Natural Numbers Estimation Pigeonhole Principle Examples Strategies Counting Counting

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2 Natural Numbers The natural numbers are the counting numbers, or the positive integers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,...

3 Definition of A rough calculation of the value, number, quantity, or extent of something. A method from moving to qualitative to quantitative thinking. Particularly powerful when combined with another method, such as the pigeonhole principle.

4 The states that if more than N objects are to go into N boxes, then at least one box must contain more than one object. It is a useful tool for drawing conclusions when the size of a collection exceeds the number of possible variations of some distinguishing trait.

5 Essential for using the principle: Objects (pigeons), and Boxes (pigeonholes). In order for the principle to apply, there must be more objects than boxes. Choose the item of which there are more to be the objects. Choose the item of which there are less to be the boxes. Then apply the principle.

6 Note: The should not be interpreted to mean that one box contains exactly two objects. For example, if we distribute 46 cupcakes to a class that has 45 students, it may happen that only one student gets two cupcakes...or it may happen that one student gets all of the cupcakes. In the latter case, someone has more than one cupcake, but there are no students who have exactly two cupcakes.

7 Example: A forest contains 100,000 pine trees. Each pine tree has no more than 80,000 needles. Show that at least two trees have the same number of needles. Here, we are given quantitative information directly from the statement of the problem. We only need to determine what should be the objects, and what should be the boxes.

8 Since there are more trees than there are possible amounts of needles, we will let the trees be the objects (pigeons), and the number of needles will be the boxes (pigeonholes). Picture 80,000 boxes (for the possible number of needles on each tree). We will place each tree in the box which corresponds to how many needles it has. There are 100,000 trees that will each go into one of the 80,000 boxes. By the, at least one box must contain at least two objects, or in terms of the wording of the problem, at least two trees must have the same number of needles.

9 Example: Eleven integers are chosen at random. Show that at least two of them have the same unit digit. We will use the quantity 11 in some capacity. In order to use the, we must be able to compare it with another quantity. What is the second quantity?

10 The second quantity is the number of possible unit digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 There are ten possible unit digits. Since the number of integers we randomly choose (11) is greater than the number of possible unit digits (10), which should be the objects and which should be the boxes?

11 Take the number of integers randomly chosen to be the objects (pigeons). Take the number of possible unit digits to be the boxes (pigeonholes). Then by the, the number of objects is greater than the number of boxes, so at least one box must contain at least two objects. Hence, at least two of the randomly chosen integers will end in the same unit digit.

12 Natural Numbers Make things quantitative. Understand simple things deeply.