1. Making Sense of Rational and Irrational Numbers
Objectives: Identify number sets.
Write decimals as fractions.
Write fractions as decimals.

2. The set of real numbers is all numbers that can be written on a number line. It consists of the set of rational numbers and the set of irrational numbers.
Irrational numbers
Rational numbers
Real Numbers
Integers
Whole
numbers

3. Recall that rational numbers can be written as the quotient of two integers (a fraction) or as either terminating or repeating decimals.
4 5 23
3 = 3.8
= 0.6
1.44 = 1.2

4. Rational Numbers Natural Numbers - Natural counting numbers.
1, 2, 3, 4 …
Whole Numbers -
Natural counting numbers and zero.
0, 1, 2, 3 …
Integers -
Whole numbers and their opposites.
… -3, -2, -1, 0, 1, 2, 3 …
Rational Numbers -
Integers, fractions, and decimals.
Ex:
-0.76, -6/13, 0.08, 2/3

5. Biologists classify animals based on shared characteristics
Biologists classify animals based on shared characteristics. The horned lizard is an animal, a reptile, a lizard, and a gecko. Rational Numbers are classified this way as well!
You already know that some numbers can be classified as whole numbers, integers, or rational numbers. The number 2 is a whole number, an integer, and a rational number. It is also a real number.
Animal
Reptile
Lizard
Gecko

6. Venn Diagram: Naturals, Wholes, Integers, Rationals
Real Numbers
Rationals
Integers
Wholes
Naturals

7. Name all the sets of numbers to which the given
number belongs. Circle the most specific set.
Integers
, Rationals
Rationals
Naturals
, Wholes
, Integers
, Rationals
Wholes
, Integers
, Rationals
Rationals

8. Irrational numbers can be written only as decimals that do not terminate or repeat. They cannot be written as the quotient of two integers. If a whole number is not a perfect square, then its square root is an irrational number. For example, 2 is not a perfect square, so is irrational.
A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits.
Caution!

9. Reminder
Real numbers are all the positive, negative, fraction, and decimal numbers you have heard of.
They are also called Rational Numbers.
IRRATIONAL NUMBERS are usually decimals that do not terminate or repeat. They go on forever.
Examples: π

10. Identify each root as rational or irrational.

11. Decimal to Fraction: A skill you will need for this unit!
To change a decimal to a fraction, take the place value and reduce!
0.5 means 5 tenths, so 5/10.
Now reduce 5/10 = ½
0.5 = 1/2

12. Converting Fractions and Decimals
To change a fraction to a decimal, take the top divided by the bottom, or numerator divided by the denominator.

13. Complete the table.
Fraction
Decimal

14. Repeating Decimals
Fraction
Decimal
Every rational number (fraction) either terminates
or repeats when written as a decimal.

15. Repeating Decimals
Fraction
Decimal

16. Repeating Decimals
Fraction
Decimal