Decimal Expansion of Rational Numbers

The chapter "Decimal Expansion of Rational Numbers" focuses on understanding how rational numbers can be expressed in decimal form. A rational number is defined as any number that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. This chapter will explore the types of decimal expansions, methods to convert rational numbers into decimal form, and the significance of terminating and non-terminating decimals.

Key Concepts

1. Definition of Rational Numbers

A rational number is a number that can be written in the form ( \frac{p}{q} ), where ( p ) and ( q ) are integers and ( q \neq 0 ). This includes integers, fractions, and finite decimals. For example, ( \frac{1}{2} ), ( 3 ), and ( -4 ) are all rational numbers.

2. Decimal Expansion

Decimal expansion refers to expressing a number in terms of powers of ten. Rational numbers can be expressed in decimal form, which can be either:

a. Terminating Decimals

A terminating decimal is a decimal that has a finite number of digits after the decimal point. For example:

• ( 0.5 ) (which is ( \frac{1}{2} ))
• ( 1.75 ) (which is ( \frac{7}{4} ))

To determine if a rational number has a terminating decimal, we check the prime factorization of the denominator (after simplifying the fraction). If the denominator has only the prime factors 2 and/or 5, the decimal will terminate.

Example:

• For ( \frac{3}{8} ): The denominator 8 can be factored as ( 2^3 ), which means the decimal will terminate. ( \frac{3}{8} = 0.375 ).

b. Non-Terminating Decimals

A non-terminating decimal is a decimal that goes on forever without repeating. There are two types of non-terminating decimals:

• Non-terminating repeating decimals: These decimals have a repeating pattern. For example:

  • ( 0.333... ) (which is ( \frac{1}{3} ))
  • ( 0.142857142857... ) (which is ( \frac{1}{7} ))

• Non-terminating non-repeating decimals: These decimals do not have a repeating pattern and cannot be expressed as a fraction. However, these are not rational numbers. An example is the decimal representation of ( \pi ).

3. Converting Rational Numbers to Decimal Form

To convert a rational number into decimal form, you can use long division. The process involves dividing the numerator by the denominator. Here’s how you can do it:

Example: Convert ( \frac{5}{12} ) to decimal form:

1. Divide 5 by 12 using long division.
2. Since 5 is less than 12, add a decimal point and a zero to make it 50.
3. 12 goes into 50 four times (12 x 4 = 48).
4. Subtract 48 from 50, which gives you 2. Bring down another zero to make it 20.
5. 12 goes into 20 one time (12 x 1 = 12).
6. Subtract 12 from 20, which gives you 8. Bring down another zero to make it 80.
7. 12 goes into 80 six times (12 x 6 = 72).
8. Subtract 72 from 80, which gives you 8. Repeat the process.

This will yield a decimal expansion of ( 0.416666... ), which is a non-terminating repeating decimal.

4. Identifying Types of Decimal Expansions

Once you have the decimal expansion, you can identify whether it’s terminating or non-terminating. If it ends or starts repeating after a certain point, it is non-terminating repeating. If it has a finite number of digits, it is terminating.

Summary

In this chapter, we learned that rational numbers can be expressed in decimal form. We categorized decimal expansions into two types: terminating and non-terminating decimals. We also discussed how to convert rational numbers into decimal form using long division. Understanding these concepts is crucial for working with rational numbers in mathematics. By mastering the decimal expansion of rational numbers, students can enhance their mathematical skills and prepare for advanced topics in mathematics.