Lisa writes the following list of numbers.
5.737737773…, 26,45 squared 0,9
Question 1.
a. Which numbers are rational?
Question 2
b. Which numbers are irrational?
The Correct Answer and Explanation is:
To determine which numbers in Lisa’s list are rational and which are irrational, let’s analyze each number:
- 5.737737773…: This number is a non-terminating decimal. However, it appears to follow a pattern, with the digits “737” repeating after the initial “5.7.” Such a number can be expressed as a fraction. For example, if we let ( x = 5.737737773…), then multiplying both sides by 1000 and subtracting the original equation can yield a rational expression. Therefore, this number is rational.
- 26: This is a whole number, which can be expressed as a fraction (e.g., ( \frac{26}{1} )). All integers are rational numbers since they can be represented as the quotient of two integers (the integer itself over one). Thus, 26 is rational.
- 45 squared (or ( 45^2 )): The square of 45 is ( 2025 ), which is also a whole number. Like 26, 2025 can be expressed as a fraction (( \frac{2025}{1} )), making it a rational number.
- 0.9: This is a terminating decimal, which can be expressed as ( \frac{9}{10} ). Since it can be represented as a fraction of two integers, 0.9 is also rational.
Summary of Answers:
- Rational Numbers: 5.737737773…, 26, ( 45^2 ) (2025), 0.9
- Irrational Numbers: None
Explanation of Rational and Irrational Numbers:
Rational numbers are defined as numbers that can be expressed as a fraction ( \frac{p}{q} ), where ( p ) and ( q ) are integers, and ( q \neq 0 ). They can be either terminating decimals (like 0.9, which terminates after one decimal place) or repeating decimals (like 5.737737773…, where “737” repeats). Because rational numbers can be represented in fraction form, they include integers and finite decimal numbers.
In contrast, irrational numbers cannot be expressed as a simple fraction. Their decimal expansions are non-terminating and non-repeating. Classic examples of irrational numbers include ( \pi ) and the square root of 2. These numbers cannot be accurately represented as fractions because they do not repeat or terminate, making their decimal forms infinite and unpredictable.
In Lisa’s list, every number presented can be expressed in fraction form, hence they are all rational. Understanding the difference between rational and irrational numbers is crucial for mathematical literacy, particularly when dealing with various numerical types and their properties.