Professional Courses
Industry-relevant training in Business, Technology, and Design to help professionals and graduates upskill for real-world careers.
Categories
Interactive Games
Fun, engaging games to boost memory, math fluency, typing speed, and English skills—perfect for learners of all ages.
Typing
Memory
Math
English Adventures
Knowledge
Enroll to start learning
You’ve not yet enrolled in this course. Please enroll for free to listen to audio lessons, classroom podcasts and take practice test.
Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
The Pigeonhole Principle and Coordinates
Unlock Audio Lesson
Welcome, everyone! Let's dive into the pigeonhole principle. Can anyone explain what this principle entails?
It’s about distributing items into containers so that at least one container must contain more than one item.
Exactly! Now, let’s apply this concept. If I have 5 distinct points with integer coordinates, what do we want to find out?
We want to show that at least one pair of points will have a midpoint with integer coordinates.
Great! How can we use the pigeonhole principle here?
We could categorize points by the evenness or oddness of their x and y coordinates.
Exactly! So we have 4 categories—xx even, xy even, and so on. What can we conclude?
Since there are 5 points, at least 2 must fall into the same category.
Well done! This means those two points must have x and y coordinates of the same parity, ensuring the midpoint’s coordinates remain integers. This is a critical insight!
Proving Sums using Pairs
Unlock Audio Lesson
Now let's look at another practical application—choosing five integers from the set {1, 2, 3, 4, 5, 6, 7, 8}. What are we trying to prove?
That there’s always a pair of integers that sums to 9!
Great! First, let's identify the pairs that yield a sum of 9.
We have (1,8), (2,7), (3,6), and (4,5).
Correct! This gives us 4 pairs. So, if we pick 5 integers, how do we categorize them?
Each integer maps to one of our pairs.
Exactly! By pigeonhole principle, at least one of these pairs must repeat among the 5 integers chosen, ensuring at least one sum of 9. Excellent reasoning!
Universal Statements and Combinatorial Proofs
Unlock Audio Lesson
Now, let’s explore a universally quantified statement. Can anyone summarize our goal concerning multiples of an integer?
We want to prove that for any integer, there exists a multiple only consisting of the digits 0 and 1.
Exactly! How do we begin to approach this proof?
By defining our numbers and their remainders when divided by the integer.
Right! We’ll create a sequence of numbers represented by only 1s. How many unique remainders can we have when divided by our integer?
There are only a limited number, up to one less than the number itself.
And with a larger number of sequences than remainders, the pigeonhole principle applies. Can anyone summarize what this leads us to conclude?
It means at least two of these numbers will have the same remainder, showing that their difference is divisible by the integer.
Exactly! This produces a number that consists only of 0s and 1s, confirming our original statement. Excellent observations today!
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
In this lecture, various examples demonstrate the use of the pigeonhole principle to establish the existence of integer coordinates ensuring distinct sums. The conceptual understanding is reinforced through interactive teaching techniques that clarify the implications of the principle.
Detailed
In this lecture, Professor Ashish Choudhry introduces key applications of the pigeonhole principle, focusing on distinct integer coordinates in two-dimensional planes. The session begins with a demonstration involving five distinct points, where it is shown that no matter how these points are chosen, at least one pair exists such that their midpoint has integer coordinates. The proof relies on determining parity (odd or even) of the x and y coordinates and mapping these combinations to ensure that a pair of points can always be found with matching properties. The lectures also extend to show that choosing five integers from one to eight will always yield at least one pair that sums to nine. The session concludes with an exploration of universally quantified statements about integers, using the pigeonhole principle to prove that for any integer, a multiple exists that consists solely of the digits 0 and 1. This session firmly establishes the effectiveness of combinatorial proofs in mathematical reasoning.
Youtube Videos
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Understanding Midpoints and Integer Coordinates
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
You are given here arbitrary distinct points in 2 dimensional planes. Each point will have an x-coordinate, y-coordinate and the points are having integer coordinates. So they are arbitrary points except that they are distinct. Our goal is to show that irrespective of the way these 5 points are chosen arbitrarily, there always exists a pair of points such that if you consider the midpoint of the line joining those 2 points, it has integer coordinates.
If you have 2 points, a point with coordinates (x1, y1) and another point with coordinates (x2, y2), then the midpoint of the line joining these 2 points is given by the formula (x1 + x2)/2, (y1 + y2)/2.
Detailed Explanation
This chunk introduces the concept of distinct points in a 2D plane with integer coordinates. We want to demonstrate that among any 5 chosen distinct integer coordinate points, there exists at least one pair whose midpoint is also an integer point. The formula for the midpoint is shown, and it shows how we can calculate the average of the coordinates.
Examples & Analogies
Imagine you have 5 unique addresses on a city map that correspond to different houses. If you randomly selected these houses, you can always find at least two houses that, when averaged together as a location, point to a location that also has an exact address – a house or a store, proving that this concept of midpoints applies in real life.
Applying the Pigeonhole Principle
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
So let us do that. Consider the set of 5 arbitrary points which are all distinct and have integer coordinates. We are trying to map these points based on the nature of their x-coordinates and y-coordinates. Depending upon whether the x-coordinate is even or odd, or whether the y-coordinate is odd or even, I have 4 possible combinations. This function maps these 5 points to the corresponding pair; say if x1 is odd and y1 is even, then I will say that f(x1, y1) is (odd, even) and so on.
Detailed Explanation
In this chunk, we apply the Pigeonhole Principle to our set of points. The principle states that if you have more 'pigeons' (5 points) than 'holes' (4 combinations of even/odd x and y coordinates), some pigeons must share a hole. Essentially, this means at least two of our selected points will have the same combination of even/odd properties. Hence, they will fulfill the condition to ensure that their midpoint is also an integer.
Examples & Analogies
Think of it like having 5 different colored balls (the points) and only 4 available baskets (the combinations of even/odd). No matter how you toss them into the baskets, at least one basket will end up with at least two balls. This illustrates that we must find at least two points that share the same even/odd property, leading to their midpoint being an integer.
Final Conclusion from the Theory
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Now we want to inspect what happens to the midpoint of the line joining these 2 points, say (x1, y1) and (x2, y2). The midpoint will be ((x1 + x2)/2, (y1 + y2)/2). Since both the points are mapped to the same ordered pair, if both x1 and x2 are odd or both are even, x1 + x2 will always be divisible by 2. Similarly, for y-coordinates, both y1 and y2 need to be either odd or even, leading to y1 + y2 being divisible by 2 as well. Therefore, the midpoint indeed has integer coordinates.
Detailed Explanation
This chunk concludes the theory by confirming that if the x-coordinates are either both odd or both even, their sum will always be divisible by 2, resulting in an integer for the x coordinate of the midpoint. The same applies to y-coordinates. This mathematical principle consolidates our argument that, given any distinct 5 integer-coordinate points, a midpoint with integers must exist.
Examples & Analogies
Imagine measuring distances between two points on a string made of beads representing integers. When you pull the string taut and find the halfway point, regardless of how you measured from those beads, as long as both start points align symmetrically (both odd or both even), you end up exactly on a bead, a whole number on your string.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Pigeonhole Principle: A counting technique showing that if more items are placed into fewer containers, at least one container must have more than one item.
-
Midpoint: The average of the x-coordinates and y-coordinates of two points, providing the center point on a line.
-
Integers: Whole numbers that can be positive, negative, or zero, without fractions or decimals.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
Choosing 5 distinct points has at least one pair with integer midpoints due to their parity.
-
Among the integers from 1 to 8, selecting any 5 guarantees at least one pair will sum to 9.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
-
Pigeonholes packed with math delight, guarantees pairs in plain sight.
📖 Fascinating Stories
-
Once in a math kingdom, five distinct points wanted to meet, they found a magical midpoint where they would greet.
🧠 Other Memory Gems
-
Pigeonhole - Pair Repeat Means Midpoint - Pairs are found when points abound.
🎯 Super Acronyms
P.M.P. - Pigeonhole, Midpoint, Pairs. Remember these steps to ensure pairs!
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Pigeonhole Principle
Definition:
A combinatorial principle that asserts if n items are put into m containers, with n > m, at least one container must contain more than one item.
-
Term: Midpoint Formula
Definition:
The formula used to calculate the midpoint of a line segment joining two points in the Cartesian plane, given by (x1 + x2)/2, (y1 + y2)/2.
-
Term: Parity
Definition:
The quality of integers being either even or odd.
-
Term: Universally Quantified Statement
Definition:
A statement in logic that asserts that a property holds for all elements in a given set.