1. Number systems
TWSSP Tuesday

2. Tuesday Agenda Create symbolic generalizations for subsets of integers
Define rational numbers
Investigate representations of rationals
Explore closure of rationals

3. Tuesday Agenda
Questions for today: How can we use one expression to denote a whole set? How can that help us prove conjectures about that set? What are the rational numbers, and under what operations are they closed?
Learning targets:
Given properties of a subset of integers, it is often possible to express that subset generally, using a single expression, or a finite set of expressions
The rationals are all numbers of the form _______
The rational numbers are closed under __________
Success criteria: I can express a subset generally. I can prove conjectures about the closure of a subset using the generalization. I can determine if a number is rational.

4. Our question from yesterday
Does closure of a subset imply closure of the larger set?

5. Choose one
Is the set of positive powers of 2, {21, 22, 23, …}, closed under multiplication? Under division?
If 3 is a divisor of two numbers, is it a divisor of their sum and their difference? If d is a divisor of two numbers, is it a divisor of their sum and difference?

6. Evens and Odds
We say an integer is even if it is divisible by 2. Otherwise, it is odd.
Since every even integer is divisible by 2, we can write every even integer in the form 2n, where n stands for any integer.
How can we denote every odd integer?
Investigate closure of the even integers under the four operations. Pay special attention to proving your claims.
Investigate closure of the odd integers under the four operations. Pay special attention to proving your claims

7. Is it odd? Let n, m, j, etc be any integer
Which of the following quantities are odd?
2j – 1
2n + 7
4n + 1
2n2 + 3
2n2 + 2n + 1
2m – 9
For each of the expressions above, determine if the expression could be used to denote all odd integers

8. Is it even? Let n, m, j, etc be any integer
Which of the following quantities are even?
2j + 4
4n + 2
2n - 2
2 – 2m
n2 + 2
For each of the expressions above, determine if the expression could be used to denote all even integers

9. Closure of subsets Choose 4 of the following to investigate:
Are integers of the form 3n + 1 closed under subtraction?
Are the integers of the form 3n + 2 closed under multiplication?
Are the integers of the form 3n closed under addition?
Are the integers of the form 6n + 1 closed under subtraction?
Are the integers of the form 6n + 1 closed under multiplication?
Are the integers NOT of the form 3n closed under multiplication?

10. The rational numbers
The rational numbers (or rationals) are all numbers which can be written in the form a/d where a and d are integers and d is not 0.
We use ℚ to denote the set of rationals
What are some examples of rational numbers? What are some non-examples?
Under which of addition, subtraction, multiplication, and division are the rational numbers closed?
Are the rational numbers closed under taking reciprocals?

11. How do you know if it’s rational?
Any rational number can be written as a terminating or infinitely periodic decimal; conversely, any terminating or infinitely periodic decimal is a rational number
Terminating: a finite number of digits in the decimal expansion
Infinitely periodic: an infinite number of digits, but digits repeating in a fixed pattern
First: Suppose a decimal is terminating. How do you know it is rational, by the definition?
Next: How do you know if a rational number (written in fraction form) will have a terminating decimal expansion?

12. How do you know if it’s rational?
Any rational number can be written as a terminating or infinitely periodic decimal; conversely, any terminating or infinitely periodic decimal is a rational number
Terminating: a finite number of digits in the decimal expansion
Infinitely periodic: an infinite number of digits, but digits repeating in a fixed pattern
Now: Consider the decimal How can you write this number as a fraction? In general, what does this suggest about periodic decimals?
Next: In general, if a fraction does not have a terminating decimal expansion, why must it have a periodic expansion?

13. Closure of Rationals
Your group will be assigned a subset of the rationals. Your task is to find one operation (or something you “do” to the numbers) under which your set is closed, and one operation under which your set is not closed.
Prepare to share your subset, your operations, and your reasoning with the whole group.

14. Exit Ticket (sort of…)
Find a subset of the integers which is closed under multiplication
Find a subset of the integers which is not closed under subtraction
Without dividing: terminating or repeating, and how do you know?
Convert to a fraction: …