Problem 43 - Algebra 2

Question

Determine whether each number is rational or irrational. 4.63

Step-by-Step Solution

Answer

4.63 is a rational number because it can be expressed as the fraction \( \frac{463}{100} \).

Step 1: Understand Rational and Irrational Numbers

A rational number is a number that can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \). An irrational number is a number that cannot be expressed as a simple fraction; it has a non-terminating and non-repeating decimal expansion.

Step 2: Identify the Number Type

The given number 4.63 is a decimal number. We need to determine if it can be expressed as a fraction.

Step 3: Express the Decimal as a Fraction

The decimal 4.63 can be rewritten as \( \frac{463}{100} \). To derive this, note that 4.63 is equivalent to \( 4 + \frac{63}{100} \), which is \( \frac{463}{100} \) when combined into a single fraction.

Step 4: Conclusion on Rationality

Since 4.63 can be expressed as the fraction \( \frac{463}{100} \), where both 463 and 100 are integers and 100 is not zero, it is a rational number.

Key Concepts

Decimal to Fraction Conversion

Converting decimals to fractions is a useful skill in mathematics, enhancing our understanding of the relationship between different types of numbers. To convert a decimal to a fraction, follow these simple steps:

• First, identify the place value of the far-right digit. For example, in the decimal 4.63, the number 3 is in the hundredths place.
• Next, rewrite the decimal as a fraction with a denominator corresponding to that place value. For 4.63, you would express it as \(\frac{463}{100}\).
• Always simplify the fraction if possible, but in this case, \(\frac{463}{100}\) is already in its simplest form.

Understanding these steps allows you to express any terminating decimal as a fraction easily.

Number Types

There are mainly two types of numbers we deal with in exercises like this: rational and irrational numbers.

• Rational numbers can be written as fractions, such as \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b eq 0\). Examples include 1/2, 5, and -3/4.
• Irrational numbers cannot be expressed as a simple fraction. They have decimals that do not terminate or repeat, like \(\pi\) or the square root of 2.

Knowing the distinction between these types helps us quickly determine how numbers can be manipulated or expressed in different forms.

Fraction Representation

Fraction representation is essential for understanding rational numbers. A fraction consists of two parts: a numerator and a denominator.

• The **numerator** is the top number and indicates how many parts are considered.
• The **denominator** is the bottom number and shows the total number of equal parts in a whole.

For example, in the fraction \(\frac{463}{100}\), the numerator is 463, and it represents how many parts we have. The denominator is 100, showing that the whole is divided into 100 equal parts.
This simple structure makes fractions an intuitive way to represent parts of a whole, mixed numbers, and even relationships between quantities.