This page is a list of mathematics articles in Mathematics and Multimedia. You might also want to check the Post List page which contains the newest posts in major categories.

The complete list of posts arranged by month can be found in the Archives page.

**NUMBERS/NUMBER THEORY**

- Division by 0
- Rational and Irrational Numbers
- Counting the Infinite: A Glimpse at Infinite Sets
- Is 0.999… really equal to 1?
- Prime Series 1: Introduction to Prime Numbers
- Prime Series 2: The Intuition Behind the Infinitude of Primes
- Prime Series 3: Proof of the Infinitude of Primes
- Making Sense of Exponential Growth
- Counting the Real Numbers
- Young Gauss and the Sum of the First n Positive Integers
- Dominoes and Mathematical Induction
- Pythagorean Triple 1: The Infinitude of Pythagorean Triples
- Pythagorean Triple 2: Generating Pythagorean Triples
- Large Numbers and One Million
- Light Year and the Mathematics of the Stars
- The Geometry of Commutativity
- The Distributive Property
- The Commutativity of Real Numbers
- Military Drills and the Closure Property of Real Numbers
- Finger Math 1: Finger Multiplication
- Finger Math 2: Why Finger Multiplication Works
- Understanding Fermat’s Last Theorem
- Irrational Numbers as Decimals
- Order of Operations: 360 or 354?
- The mysterious 495
- The mystery of 495 exlpained
- The Golden Ratio
- The Kaprekar constant 6714
- Operations on Integers – Addition
- Operations on Integers – Subtraction
- Real Numbers: A summary
- Is 0 a natural number?
- Multiplication of Fractions
- Linking sum of counting numbers to triangle area
- Pi to 10 Trillion Digits
- Using Area to Prove the Arithmetic Mean-Geometric Mean Inequality
- Another Proof of the Sum of the First n Positive Integers
- Counting the Number of Squares on a Chess Board
- Prime or Not: Determining Primes Through Square Root
- Triangular Numbers and the Sum of the First n Positive Integers
- Introduction to Clock Arithmetic and Modular Systems
- Clock Arithmetic and Modular Systems Part 2
- Introduction to Number Bases
- Why Expressions with Negative Exponents Equal their Reciprocals
- Finding the General Formula for the kth Polygonal Number
- Why are Non-terminating, Repeating Decimals Rational
- Are all objects with irrational lengths measureable?
- Approximating Square Root of Numbers by Averaging
- How Big is Infinity?
- The Definition of Congruence in the Modular Systems
- The Clock Arithmetic and Modular Systems Series
- Guest Post: An Interesting Property of Prime Numbers
- Why The Addition Number Trick Works
- Properties of Addition and Multiplication of Real Numbers
- Are all fractions rational numbers?

**Complex Numbers Series**

- What is i? What are imaginary numbers?
- Complex Numbers and Their Properties
- Demystifying Complex Conjugates
- Introduction to the Complex Plane

** The Divisibility Series**

- Divisibility by 2
- Divisibility by 4
- Divisibility by 5 and 10
- Divisibility by 3
- Divisibility by 7 and its Proof
- Divisibility by 8
- Divisibility by 9

**ALGEBRA**

- Making Connection Between Distance and Absolute Value
- Letters, Variables and Algebraic Expressions
- Matchsticks, Linear Relations, and Multiple Representations
- Equation of a line: The derivation of y = mx + b
- Slope Concept 1 – Understanding the Basic Concepts of Slope
- Slope Concept 2 – Slope of the Graph of a Linear Function
- Slope Concept 3 – Slopes of Vertical and Horizontal Lines
- Arithmetic Sequences and Linear Functions
- Derivation of the Quadratic Formula
- Understanding the Meaning of Discriminant
- Geometric Sequences and Series
- Why do we reverse the inequality sign?
- Why is any number raised to 0 equals 1?
- Why is zero factorial equals 1?
- How to use the summation symbol
- Compound Interest and the Irrational Number e
- Functions 1: Introduction to the Concept of Functions
- Functions 2: Basic Concepts of Functions
- Vectors, Parallelograms and Commutativity
- Simon’s Favorite Factoring Trick
- Addition of Coordinates
- The Handshake Problem and Its Multiple Solutions
- The Multiplication Parabola
- Mathematical Equations and Their Properties
- The 12 days of Christmaths and the Triangular Numbers
- Why is the slope of a vertical line undefined?
- Deceptive and Misleading Mathematical Patterns
- Visualizing the Difference of Two Squares
- Why is negative times negative equals positive?
- Guest Post: Why Study Algebra
- The Polynomials that Generate Prime Numbers

**GEOMETRY**

- Chess and Axiomatic Systems
- Angle Sum of Polygons
- Why is the angle sum of the interior angles of a triangle equal to 180 degrees?
- Parallel Lines and Transversals
- The Exterior Angle Theorem
- Is there an SSA Congruence?
- Making Connections among Pythagorean Theorem, Distance Formula and Equation of a Circle.
- The Algebraic and Geometric Proof of Pythagorean Theorem
- Tessellation: The Mathematics of Tiling
- Is a square a rectangle?
- The Inscribed Angle Theorem
- Demystifying Triangle Inequality
- Constructing Segments with Irrational Lengths
- Similarity 1: Introduction to Similarity
- Similarity 2: Properties of Similar Triangles 1
- Similarity 3: Properties of Similar Triangles Part 2
- A Closer Look at the Midpoint Formula
- Eratosthenes and the Earth’s Circumference
- Platonic Solids: Why only five of them?
- Regular Tessellations: Why only three of them?
- The Seven Bridges of Konigsberg
- Discovering Properties of Parallelograms
- The Marvels of the Nine-Point Circle
- Introduction to Coordinate Geometry
- Undefined terms: What are they?
- Diagonals of a Parallelogram
- A US President’s Proof of the Pythagorean Theorem
- Which is the most pleasing rectangle?
- Understanding the Meaning of Correspondence
- Introduction to Geometric Construction

**Area Tutorial Series**

- Area Tutorial 1 – Introduction to the Concept of Area
- Area Tutorial 2 – Area of a Triangle
- Area Tutorial 3 – Area of a Parallelogram
- Area Tutorial 4 – Area of a Circle
- Area Tutorial 5 – Area of a Trapezoid

**TRIGONOMETRY AND PRECALCULUS**

- Demystifying Degree and Radian Measures
- An elementary proof of the cosine law
- Folding a Hyperbola
- The Complements Theorem
- Pi and the Circle
- Jeepneys and Step Functions
- The Proof of the Tangent Half-Angle Formula
- Understanding Decreasing and Increasing Functions

**Curve Sketching (New!)**

- Graph Sketching 1: 4 Easy Ways to Graph a Linear Function
- Curve Sketching 2: Graphing Quadratic Functions
- Curve Sketching 3: Understanding Vertical and Horizontal Asymptotes
- Curve Sketching 4: Identifying Oblique Asymptotes

**CALCULUS**

- An Intituive Introduction to Limits
- An extensive explanation about the definition of limits
- Limit by proof: Example 1
- Limit by proof: Example 2
- Derivative Concept 1: The Algebraic and Geometric Meaning of Derivative
- Derivative Concept 2: Derivative in Real Life Context
- Derivative Concept 3: Derivative and the Maximum Area Problem
- Calculating the Area Under a Curve: An Intuitive Introduction

**The Calculus Primer Series**

**STATISTICS, PROBABILITY AND COMBINATORICS**

- Introduction to Permutations
- Introduction to Combinations
- Binomial Theorem Series 1: Milkshakes, Beads, and Pascal’s Triangle
- Binomial Theorem Series 2: The Binomial Expansion
- A Gentle Introduction to Probability
- The Unfinished Game Problem
- Using Mathematics to Win the Lottery
- Storytelling and Statistics at their finest
- Ben Goldacre and Bad Statistics
- Understanding the Multiplication Principle
- Probability Terminologies and Notations
- Wedding Guests and Circular Permutations
- Understanding Sample Space and Sample Points

**Experimental and Theoretical Probability Series**

**PROOFS**

- Proof Tutorial 1: Introduction to Mathematical Proofs
- Proof Tutorial 2: Proving the Irrationality of Square Root of 2 by Contradiction
- More examples of Proof by Contradiction
- Proof that log 2 is irrational
- Proof by Contradiction: Knights and Liars
- Another Proof of the Sum of the First n Positive Integers
- The Proof of the Tangent Half-Angle Formula
- A Simple Proof of the Arithmetic Mean Geometric Mean Inequality
- Using Similarity to Prove the Pythagorean Theorem
- Proof of the Sum of Square Numbers
- Mathematical Proofs Without Words: What are they?
- A Proof that the Vertex Angle Sum of a Pentagram is 180 degrees
- Proof without Words: Odds and Squares
- Understanding If Then Statements and Counterexamples

**MATH LITE (**Fun, Games, Puzzles, Trivia**)**

- Funny Math – 10 Steps you Need to Solve a Calculus Problem
- The Square Root of Three
- Why Mathematics Teachers Drink Part 1
- Why Mathematics Teachers Drink Part 2
- Men, Women, and Mathematics
- Amazing Number Patterns
- Simba sings math
- Math is …
- Nature by Numbers: Watch and fall in love with math
- Potato chips and mathematics
- Math Easter Egg
- Calculus Rhapsody
- Three
- Math Words in Filipino
- Calculating the volume of a pizza
- Math, Santa and Venn Diagrams
- Tangled and the Mathematics of Hairdressing
- The Beauty of Narcissistic Numbers
- Heart Graph for the Mathematically In Love
- Would you have given a different mark?
- Why Every Proof that 0.999… = 1 is Wrong
- Mathematics and Perspective
- Can you solve the hardest Sudoku puzzle yet?
- 11 Mathematical Proof Techniques Liberal Arts Majors Should Know
- Apple and the Golden Ratio?
- The Musical Beauty of Tau
- The Magic Circle Puzzle
- Teaching Math Like a Boss
- Who’s the Winner
- The Sound of Pi
- Grading Test Papers Like a Boss
- Watch Exceptional Fast-Paced Math Videos at Vi Hart
- How many squares are there?
- Pi In The Sky, Literally
- What if your paycheck…
- Matchsticks and Squares
- Origami and Its Surprising and Stunning Applications
- The Mathematics of Christmas Wrapping
- Taylor, Mathematics, and Perspective
- The Cycloid and the Pendulum Clock

**MISCELLANEOUS** Math Word Problem Solving Series **Age Problems**

**Motion Problems**

**Vedic Mathematics **(by Sanjay Gulati)

- Introduction to Vedic Mathematics
- Squaring Numbers in 5
- Multiplication with a series of 9s
- Nikhilam Method of Multiplication
- Nikhilam Method of Multiplication 2
- Squaring Numbers: Part 1

**Paper Folding**

- Getting the square root of any number on a number line
- Paper Folding: Extracting the cube root of a number
- Paper Folding: The proof of the cube root applet
- Origami and GeoGebra — a slightly different version

**Set Theory**

- Sets: A Gentle Introduction
- Subset: A set contained in a set
- Sets: Terminologies, Notations, and Operations
- Counting the Number of Subsets in a Set
- Are all infinities created equal?

**General Information/Miscellany**

- The Mystery Behind the Proof of 1 = 2
- The Search for Magic Squares
- The definition of Undefined
- Is mathematics and exact science?
- Steven Strogatz
- Making Mathematicians I
- How do mathematicians greet happy valentines?
- SOMEBODY up there pushed me
- Would you have made it into Harvard in 1869?
- All time Top Mathematics Posts
- A Hundred Years Down the Road
- The man who conned a mathematician
- 10 Ways to Think Like a Mathematician
- Applying Mathematics to Web Design
- Mathematics and the Love of God?
- 25 Signs You’re a Future Mathematician
- Who Owns the Zebra?
- Lottery Math — If you play, you should know your odds
- The Mathematics of Leap Years Explained
- The 10 Misfortunes of Evariste Galois
- Where is the Nobel Prize in Mathematics?
- Millenium Prize Problems: Problems Worth $1M Each
- 7 Extraordinary Mathematicians You Should Know About
- The Actual Fermat-Pascal Correspondence

**MATH RESOURCES/FREEBIES**

- More than 20,000 contest problems and solutions
- 8000+ Math, Physics, and Engineering applets
- Calculus 1 Video Lectures
- 8 Free Online Graphing Tools
- Mu Alpha Theta: 10,000+ Math Contest Problems with Solutions
- Wholemovement and The Art of Folding Circles
- The Prime Composite Dance

**MATH TEACHING AND PEDAGOGY**

- Math Exercise, Problem Solving, and Investigation
- The Inverted Classroom
- Conrad Wolfram: Teaching kids real math with computers
- When do mathematical definitions matter?
- Exploring Computational Thinking
- Math Curriculum Makeover
- What is Universal Design Learning
- Popularizing Lesson Study
- Using Paper Folding to Introduce the Notion of Proof
- 10 Tips in Studying School Mathematics
- Constructing Open -Ended Math Problems
- A Mathematician’s Lament
- 10 Reasons Why Mathematics Teachers Should Blog
- An Open Letter to New Teachers
- Common Errors in MDAS Operations
- PEMDAS and the Common Errors in the Order of Operations
- 4 Common Errors in Calculating Expressions with Exponents
- The Trigonometric Ratio Hexagon

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Why 0.999999999… = 1?

This equality usually surprises everybody. This is a short explanation:

Clearly

0.3333333333… = 1/3

Multiply both members by 3:

3*0.3333333333… = 0.99999999999… = 3*(1/3) = 1

Another way to go…

x = 0.999999999…

10x = 9.999999999…

Subtract the two and the infinite decimals cancel out

10x – x = 9x =9

x = 1

Hence 0.999999999… = 1

Right you are Karl. Thanks. 🙂

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sir saan po b pwede ma view ung mga sagot sa questionare ng math sa ALL-UP workers uniion CSC review..

maraming salamat po!

naipost ko na sa http://mathandmultimedia.com/2012/04/25/up-union-civil-service-review/

I am a Math Teacher and I want to learn more tricks and new techniques in teaching mathematics. So that, I can make teaching and learning more fun and enjoyable. Through this website you really help us teachers learn more and make learning mathematics easier to our students. Good job!

Thank you Ms. Gopez.

To Whom It May Concern,

BSE grad po ako major in Math…To be honest po..tagal ko na pong di nagturo ..last turo ko po ay year 2000 pa and di ko pa po nagagamit ang major ko sa pagtuturo..Nag try po ako na mag apply sa public para magturo but they have told me na I need to take refresher course….I was thinking na instead of taking refresher course…will enter private school, and when I’ve found this site…so thankful …dahil malaki pong bagay ang ginawa niya to those teachers na need ng references na mapagaaralan…oo nga po pala..may teaching demo po ako this coming Tuesday (April 29, 2014)..the lesson is about ” SOLVING QUADRATIC EQUATION BY FACTORING”…dahil sa tagal ko na po di nagturo…yung style po ng lesson plan na alam ko ay old style na po..puwede niyo po ba ako i guide..for sample lesson plan about this topic? Your help would be highly appreciated..About po dun sa lesson…alam ko na po yung subject…kaya lang po ung outline ng lesson plan ang wala ho ako idea…Please do help me..thanks po

Ms. Carmela,

Honestly, I don’t have any idea of what kind of lesson plans DepEd requires. I don’t teach in basic education, I am working in a teacher training institute, so our lesson plans are really for training and not for classroom use. In my opinion, hindi naman ganun ka-crucial ang lesson plan sa demo, ang importante siguro ay madeliver nyo ng maayos. However, if you are concerned about it, maybe you can ask high school teachers.

If I will be observing you, siguro hahanapin ko sa lesson plan mo ang problem, possible solutions, at guide questions na itatanong mo during the demo. Hindi ko lang alam kung ganon dun ang iniisip ng magoobserve sayo. 🙂

Guillermo