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Chapter 5: Problem 51

For Exercises \(47-58,\) identify the number as rational or irrational. $$3.5$$

Short Answer

Expert verified
Rational

Step by step solution

01

Understand the Type of Number

A rational number is a number that can be expressed as a fraction \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b \) is not zero. An irrational number cannot be written as a simple fraction; it has a non-repeating, non-terminating decimal expansion.
02

Analyze the Given Number

The given number is \( 3.5 \). Write it in fractional form. \( 3.5 \) can be expressed as \( \frac{35}{10} \).
03

Simplify the Fraction

Simplify the fraction \( \frac{35}{10} \). This simplifies to \( \frac{7}{2} \), since both 35 and 10 are divisible by 5.
04

Conclusion

Since \( 3.5 \) can be expressed as a fraction \( \frac{7}{2} \), it is a rational number because it meets the criteria of being written as a fraction.

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Key Concepts

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

fraction
A fraction represents a part of a whole. It's written as two numbers separated by a slash: the top number is the numerator, and the bottom number is the denominator. For example, in the fraction \(\frac{35}{10}\), 35 is the numerator and 10 is the denominator.

Fractions are used to express quantities that are not whole numbers. They're found in everyday life, like dividing a pizza into slices or measuring flour for a recipe.

To understand fractions, it's important to know two things:
  • The numerator (upper part) indicates how many parts are considered.
  • The denominator (lower part) shows the total number of equal parts the whole is divided into.
rational numbers
A rational number is any number that can be written as a fraction \(\frac{a}{b}\), where both \(a\) and \(b\) are integers and \(b e 0\). Examples include whole numbers like 4 (which can be written as \(\frac{4}{1}\)), negative numbers like -3 (written as \(\frac{-3}{1}\)), and fractions like \(\frac{1}{2}\) or \(\frac{35}{10}\).

Rational numbers have either terminating decimals, like 0.75, or repeating decimals, like 0.333... (which can be written as \(\frac{1}{3}\)).

In identifying whether a number is rational:
  • Check if you can write the number as a fraction with integers.
  • If you can, then it's a rational number.
irrational numbers
Irrational numbers are numbers that cannot be written as simple fractions. Their decimal expansions are non-terminating (they go on forever) and non-repeating.

Common examples include \(\text{\(\pi\)}\) (pi), which is approximately 3.14159, and \(\text{\(\sqrt{2}\)}\) (the square root of 2), which is approximately 1.41421.

To determine if a number is irrational:
  • Check if its decimal form doesn't terminate or repeat.
  • If it can't be expressed as a fraction of integers, it's irrational.
simplified fractions
A simplified fraction is a fraction where the numerator and the denominator have no common factors other than 1. For instance, \(\frac{35}{10}\) simplifies to \(\frac{7}{2}\) because 35 and 10 can both be divided by 5.

Steps to simplify a fraction:
  • Find the greatest common divisor (GCD) of the numerator and denominator.
  • Divide both the numerator and the denominator by their GCD.
In our example \(\frac{35}{10}\), the GCD of 35 and 10 is 5. Divide both by 5 to get \(\frac{7}{2}\).

Simplified fractions are easier to work with and understand.

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