rational vs. irrational making sense of rational and irrational numbers
TRANSCRIPT
Rational Vs. Irrational Rational Vs. Irrational
Making sense of rational and Irrational numbers
AnimalReptile
Biologists classify animals based on shared characteristics. The horned lizard is an animal, a reptile, a lizard, and a 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.
LizardGecko
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 numbersRational numbers
Real Numbers
Integers
Wholenumbers
Recall that rational numbers can be written as the quotient of two integers (a fraction) or as either terminating or repeating decimals.
3 = 3.84 5
= 0.623
1.44 = 1.2
A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits.
Caution!
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.
Additional Example 1: Classifying Real NumbersWrite all classifications that apply to each number.
5 is a whole number that is not a perfect square.
5
irrational, real
–12.75 is a terminating decimal.–12.75rational, real
16 2
whole, integer, rational, real
= = 24 2
16 2
A.
B.
C.
Check It Out! Example 1
Write all classifications that apply to each number.
9
whole, integer, rational, real
–35.9 is a terminating decimal.–35.9rational, real
81 3
whole, integer, rational, real
= = 39 3
81 3
A.
B.
C.
9 = 3
A fraction with a denominator of 0 is undefined because you cannot divide by zero. So it is not a number at all.
State if each number is rational, irrational, or not a real number.
21
irrational
0 3
rational
0 3
= 0
Additional Example 2: Determining the Classification of All Numbers
A.
B.
not a real number
Additional Example 2: Determining the Classification of All Numbers
4 0
C.
State if each number is rational, irrational, or not a real number.
23 is a whole number that is not a perfect square.
23
irrational
9 0
undefined, so not a real number
Check It Out! Example 2
A.
B.
State if each number is rational, irrational, or not a real number.
64 81
rational
8 9
=8 9
64 81
C.
Check It Out! Example 2
State if each number is rational, irrational, or not a real number.
The Density Property of real numbers states that between any two real numbers is another real number. This property is not true when you limit yourself to whole numbers or integers. For instance, there is no integer between –2 and –3.
Additional Example 3: Applying the Density Property of Real Numbers
2 5
3 + 3 ÷ 23 5
There are many solutions. One solution is halfway between the two numbers. To find it, add the numbers and divide by 2.
5 5
= 6 ÷ 21 2
= 7 ÷ 2 = 3
31 2
3 3 31 5
2 5 43 33
54 5
Find a real number between 3 and 3 .
3 5
2 5
A real number between 3 and 3 is 3 .3 5
2 5
1 2
Check: Use a graph.
Check It Out! Example 3
3 7
4 + 4 ÷ 24 7
There are many solutions. One solution is halfway between the two numbers. To find it, add the numbers and divide by 2.
7 7= 8 ÷ 2
1 2= 9 ÷ 2 = 4
41 2
4 44 4 4 42 7
3 7
4 7
5 7
1 7
6 7
Find a real number between 4 and 4 .
4 7
3 7
A real number between 4 and 4 is 4 .4 7
3 7
1 2
Check: Use a graph.
Lesson Quiz
Write all classifications that apply to each number.
1. 2. –
State if each number is rational, irrational, or not a real number.
3. 4.
Find a real number between –2 and –2 .3 8
3 4
5.
2
4 • 9
16 2
25 0
not a real number rational
real, irrational real, integer, rational
Possible answer: –2 5 8