical relationships to determine the values within a solution space. This requires a process that manipulates symbolic variables according to immutable rules

Some rules are set in stone!

Muhammad bin Musa al-Khwarizmi

  • A Persian mathematican (780-850 AD) considered to be the father of Algebra
  • Born in the city of Khwarezm (modern-day Khiva, Uzbekistan)
  • Became geographer and astronomer by profession
  • Developed the Arabic numerals based on the Hindu numerals used in Indian mathematics.

Numerals
  • appointed the head of the House of Wisdom (Darul Hikma) in Baghdad in 820 AD
  • introduced the concept of algorithms
  • defined algebra as an algorithm to solve linear and quadratic equations.
  • Master piece was a handbook on algebra entitled: Calculation by Completion and Balancing
  • Improved the accuracy of sine and cosine tables based on Persian number system based on Base 60. (60 sec = 1 minute, 60 minutes = 1 degree, 6 sets of 60 degrees = 1 circle)
  • The second major book revised the concepts of geography
  • Revised the system of lattitude and longitude assuming a spherical earth
  • Revised maps to correctly map major cities in 3D using lattitude and longitude.

Muhammad bin Musa al-Khwarizmi

The Key to Algebra

  • EQUALITY RULE:

    • The Equal sign: Whatever you do on one side of the equal sign, you must do on the other.
    • The Division line: Whatever you do to the numerator, you must do to the denominator.

  • OVERRIDING STRATEGY USING RULES

    1. Segregate variables from constants
    2. Reduce the number of variable
    3. Back substitution

Examples

Ex. 1: Transform, Combine, Re-arrange and Reduce

\[\large\frac{-4x}{x-1} + \frac{4}{x+1} = \frac{-8}{x^2-1}\]

Solution

\[\begin{eqnarray} \frac{-4x}{x-1} + \frac{4}{x+1} &=& \frac{-8}{x^2-1}\cr\cr \frac{-4x}{x-1}\cdot \color{#ff0000}{\frac{x+1}{x+1}} + \frac{4}{x+1}\cdot \color{#ff0000}{\frac{x-1}{x-1}}&=&\frac{-8}{x^2-1}\cr\cr \color{#ff8800}{\frac{-4x^2-4x}{x^2-1} + \frac{4x-4}{x^2-1}} &=& \frac{-8}{x^2-1}\cr\cr \frac{(-4x^2-4x) + (4x-4)}{x^2-1} &=& \frac{-8}{x^2-1}\cr\cr \frac{-4x^2 + -4x + 4x -4}{x^2-1} &=& \frac{-8}{x^2-1}\cr\cr \frac{-4x^2 + \color{#ff8800}{-4x + 4x} -4}{x^2-1} &=& \frac{-8}{x^2-1}\cr\cr \frac{-4x^2 -4}{x^2-1} &=& \frac{-8}{x^2-1}\cr\cr \color{#ff0000}{\frac{x^2-1}{-4}}\cdot\left(\frac{-4x^2- 4}{x^2-1}\right) &=&\color{#ff0000}{\frac{x^2-1}{-4}}\cdot \frac{-8}{x^2-1}\cr\cr x^2+1 &=&2\cr\cr x^2+1\color{#ff0000}{-2} &=&2\color{#ff0000}{-2}\cr\cr \color{#00ff00}{x^2-1} &=&0\cr\cr (x-1)(x+1) &=&0\cr\cr x &=& \{-1,1\}\cr \end{eqnarray}\]

Ex. 2: Substitute, solve, revert

\[\large \left(x+2\right)^2 + 11\left(x+2\right) - 12 = 0\]

Ex. 2 Solution

\[\begin{eqnarray}\left(x+2\right)^2 + 11\left(x+2\right) - 12 &=& 0\cr\cr \color{#00ff00}{x+2} &\color{#00ff00}{=}&\color{#00ff00}{u}\cr\cr u^2 + 11u - 12 &=& 0\cr\cr (u+12)(x-1)&=& 0\cr\cr u &=& \{-12,1\}\cr\cr \therefore\quad x& =& u -2\cr\cr x&=& \{-14,-1\}\cr\cr \end{eqnarray}\]

Ex. 3 Simplify by substitution

\[\large 3x^4 - 2x^2 -1 = 0\]

Ex. 3 Solution

\[\begin{eqnarray} 3x^4 - 2x^2 -1 &=& 0\cr\cr \color{#00ff00}{u} &\color{#00ff00}{=}& \color{#00ff00}{x^2}\cr\cr 3u^2 -2u -1 &=& 0\cr\cr (3u+1)(u-1) &=& 0\cr\cr u &=& \left\{-\frac{1}{3},1\right\}\cr\cr x&=&\sqrt{u}=\left\{-i\sqrt{\frac{1}{3}},i\sqrt{\frac{1}{3}},-1,1\right\} \end{eqnarray}\]

Ex.4 Order of Presendence

\[x = 20\times 3+ 3(2-1) + 54/6 +3\times4\] \[\eqalign{ &Please & Parenthesis \\ &Excuse & Exponentation\\ &My & Multiplication \\ &Aunt & Addition \\ &Sally & Substraction\\ }\]

Ex. 4 Solution

\[\begin{eqnarray} x &=& 20\times 3+ 3(2-1) + 54 /6 +3\times4\cr\cr x &=& 20\times 3+ 3\times3 + 54 /6 +3\times4\cr\cr x &=& 20\times 3+ 3\times3 + 54 /6 +3\times4\cr\cr x &=& 60 + 9 + 54/6 +12\cr\cr x &=& 60 + 9 + 9 +12\cr\cr x &=& 90 \cr\cr \end{eqnarray}\]

Ex. 5 Cereal box challenge

Symbolic math