Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-10,\) tell whether each represents a number that is rational, irrational, or neither. $$ \sqrt{25} $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Expression

We begin by examining the given expression: \( \sqrt{25} \). This is a square root operation.

Step 2: Calculating the Square Root

The square root of a number \( n \) is a value that, when multiplied by itself, gives the number \( n \). In this case, \( \sqrt{25} \) means finding a number that when squared equals 25.

Step 3: Finding the Square Root

The square root of 25 is 5, because \( 5 \times 5 = 25 \). Thus, \( \sqrt{25} = 5 \).

Step 4: Identifying the Number Type

A rational number is any number that can be expressed as a fraction \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b eq 0 \). The number 5 can be expressed as \( \frac{5}{1} \), which is a fraction involving integers.

Step 5: Conclusion

Since 5 is a rational number, we can conclude that \( \sqrt{25} \) represents a rational number.

Key Concepts

Understanding Square Roots

When you see a square root symbol, it can sometimes seem a bit confusing. But don't worry, it's a very handy tool in mathematics. The square root of a number is a special value that, when multiplied by itself, gives back the original number. Think of the square root as the opposite of squaring a number.
Let's say you have the number under the square root symbol, 25. You are looking for a number that, when you square it (or multiply it by itself), equals 25. In our example, this number is 5, because when you calculate 5 times 5, you get 25. So we say that the square root of 25 is 5. It's important to remember:

• The square root of a perfect square (like 25) is always a whole number.
• Perfect squares are numbers like 1, 4, 9, 16, 25, etc. because they have whole numbers as their square roots.

The Basics of Number Classification

Numbers come in different types, and classifying them helps us understand their properties and relationships. There are several categories to know.

• Natural Numbers: These are all the positive, non-zero whole numbers, like 1, 2, 3, and so on.
• Whole Numbers: This group includes all natural numbers and the number zero.
• Integers: These include all whole numbers as well as their negative counterparts. So, they look like -3, -2, -1, 0, 1, 2, 3, etc.
• Rational Numbers: Think of these as numbers that can be expressed as a fraction of two integers, such as 1/2 or 5/1.

Now, whenever you're working on math problems, try to identify what type of number you have. Is it a whole number? An integer? This will help you understand how to work with it.

Rational and Irrational Numbers

In the world of numbers, rational and irrational numbers take on a very important role. Here’s what you need to know:
**Rational Numbers**
These are numbers that you can write as a simple fraction where both the numerator (top number) and the denominator (bottom number) are integers (whole numbers). The denominator must not be zero. For example, 5 can be written as 5/1, making it rational.
**Irrational Numbers**
These numbers cannot be written as a simple fraction. They have non-repeating, endless decimal parts. For instance, the number π (pi) is a famous irrational number due to its never-ending, non-repeating decimal form.
It's vital to distinguish between the two:

• If a number is clean and can settle nicely into a fraction, it's rational.
• If it has a never-ending, unpredictable form, it's probably irrational.

In the case of \( \sqrt{25} \), it simplifies to 5, which is neat and rational. As you explore more complex roots like \( \sqrt{2} \), you'll find they don't turn into tidy fractions, marking them as irrational.