Problem 45 - Algebra 2

Question

Determine whether each number is rational or irrational. \(\frac{16}{3}\)

Step-by-Step Solution

Answer

\( \frac{16}{3} \) is a rational number.

Step 1: Define Rational and Irrational Numbers

A rational number can be expressed as the quotient or fraction of two integers, where the numerator is an integer and the denominator is a non-zero integer. An irrational number cannot be written as a simple fraction; its decimal form is non-terminating and non-repeating.

Step 2: Examine the Given Number

The given number is \( \frac{16}{3} \). This is a fraction composed of two integers: 16 as the numerator and 3 as the denominator.

Step 3: Check the Denominator

The denominator in the fraction \( \frac{16}{3} \) is 3, which is a non-zero integer. Thus, the fraction is defined, making \( \frac{16}{3} \) a rational number.

Step 4: Conclusion

Since \( \frac{16}{3} \) can be expressed as a quotient of two integers, it is a rational number.

Key Concepts

Integer

An integer is a whole number that can be positive, negative, or zero. It does not contain any fractional or decimal component. Integers are a fundamental part of rational numbers because rational numbers can be expressed as fractions or quotients of integers.

For example:

• Positive integers: 1, 2, 3, 4, 5, ...
• Negative integers: -1, -2, -3, -4, -5, ...
• Zero: 0

Integers are crucial for constructing fractions, as both the numerator and the denominator in a fraction must be integers, with the denominator being non-zero. This property helps classify a number like \( \frac{16}{3} \) as a rational number since both 16 and 3 are integers.

Fraction

A fraction represents a part of a whole or a division between two numbers. It is expressed as \( \frac{a}{b} \), where \( a \) is the numerator and \( b \) is the denominator. In rational numbers, fractions are a way to express numbers that are not whole. Fractions can simplify complex numbers into understandable parts.

For example, the fraction \( \frac{16}{3} \) indicates that 16 is being divided by 3. The fraction itself shows that the number is made by splitting 16 into 3 equal parts. This division is what makes \( \frac{16}{3} \) a rational number because it is shown as a ratio or quotient of integers.

Denominator

The denominator is the bottom part of a fraction, denoted as \( b \) in \( \frac{a}{b} \). It signifies the total number of equal parts into which the whole is divided. A non-zero denominator is vital for a fraction to represent a rational number.

In our example of \( \frac{16}{3} \):

• The denominator is 3, which means the quantity 16 is divided into 3 parts.

For the fraction to be valid and rational, the denominator must never be zero, as division by zero is undefined. This rule is why \( \frac{16}{3} \) is a rational number; it has a valid, non-zero denominator, confirming its property as a rational number.

Quotient

A quotient is the result of a division operation. In the context of rational numbers, it represents the fraction or division of two integers. The quotient is simply the outcome you get when you divide the numerator by the denominator.

For \( \frac{16}{3} \), the quotient is what results when 16 is divided by 3. Although \( \frac{16}{3} \) may not divide perfectly into an integer, it is a rational number because the division results in a value that can still be expressed as a fraction of two integers: 16 and 3.

Rational numbers are characterized by their ability to be expressed as such quotients. Thus, understanding the quotient helps identify whether a number is rational, as in \( \frac{16}{3} \), which confirms its status as a rational number through its expressible quotient form.