section 1.2 the real number line. 1.2 lecture guide: the real number line objective: identify...
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Section 1.2
The Real Number Line
1.2 Lecture Guide: The Real Number Line
Objective: Identify additive inverses.
Opposites or Additive Inverses:
Algebraically
Every real number has an additive inverse. This concept is important when we begin to look carefully at subtraction.
Verbally
Numerical Examples
If a is a real number, the opposite of a is _____.
Except for zero, the additive inverse of a real number is formed by changing the ______ of the number.
____ is the opposite of 3____ is the opposite of –30 is the opposite of 0
Graphical Example
−3 30
Opposites
Opposites or Additive Inverses:
Algebraically
Verbally The sum of a real number and its additive inverse is zero.
Numerical Examples
______a a
3 3 0
3 ____ 0
0 0 0
Write the additive inverse of each number:
1. −2 Number:
Additive Inverse: ______
Write the additive inverse of each number:
2. 5Number:
Additive Inverse: ______
Write the additive inverse of each number:
3. 13
Number:
Additive Inverse: ______
Write the additive inverse of each number:
4. Number:
Additive Inverse: ______
0
5.
One number is graphed on each of the following number lines. Graph the additive inverse of each number on the same number line.
0
6.0
−4
5
Algebraically
Verbally
The opposite of the additive inverse of a is a.
Numerical Examples
Double Negative Rule:
For any real number a, a a
7 ______
Simplify each expression.
7. 5.1 5.1
Simplify each expression.
8. 3.2 3.2
Simplify each expression.
9. 6.4 0
Simplify each expression.
10. 3
Simplify each expression.
11. 3
Objective: Evaluate absolute value expressions.
Absolute Value:
Algebraically
Verbally
The absolute value of x is the distance between 0 and x.
Numerical Example
if is nonnegative
if is negative
x xx
x x
2 2
2 2
Graphical Example
−2 20
2 units left 2 units right
For each number graphed below, determine the distance from this point to the origin, and give the absolute value of this number.
12.
Distance: ______ Absolute value: ______
0−6
For each number graphed below, determine the distance from this point to the origin, and give the absolute value of this number.
13. 0 2
Distance: ______ Absolute value: ______
The absolute value of a nonzero number is always a positive value since distance is never negative.
Evaluate each absolute value expression and check your results on a calculator.
14. 8
1215.
2.1 5.8 16.
2.1 5.8 17.
The absolute value of a nonzero number is always a positive value since distance is never negative.
Evaluate each absolute value expression and check your results on a calculator.
18. If x is positive, the numerical value of the absolute value of x is negative / zero / positive (Circle the best choice) and x could be represented algebraically by− x / x (Circle the best choice).
19. If x is 0, the absolute value of x is ______.
− x / x (Circle the best choice).
20. If x is negative, the numerical value of the absolute value of x is negative / zero / positive (Circle the best choice) and x could be represented algebraically by
21. Fill in the blanks to explain why the absolute value of x is defined in two parts. Since distance is never negative, the absolute value of x requires a change in sign for values that are __________________ and does not change the sign for values that are zero or __________________.
Objective: Use interval notation. Can you list all integers between 1 and 10? Yes, this set of integers is {2, 3, 4, 5, 6, 7, 8, 9}. Can you list all real numbers between0 and 1? No, because this set has an infinite number of real
numbers including 1 1 1 1
, , ,2 3 4 5
. Thus we often use interval
notation as a concise way of referring to a set of real numbers.
22. The table below contains four ways to refer to a set of real numbers. Complete the following table by filling in the missing two columns from each row.
Verbal Description Inequality Notation
Number Line Graph Interval Notation
4x
( ,4]
It really helps to understand a symbolic notation if you can say the verbal description to yourself.
22. The table below contains four ways to refer to a set of real numbers. Complete the following table by filling in the missing two columns from each row.
Verbal Description Inequality Notation
Number Line Graph Interval Notation
x is greater than three.
0 3(
It really helps to understand a symbolic notation if you can say the verbal description to yourself.
22. The table below contains four ways to refer to a set of real numbers. Complete the following table by filling in the missing two columns from each row.
Verbal Description Inequality Notation
Number Line Graph Interval Notation
1 6x
-1 6( )
It really helps to understand a symbolic notation if you can say the verbal description to yourself.
18. The table below contains four ways to refer to a set of real numbers. Complete the following table by filling in the missing two columns from each row.
Verbal Description Inequality Notation
Number Line Graph Interval Notation
x is greater than or equal to −5 and less than 2.
[ 5,2)
It really helps to understand a symbolic notation if you can say the verbal description to yourself.
Insert <, =, or > in the blank to make each statement true.
23. 4 ____ 5
Insert <, =, or > in the blank to make each statement true.
24.3 3
_____4 5
Insert <, =, or > in the blank to make each statement true.
25. 4 _____ 4
Insert <, =, or > in the blank to make each statement true.
26. 5 _____ 5
Insert <, =, or > in the blank to make each statement true.
27. 9 _____ 9
Objective: Estimate and approximate square roots.
28. Complete the following table of common square roots. To estimate a square root of a number, it is extremely helpful to first think of a perfect square near that number.
15
1 1
4 2
3
16 4
25 5
6
49
64 8
81
100 10
121
12
169
196
Determine without a calculator the exact value to complete each equation.
Estimate the following square roots to the nearest integer and fill in the relationship between the number and your estimate with either < or >.
Use your calculator to approximate the following square roots to the nearest hundredth.
11 3
11 3.32
48 7
48 6.93
50
50
Estimate the following square roots to the nearest integer and fill in the relationship between the number and your estimate with either < or >.
Use your calculator to approximate the following square roots to the nearest hundredth.
125
125
99
99
Estimate the following square roots to the nearest integer and fill in the relationship between the number and your estimate with either < or >.
Use your calculator to approximate the following square roots to the nearest hundredth.
192
170
192
170
29. Use a calculator to complete the following table. (Hint: See Calculator Perspective 1.2.4.)
Objective: Identify natural numbers, whole numbers, integers, rational numbers, and irrational numbers. All real numbers are either rational or irrational.
A real number x is rational if a
xb
for integers a and b, with 0b
NumericallyIn decimal form, a rational number is either a terminating decimal or an infinite repeating decimal.
Rational and Irrational Numbers
RationalAlgebraically
Numerical Examples Verbal Examples1
0.52
12
in decimal form is a terminating decimal.
10.333... 0.3
3
13
in decimal form is a repeating decimal.
50.151515... 0.15
33
533
in decimal form is a repeating decimal.
NumericallyIn decimal form, an irrational number is an infinite non-repeating decimal.
IrrationalAlgebraically
Numerical Examples Verbal Examples
A real number x is irrational if it cannot be written as a
xb
for integers a and b.
2 1.414213 2 cannot be written as a rational fraction – it is an infinite non-repeating decimal.
3.141593 cannot be written as a rational fraction – it is an infinite non-repeating decimal.
0.1010010001... This irrational number does exhibit a pattern but it does not terminate and it does not repeat.
30. Can you express the number 3 as a fraction and in decimal form? If so, provide an example.
31. Give the definitions of the integers, the whole numbers, and the natural numbers.
Integers:
Whole Numbers:
Natural Numbers:
32. Is the square root of 4 a rational number?
33. Is the square root of 5 a rational number?
The following diagram may be helpful to visualize how the subsets of the real numbers are related.
IrrationalNumbers
Rational Numbers
Integers
Whole Numbers
Natural Numbers
The Real Numbers
34. Place a check beneath each column to which each numbers belongs.Number Natural Whole Integer Rational Irrational Real
0
32
5233
24
16
15
1.234
1.234 1.234234234... means
35. Try evaluating 4 and 30
on your calculator.
What happens?