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Chapter 1: Problem 2

Explain why \(0.3\) is a rational number.

Short Answer

Expert verified
0.3 is a rational number because it can be written as the fraction \( \frac{3}{10} \).

Step by step solution

01

Understand the Definition of Rational Numbers

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\.
02

Express the Decimal as a Fraction

Write \(0.3\) as \( \frac{3}{10} \).
03

Verify the Fraction

Check if the fraction \( \frac{3}{10} \) satisfies the condition that both the numerator and the denominator are integers and the denominator is not zero. Here, \(3\) is an integer and \(10\) is an integer, and \(10 eq 0\).
04

Conclusion

Since \(0.3\) can be written as \( \frac{3}{10} \), which is a fraction of two integers, \(0.3\) is a rational number.

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Key Concepts

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

decimal to fraction
Decimals and fractions are two ways of representing numbers. Converting a decimal to a fraction can make it easier to understand and work with. To convert a decimal like 0.3 to a fraction, follow these simple steps:
  • Write down the decimal divided by 1. For example, \(0.3\) becomes \( \frac{0.3}{1} \).
  • Now, multiply both the numerator and the denominator by 10 (or by a power of 10 that makes the numerator a whole number). So you get \( \frac{0.3 \times 10}{1 \times 10} = \frac{3}{10} \).
After these steps, your decimal is now a fraction. In this case, 0.3 is now \( \frac{3}{10} \). So whenever you see a decimal, just remember you can convert it to a fraction easily.
integer
An integer is a whole number that can be positive, negative, or zero. This means that numbers like -3, 0, and 5 are all integers. When we write fractions, the numbers in the numerator and the denominator must be integers.
For example, in the fraction \( \frac{3}{10} \), both 3 and 10 are integers. It's important to use integers when expressing fractions because it helps to clearly define the parts of the fraction.
If you have a fraction with non-integer components, it鈥檚 advisable to multiply by a sufficiently large power of 10 until you get integers in the numerator and the denominator. This ensures the fraction is properly represented.
non-zero denominator
A denominator is the bottom part of a fraction, representing how many equal parts the whole is divided into. It is crucial that the denominator is not zero.
A fraction with a zero denominator is undefined because division by zero is not possible. For example, \( \frac{3}{0} \) is not valid.
Always check that your denominator is a non-zero integer to ensure your fraction is valid and meaningful.
In the example \( \frac{3}{10} \), the denominator 10 is a non-zero integer, which confirms that this fraction is properly formatted and valid.

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