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Chapter 9: Problem 42

Classify the number as rational or irrational. $$\sqrt{25}$$

Short Answer

Expert verified
The number \(\sqrt{25}\) or 5 is a rational number.

Step by step solution

01

Identify the number

The number given is \(\sqrt{25}\).
02

Compute the square root

Calculate the square root of 25, which equals 5.
03

Classify the number

Since 5 can be expressed as a fraction of two integers (5/1), it is a rational number.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Roots
The concept of square roots revolves around finding a number that, when multiplied by itself, gives a particular product. For example, the square root of 25, written as \( \sqrt{25} \), results in 5. This is because \( 5 \times 5 = 25 \).
Square roots are particularly important in mathematics because they reverse the operation of squaring a number.
  • The square root symbol: \( \sqrt{} \) denotes the operation, aiming to determine the principal square root, meaning the non-negative root.
  • Not all numbers have a neat whole number as their square root鈥攐nly perfect squares do. Numbers like 9, 16, 25, and 36 are examples of perfect squares.
  • When dealing with non-perfect square numbers, square roots are usually irrational, meaning their exact value cannot be neatly written as a simple fraction or whole number, resulting in a decimal that neither ends nor repeats.
Number Classification
Numbers can generally be classified into rational and irrational categories. Understanding these categories helps in recognizing the kinds of numbers we work with.
  • Rational Numbers: A rational number is any number that can be expressed as the quotient or fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \). These include integers, fractions, and exact decimals.
  • Irrational Numbers: Irrational numbers cannot be expressed as a simple fraction. Their decimal form is non-terminating and non-repeating. Examples include \( \pi \) and \( \sqrt{2} \).
When we look at \( \sqrt{25} \), we find that it equals 5, an integer which can be written as 5/1. This shows that \( \sqrt{25} \) is rational because it simplifies to a number that fits the rational number definition.
Fractions
Fractions are a way of representing parts of a whole and are essential in mathematics for understanding ratios, probabilities, and precise calculations. A fraction consists of a numerator, the top part, and a denominator, the bottom part. For instance, in the fraction \( \frac{a}{b} \), "a" is the numerator and "b" is the denominator.
Fractions help in expressing rational numbers. Here are a few key points about fractions:
  • Any whole number can be written as a fraction with the number 1 as the denominator (e.g., 5 can be written as \( \frac{5}{1} \)).
  • Fractions can be simplified or made equivalent by finding common factors of the numerator and the denominator.
  • Addition, subtraction, multiplication, and division of fractions involve aligning numerators and denominators appropriately to solve problems accurately.
In the context of \( \sqrt{25} \), when we find it equals 5, we can express this as \( \frac{5}{1} \), supporting our understanding of 5 as a rational number.

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