Transcript Rational Vs. Irrational Making sense of rational and Irrational numbers

Rational Vs. Irrational
Making sense of rational and
Irrational numbers
Biologists classify animals based on shared
characteristics. The horned lizard is an animal, a
reptile, a lizard, and a gecko.
Animal
Reptile
Lizard
Gecko
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.
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.
Real Numbers
Rational numbers
Integers
Whole
numbers
Irrational numbers
Recall that rational numbers can be written as
the quotient of two integers (a fraction) or as
either terminating or repeating decimals.
4
2
3
= 3.8
= 0.6
1.44 = 1.2
5
3
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 2 is irrational.
Caution!
A repeating decimal may not appear to
repeat on a calculator, because
calculators show a finite number of digits.
Additional Example 1: Classifying Real Numbers
Write all classifications that apply to each number.
A.
5 is a whole number that is
not a perfect square.
irrational, real
5
B. –12.75 –12.75 is a terminating decimal.
rational, real
C.
16
2
16
4
=
=2
2
2
whole, integer, rational, real
Check It Out! Example 1
Write all classifications that apply to each number.
A.
9
9
=3
whole, integer, rational, real
B.
C.
–35.9
–35.9 is a terminating decimal.
rational, real
81
81
9
=
=3
3
3
3
whole, integer, rational, real
A fraction with a denominator of 0 is undefined
because you cannot divide by zero. So it is not a
number at all.
Additional Example 2: Determining the
Classification of All Numbers
State if each number is rational, irrational,
or not a real number.
A.
21
irrational
B.
0
3
rational
0
=0
3
Additional Example 2: Determining the
Classification of All Numbers
State if each number is rational, irrational,
or not a real number.
C. 4
0
not a real number
Check It Out! Example 2
State if each number is rational, irrational,
or not a real number.
A.
23
23 is a whole number that
is not a perfect square.
irrational
B.
9
0
undefined, so not a real number
Check It Out! Example 2
State if each number is rational, irrational,
or not a real number.
C.
64
81
rational
8
9
8
64
=
9
81
Lesson Quiz
Write all classifications that apply to each number.
1.
2
2. – 16
2
real, integer, rational
real, irrational
State if each number is rational, irrational, or not
a real number.
3. 25
0
not a real number
4.
4 •
rational
9