Problem 8 - Algebra 1: Concepts and Skills

Question

Determine whether each expression is rational or irrational. $$ \sqrt{25} $$

Step-by-Step Solution

Answer

The expression \( \sqrt{25} \) is rational.

Step 1: Find the Square Root

Find the square root of 25. Using a calculator or knowing our multiplication tables, it can be determined that the square root of 25 is 5.

Step 2: Check for Rationality

Check if the result can be expressed as a fraction where both the numerator and denominator are integers and if the decimal representation terminates or repeats. In this case, 5 can be expressed as \( \frac{5}{1} \) , where both 5 and 1 are integers. The decimal representation of 5 is 5.0, which is a terminating decimal.

Key Concepts

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, when we are looking for the \( \sqrt{25} \) we want to find a number which, when squared, equals 25. This is a fundamental concept in mathematics, often visualized on the number line or within the context of area calculations. In our example, the square root of 25 is 5, because \( 5 \times 5 = 25 \). Finding square roots is not only about getting the answer but also understanding the process, which often involves prime factorization or estimating with smaller squares for less apparent square roots.

Some square roots don't result in a whole number. For example, \( \sqrt{2} \) does not have a neat and tidy answer. This leads to the conversation about rational versus irrational numbers, which is closely tied to the properties of square roots. Knowing how to properly determine a square root can expand understanding of the deeper number properties, which is why it is crucial to cover this topic thoroughly.

Rationality of Numbers

The rationality of numbers is a classification that hinges on whether a number can be written as a fraction of two integers - where the numerator is an integer and the denominator is a non-zero integer. Therefore, a rational number can be expressed in the form of \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b \eq 0 \). Rational numbers include integers, fractions, and even repeating or terminating decimals.

For example, the number 5 from our square root exercise is rational because it can be written as \( \frac{5}{1} \) which satisfies the definition of a rational number. Understanding rationality is key when differentiating between numbers that can be precisely defined and those that can't, like irrational numbers which cannot be accurately expressed as a simple fraction or a repeating or terminating decimal.

Terminating Decimals

When we discuss terminating decimals, we are referring to decimal numbers that end or have a finite number of digits after the decimal point. For instance, when we convert the fraction \( \frac{5}{1} \) to a decimal, we get 5.0. This decimal representation ends cleanly without going on indefinitely, hence it 'terminates'.

Understanding terminating decimals is significant because they are an indicator of rational numbers. If a decimal does not go on forever, then the number can be written as a fraction of two integers. This feature of terminating decimal is a clear sign of a number's rationality and is vital for students to grasp when learning about different types of numbers. It provides a more concrete understanding of what makes a number rational and how it relates to everyday math, like money and measurements, where non-repeating and non-terminating decimals can denote a level of precision or the need for estimation.