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Rational and Irrational Numbers9 2.
Recall that a rational number is a number that can be written as a quotient ,
where a and b are integers and b ≠ 0.
ab
An irrational number is a number that cannot be written as a quotient of two integers.
If n is a positive integer and is not a perfect square, then n and – n are
irrational numbers.
Rational and Irrational Numbers9 2.
Recall that a rational number is a number that can be written as a quotient ,
where a and b are integers and b ≠ 0.
ab
Together, rational numbers and irrational numbers make up the set of real numbers.
An irrational number is a number that cannot be written as a quotient of two integers.
If n is a positive integer and is not a perfect square, then n and – n are
irrational numbers.
Rational and Irrational Numbers9 2.
Irrational numbersRational numbers
Integers
Whole numbers
The decimal form of a rational number is either terminating or repeating.
The decimal form of an irrational number does not terminate or repeat.
Together, rational numbers and irrational numbers make up the set of real numbers.
The Venn diagram shows the relationships among numbers in the real number system.
EXAMPLE 1 Classifying Real Numbers
Number
Rational and Irrational Numbers9 2.
Rational or Irrational and why?Decimal Form Type of Decimal
34
111
3
144
25.
Evaluate the expression when a = 9 b = 15 and c = 16. Tell whether the answer is rational or irrational
5.
6.
7.
ca
ac
ca
Stop here for day 1
EXAMPLE 2 Comparing Real Numbers
Graph the pair of numbers on a number line. Then copy and complete the statement with <, >, or =.
Rational and Irrational Numbers9 2.
Use a calculator to approximate the square root and write any fractions as decimals. Then graph the numbers on a number line and compare.SOLUTION
1.51.1 1.2 1.3 1.4 1.71.61 1.8 21.9 2.1
2 ≈ 1.4142. So, 2 < 2.
2 ?
2
2 .
EXAMPLE 3 Comparing Real Numbers
Graph the pair of numbers on a number line. Then copy and complete the statement with <, >, or =.
Rational and Irrational Numbers9 2.
?12
12
0.50.1 0.2 0.3 0.4 0.70.60 0.8 10.9 1.1
≈ 0.707112
..
= 0.512
So, > .12
12
EXAMPLE 3 Ordering Decimals
Rational and Irrational Numbers9 2.
Write each decimal out to six decimal places.1
2
Notice that the first two digits after the decimal point are the same for each number.
0.477
Order the decimals 0.47, 0.474, 0.47, and 0.477 from least to greatest.
0.47 =
0.474 =
0.47 =Use the second pair of digits
to order the decimals.
From least to greatest, the order of the numbers is 0.474474…, 0.474747…, 0.4770, and 0.477777….
Rational and Irrational Numbers9 2.
Practice
Tell whether the number is rational or irrational. 1. 2.
Graph the pair of numbers on a number line. Then copy and complete the statement with <, >, or =.
3.
Order the numbers from least to greatest.
4.
3___10
169 17
3.22.82.9 3 3.1 3.43.32.7 3.5 3.73.6 3.8
5.2,2
25,4,10
EXAMPLE 4 Using an Irrational Number
Rational and Irrational Numbers9 2.
SOLUTION
The wind speed must be about 22.36 knots. ANSWER
s = h0.019
Waves For large ocean waves, the wind speed s in knots and the height
of the waves h in feet are related by the equation s = . If the
waves are about 9.5 feet tall, what must the wind speed be? (1 knot is
equivalent to 1.15 miles per hour.)
h0.019
9.50.019 =
500 =
≈ 22.36
Write original equation.
Substitute 9.5 for h.
Divide.
Approximate square root.
