Download rational numbers

NUMBER SYSTEMS
TWSSP Tuesday
Tuesday Agenda
• Create symbolic generalizations for subsets of integers
• Define rational numbers
• Investigate representations of rationals
• Explore closure of rationals
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.
Our question from yesterday
• Does closure of a subset imply closure of the larger set?
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?
Evens and Odds
• We say an integer is even if it is divisible by 2. Otherwise,
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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
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
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
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
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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?
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?
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?
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 28.123456. 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?
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.
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
173
you know?
360
• Convert to a fraction: 1.112112112…