1. Set includes rational numbers and irrational numbers.
You need to understand the definition of each set of numbers and whether or not the sets are CLOSED under certain operations.
That means: when you perform an operation on a
specific set of numbers, and the answer remains
within that set, it is CLOSED under that operation.
To prove that a set is NOT CLOSED under an
operation, you need to find only one counterexample.
1. Give a counterexample to show why the
set of integers is not closed under division.
2. Give a counterexample to show why the
set of irrational numbers is not closed under
multiplication.
Real Numbers
Set includes rational numbers
and irrational numbers.
Rational Numbers
Irrational Numbers
Any number you
can write in the form
In decimal form it either
terminates or repeats.
Cannot be expressed
in the form
Decimal forms are non-repeating or non-terminating.
Integers
Whole Numbers
Natural Numbers

2. 3. The irrational numbers are closed under multiplication.
These are Practice Exam type questions {mainly 42-47}.
3. The irrational numbers are
closed under multiplication.
. True B. False
For questions 4-5, classify each
number as rational or irrational.
A. Rational A. Rational
B. Irrational B. Irrational
6. Answer each part.
A. What is an irrational number?
B. Explain why is an irrational number.
7. In each part, provide an example of the statement.
A. The sum of two rational numbers is rational.
B. The product of a rational number and
an irrational number is irrational.
8. Answer each part.
A. Write as the product of a
rational and an irrational number.
B. Give an example where the product of two
irrational numbers is a rational number.
C. Explain why the sum of a rational number
and an irrational number must be irrational.