Book cover for Prealgebra

Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

ISBN #9781938168994

1st Edition

5,729 Questions

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210,912 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces the classification and properties of real numbers, focusing on rational and irrational numbers. It emphasizes that rational numbers can be expressed as a ratio of two integers with decimals that terminate or repeat, while irrational numbers do not follow this pattern. Additionally, the commutative and associative properties provide foundational tools to rearrange and simplify expressions, ensuring that the order of operations does not affect the outcome. Mastering these concepts is critical for a solid algebra foundation.

Learning Objectives

1

Identify and classify different types of real numbers, including rational and irrational numbers.

2

Convert decimals and integers to fractions to demonstrate their rational form.

3

Apply the method of classification to decide whether a given number is rational or irrational.

4

Utilize the commutative and associative properties to simplify and evaluate algebraic expressions.

Key Concepts

CONCEPT

DEFINITION

Rational Number

A number that can be expressed as a ratio p/q, where p and q are integers and q ≠ 0. Its decimal form either terminates or repeats.

Irrational Number

A number that cannot be written as the ratio of two integers. Its decimal form is non-terminating and non-repeating.

Real Numbers

The set of numbers that can be classified as either rational or irrational.

Commutative Property

A property of addition and multiplication stating that the order of the numbers does not change the result (e.g., a + b = b + a and a · b = b · a).

Associative Property

A property that indicates that when adding or multiplying three or more numbers, the grouping (association) of the numbers does not affect the sum or product (e.g., (a + b) + c = a + (b + c)).

Integer

A whole number, which can be positive, negative, or zero. All integers are rational since they can be written as a fraction with denominator 1.

Example Problems

Example 1

In the following exercises, write as the ratio of two integers. $$ (a)5 $$ $$ (b)3.19 $$

Example 2

In the following exercises, write as the ratio of two integers. $$ (a)8 $$ $$ (b)-1.61 $$

Example 3

In the following exercises, write as the ratio of two integers. $$ (a)-12 $$ $$ (b)9.279 $$

Example 4

In the following exercises, write as the ratio of two integers. $$ (a)-16 $$ $$ (b)4.399 $$

Example 5

In the following exercises, determine which of the given numbers are rational and which are irrational. $$ 0.75,0.22 \overline{3}, 1.39174 \ldots $$
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Step-by-Step Explanations

QUESTION

How can you write the decimal 7.3 as a fraction?

STEP-BY-STEP ANSWER:

Step 1: Recognize that 7.3 has one digit after the decimal point, so the denominator will be 10.
Step 2: Write 7.3 as 73/10 by removing the decimal point and placing it over 10.
Step 3: Simplify the fraction if possible. In this case, 73/10 is already in simplest form.
Final Answer: 73/10.

Converting a Terminating Decimal to a Fraction

QUESTION

How can you use the commutative property to rewrite the expression 4 · 9?

STEP-BY-STEP ANSWER:

Step 1: Recall the commutative property of multiplication: a · b = b · a.
Step 2: Swap the order of the factors. Thus, 4 · 9 can be rewritten as 9 · 4.
Step 3: Verify that multiplication is unaffected by the order, ensuring the product remains the same.
Final Answer: 9 · 4.

Applying the Commutative Property

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Common Mistakes

  • Assuming all non-terminating decimals are irrational, without checking for repeating patterns.
  • Confusing the properties of rational numbers with those of irrational numbers by using improper fraction conversion.
  • Neglecting to apply the commutative property correctly when rearranging terms in an expression.
  • Forgetting that an integer can always be expressed as a fraction with a denominator of one.