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Section 1.3

Section 1.3. The Real Numbers. Objective 1. Define the sets that make up the real numbers. Sets. In this section, we will look at some number sets. Before we do that, we should consider the idea of a set .

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Section 1.3

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  1. Section 1.3 • The Real • Numbers

  2. Objective 1 • Define the sets that make up the real numbers.

  3. Sets • In this section, we will look at some number sets. Before we do that, we should consider the idea of a set. • A set is a collection of objects whose contents can be clearly determined. The objects in the set are called the elements of the set.

  4. Sets • The set of numbers used for counting can be • represented by: • The braces, { }, indicate that we are representing a • set. This form of representing a set uses commas • to separate the elements of the set.

  5. Number Sets • Counting numbers or natural numbers: • Whole numbers: • Integers:

  6. Objective 1: Examples • 1a. Write a positive or negative integer that describes the following situation. • A debt of

  7. Objective 1: Examples • 1b. Write a positive or negative integer that describes the following situation. 282 feet below sea level.

  8. Objective 2 • Graph numbers on a number line.

  9. Number Line • The number line is the graph we use to visualize the set of integers as well as other sets of numbers. The number line extends indefinitely in both directions. Zero separates the positive numbers from the negative numbers on the number line. The positive integers are located to the right of 0 and the negative integers are located to the left of 0. Zero is neither positive nor negative.

  10. Number Line • For every positive integer on a number line, there is a corresponding negative integer on the opposite side of 0.

  11. Objective 2: Examples • 2a.Graph:

  12. Rational Numbers • The set of rational numbers is the set of all numbers that can be expressed in the form , where a and b are integers and b is not equal to zero, written . The integer a is called the numerator and the integer b is called the denominator.

  13. Objective 2: Examples • 2b. Graph:

  14. Objective 3 • Express rational numbers as decimals.

  15. Express Rational Numbers as Decimals • Every rational number can be expressed as a fraction and as a decimal. To express the fraction as a decimal, divide the denominator, b, into the numerator, a. • Any rational number can be expressed as a decimal. The resulting decimal will either terminate (stop), or it will have a digit that repeats or a block of digits that repeat.

  16. Objective 3: Examples • 3a. Express the rational number as a decimal:

  17. Objective 3: Examples • 3b. Express the rational number as a decimal:

  18. The Irrational Numbers The set of irrational numbers is the set of all numbers whose decimal representations are neither terminating nor repeating. Irrational numbers cannot be expressed as a quotient of integers. Each of the following three numbers is an irrational number.

  19. Objective 4 • Classify numbers as belonging to one or more sets of the real numbers.

  20. The Set of Real Numbers • All numbers that can be represented by points on the number line are called real numbers. Thus, the set of real numbers is formed by combining the rational numbers and the irrational numbers. • Every real number is either rational or irrational.

  21. Number Sets

  22. Number Sets

  23. Objective 4: Examples • 4. List all numbers from the given set that are: a. natural numbers, b. whole numbers, c.integers,d. rational numbers, e. irrational numbers,f.real numbers. • a. natural numbers: • b. whole numbers:

  24. Objective 4: Examples (cont) • c. integers: • d. rational: • e. irrational: • f. real:

  25. Objective 5 • Understand and use inequality symbols.

  26. Ordering the Real Numbers

  27. Objective 5: Examples • 5a. Insert either to make the statement true. • Since is to the left of on the number line, then

  28. Ordering the Real Numbers

  29. Objective 5: Examples • 5b.Determine if the inequality is true or false. • Because is true, then • is true.

  30. Objective 5: Examples • 5c. Determine if the inequality is true or false. • Because neither nor is true,then is false.

  31. Objective 6 • Find the absolute value of a real number.

  32. Absolute Value • The absolute value of a real number a, denoted , • is the distance from 0 to a on the number line. This • distance is always nonnegative. • The absolute value of is 5because is 5 units from 0 on the number line. • The absolute value of 3 is because 3 is 3 units from 0 on the number line.

  33. Objective 6: Examples • 6a. Find the absolute value: • is 4 units from 0. • Thus .

  34. Objective 6: Examples • 6b. Find the absolute value: • is units from 0. • Thus

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