Section 1.2 The Real Number Line

1.2 Lecture Guide: The Real Number Line Objective: Identify additive inverses.

Opposites or Additive Inverses: Every real number has an additive inverse. This concept is important when we begin to look carefully at subtraction. Opposites or Additive Inverses: Algebraically If a is a real number, the opposite of a is _____. Verbally Except for zero, the additive inverse of a real number is formed by changing the ______ of the number. Numerical Examples Graphical Example ____ is the opposite of 3 ____ is the opposite of –3 0 is the opposite of 0 −3 3 Opposites

Opposites or Additive Inverses: Algebraically Verbally The sum of a real number and its additive inverse is zero. Numerical Examples

Write the additive inverse of each number: 1. Number: −2 Additive Inverse: ______

Write the additive inverse of each number: 2. Number: Additive Inverse: ______

Write the additive inverse of each number: 3. Number: Additive Inverse: ______

Write the additive inverse of each number: 4. Number: Additive Inverse: ______

One number is graphed on each of the following number lines One number is graphed on each of the following number lines. Graph the additive inverse of each number on the same number line. 5. −4 6. 5

Double Negative Rule: Algebraically For any real number a, Verbally The opposite of the additive inverse of a is a. Numerical Examples

Simplify each expression. 7.

Simplify each expression. 8.

Simplify each expression. 9.

Simplify each expression. 10.

Simplify each expression. 11.

Objective: Evaluate absolute value expressions.

The absolute value of x is the distance between 0 and x. Algebraically Verbally The absolute value of x is the distance between 0 and x. Numerical Example Graphical Example −2 2 2 units left 2 units right

Distance: ______ Absolute value: ______ For each number graphed below, determine the distance from this point to the origin, and give the absolute value of this number. 12. −6 Distance: ______ Absolute value: ______

Distance: ______ Absolute value: ______ For each number graphed below, determine the distance from this point to the origin, and give the absolute value of this number. 13. 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. 15.

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. 16. 17.

18. If x is positive, the numerical value of the absolute value of x is negative / zero / positive (Circle the best choice) and could be represented algebraically by − x / x (Circle the best choice). 19. If x is 0, the absolute value of x is ______.

20. If x is negative, the numerical value of the absolute value of x is negative / zero / positive (Circle the best choice) and could be represented algebraically by − x / x (Circle the best choice). 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 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 between 0 and 1? No, because this set has an infinite number of real numbers including . 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 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 x is greater than three. Number Line Graph Interval Notation 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 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 x is greater than or equal to −5 and less than 2. Number Line Graph Interval Notation 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.

Insert <, =, or > in the blank to make each statement true. 24.

Insert <, =, or > in the blank to make each statement true. 25.

Insert <, =, or > in the blank to make each statement true. 26.

Insert <, =, or > in the blank to make each statement true. 27.

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. 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.

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.

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.

29. Use a calculator to complete the following table 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.

Rational and Irrational Numbers Algebraically A real number x is rational if for integers a and b, with Numerically In decimal form, a rational number is either a terminating decimal or an infinite repeating decimal. Numerical Examples Verbal Examples in decimal form is a terminating decimal. in decimal form is a repeating decimal. in decimal form is a repeating decimal.

Irrational Algebraically A real number x is irrational if it cannot be written as for integers a and b. Numerically In decimal form, an irrational number is an infinite non-repeating decimal. Numerical Examples Verbal Examples cannot be written as a rational fraction – it is an infinite non-repeating decimal. cannot be written as a rational fraction – it is an infinite non-repeating decimal. 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.

32. Is the square root of 4 a rational number?

The following diagram may be helpful to visualize how the subsets of the real numbers are related. Irrational Numbers Rational Numbers Integers Whole Numbers Natural Numbers The Real Numbers

34. Place a check beneath each column to which each numbers belongs. Natural Whole Integer Rational Irrational Real means

35. Try evaluating and on your calculator. What happens?