course 3 4-7 the real numbers warm up1-28-09 each square root is between two integers. name the two...
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Course 3
4-7 The Real Numbers
Warm Up 1-28-09Each square root is between two integers. Name the two integers.
Estimate each value. Round to the nearest tenth.
10 and 11
–4 and –3
1.4
–11.1
1. 119
2. – 15
3. 2
4. – 123
Course 3
4-7 The Real 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
Course 3
4-7 The Real Numbers
Recall that rational numbers can be written as fractions. Rational numbers can also be written as decimals that either terminate or repeat.
3 = 3.84 5
= 0.623
1.44 = 1.2
Course 3
4-7 The Real Numbers
Irrational numbers can only be written as decimals that do not terminate or repeat. If a whole number is not a perfect square, then its square root is an irrational number.
2 ≈1.4142135623730950488016…
A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits.
Caution!
Course 3
4-7 The Real Numbers
The set of real numbers consists of the set of rational numbers and the set of irrational numbers.
Irrational numbersRational numbers
Real Numbers
Integers
Wholenumbers
Course 3
4-7 The Real Numbers
Rationals
Integers
Wholes
Naturals
Irrationals
What are the different types of numbers?Real Numbers
1
3
Course 3
4-7 The Real Numbers
9
Fill In Your Real Number Chart
Counting “Natural” Numbers: 1, 2, 3, 4, 5, 6, . . .
Whole Numbers: 0, 1, 2, 3, 4, . . .
Integers: . . . -3, -2, -1, 0, 1, 2, 3, 4. . .
Rational Numbers: 0, …1/10, …1/5, …1/4, ... 33, …1/2, …1, perfect squares
Real Numbers: all numbers
Irrationals: π, non-repeating decimal, nonperfect squares
Course 3
4-7 The Real Numbers
Classifying Real Numbers
Write all names that apply to each number (whole, integer, rational,
irrational, real)
Course 3
4-7 The Real Numbers
Example 1
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.
Course 3
4-7 The Real Numbers
Example 2
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
Course 3
4-7 The Real Numbers
Determining the Classification of All
NumbersState if each number is rational, irrational,
or not a real number.
Course 3
4-7 The Real Numbers
not a real number
Example 3 continued..
–4
4 9
rational
2 3
=2 3
4 9
C.
D.
Course 3
4-7 The Real Numbers
23 is a whole number that is not a perfect square.
23
irrational
9 0
not a number, so not a real number
Example 4
A.
B.
Course 3
4-7 The Real Numbers
not a real number
–7
64 81
rational
8 9
=8 9
64 81
C.
D.
Example 4 Continued…
Course 3
4-7 The Real Numbers
The Density Property of real numbers states that between any two real numbers is another real number. This property is also true for rational numbers, but not for whole numbers or integers. For instance, there is no integer between –2 and –3.
Course 3
4-7 The Real Numbers
Find a real number between a set of numbers
There are many solutions. Let’s try to find the solution that is halfway
between the two numbers
Course 3
4-7 The Real Numbers
Example 5
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
Course 3
4-7 The Real Numbers
Example 6
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
Course 3
4-7 The Real NumbersLesson Summary
Write all names 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