Convert 0.72 To A Rational Number In Simplest Form

Converting 0.72 to a Rational Number in Simplest Form: A Comprehensive Guide

This article provides a detailed explanation of how to convert the decimal number 0.72 into a rational number in its simplest form. We will explore the underlying concepts of rational numbers, decimals, and fraction simplification, ensuring a thorough understanding of the process. This guide is ideal for students learning about rational numbers and fraction manipulation, as well as anyone seeking a refresher on these fundamental mathematical concepts.

Understanding Rational Numbers

A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator 'p', and a non-zero denominator 'q'. In simpler terms, it's any number that can be written as a fraction. Examples of rational numbers include 1/2, 3/4, -2/5, and even whole numbers like 5 (which can be written as 5/1). Decimal numbers that terminate (end) or repeat in a pattern are also rational numbers. 0.72 is a terminating decimal, making it a rational number.

Converting Decimals to Fractions

The conversion of a decimal to a fraction involves identifying the place value of the last digit. In 0.72, the last digit, 2, is in the hundredths place. This indicates that the denominator of our fraction will be 100.

Step 1: Write the decimal as a fraction with a denominator of a power of 10.

0.72 can be written as 72/100. This is because the decimal represents 72 hundredths.

Step 2: Simplify the fraction to its lowest terms.

This is the crucial step in expressing the rational number in its simplest form. To simplify a fraction, we find the greatest common divisor (GCD) of the numerator and the denominator and divide both by it.

The GCD is the largest number that divides both 72 and 100 without leaving a remainder. We can find the GCD using several methods:

Listing Factors: List the factors of 72 (1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72) and 100 (1, 2, 4, 5, 10, 20, 25, 50, 100). The greatest common factor is 4.
Prime Factorization: Find the prime factorization of both numbers.
- 72 = 2³ x 3²
- 100 = 2² x 5²

The GCD is found by taking the lowest power of each common prime factor. In this case, the only common prime factor is 2, and the lowest power is 2². Therefore, the GCD is 2² = 4.

Step 3: Divide both the numerator and the denominator by the GCD.

Dividing both 72 and 100 by the GCD (4), we get:

72 ÷ 4 = 18 100 ÷ 4 = 25

Therefore, the simplified fraction is 18/25.

0.72 as a Rational Number in Simplest Form: Conclusion

The decimal 0.72, when converted to a rational number in its simplest form, is 18/25. This fraction cannot be simplified further because the greatest common divisor of 18 and 25 is 1. This process demonstrates the fundamental relationship between decimals and fractions, showcasing how terminating decimals can always be represented as rational numbers.

Further Exploration of Rational Numbers

Now that we've successfully converted 0.72, let's delve deeper into the world of rational numbers and explore related concepts:

Types of Decimals and their Rational Equivalents

Understanding the different types of decimals is crucial for converting them into fractions. There are three main types:

Terminating Decimals: These decimals have a finite number of digits after the decimal point, like 0.72, 0.5, and 0.375. They can always be expressed as a fraction with a denominator that is a power of 10.
Recurring Decimals (Repeating Decimals): These decimals have a digit or a sequence of digits that repeat infinitely, like 0.333... (1/3) or 0.142857142857... (1/7). Converting these to fractions requires a slightly more complex method which often involves algebraic manipulation.
Non-Recurring, Non-Terminating Decimals: These decimals have an infinite number of digits after the decimal point, and the digits do not repeat in a pattern. These are irrational numbers, and they cannot be expressed as a fraction of two integers. Examples include π (pi) and √2.

Working with Fractions: Addition, Subtraction, Multiplication, and Division

Understanding fraction arithmetic is essential for working effectively with rational numbers. Here's a brief overview:

Addition and Subtraction: To add or subtract fractions, you need a common denominator. Find the least common multiple (LCM) of the denominators, convert the fractions to equivalent fractions with the LCM as the denominator, and then add or subtract the numerators.
Multiplication: Multiply the numerators together and multiply the denominators together. Simplify the resulting fraction if possible.
Division: To divide fractions, invert the second fraction (reciprocal) and then multiply.

Applications of Rational Numbers in Real Life

Rational numbers are fundamental in many aspects of daily life:

Cooking and Baking: Recipes often involve fractions (e.g., 1/2 cup of sugar).
Construction and Engineering: Accurate measurements and calculations require the use of fractions and decimals.
Finance: Dealing with money involves fractions (e.g., calculating interest rates).
Science: Many scientific formulas and measurements utilize rational numbers.

Advanced Topics: Continued Fractions and Decimal Expansion

For a deeper dive, explore the concepts of continued fractions. A continued fraction represents a rational number as a sequence of integers, offering another way to understand and manipulate rational numbers. Also, understanding the decimal expansion of a rational number helps in converting repeating decimals into fractions. This involves setting up and solving an equation using algebra.

This comprehensive guide provides a solid foundation in converting decimals to rational numbers, highlighting the importance of understanding rational numbers and their applications in various fields. By mastering these concepts, you will enhance your mathematical skills and problem-solving abilities. Remember, practice is key to solidifying your understanding of these fundamental concepts. Work through numerous examples to build your confidence and expertise in handling rational numbers.