Problem 45 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find a positive rational number and a positive irrational number both smaller than \(0.00001\).

Step-by-Step Solution

Answer

Rational: 0.000001; Irrational: 0.00000141421...

Step 1: Identify a Positive Rational Number

To find a positive rational number smaller than 0.00001, you can take any fraction of two integers that results in a value less than 0.00001. A simple example is \( \frac{1}{1000000} = 0.000001 \), which is smaller than 0.00001. In terms of finding a rational number, a rational number is simply any number that can be expressed as a fraction \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b eq 0 \). \( 0.000001 \) fits this criteria as it can be written as \( \frac{1}{1000000} \).

Step 2: Identify a Positive Irrational Number

To find a positive irrational number smaller than 0.00001, consider an irrational number that you can modify to meet the condition. For example, \( \sqrt{2} \) is irrational. However, its values cannot be used directly without modification. Instead, take \( \sqrt{2} \) and multiply it by a very small rational number to ensure its value is less than 0.00001. In this case, \( 0.000001 \times \sqrt{2} \approx 0.00000141421 \) which is smaller than 0.00001. Since it involves an irrational component (\( \sqrt{2} \)), this number remains irrational.

Key Concepts

Positive Rational Number

A positive rational number is any number that can be expressed as a fraction of two integers, where the numerator and denominator are both positive and the denominator is not zero. For example, numbers like \( \frac{1}{2} \, \text{or} \, \frac{3}{4} \) are rational. These are numbers that have a simple decimal expansion or can be expressed as a repeating or terminating decimal.
A rational number smaller than 0.00001 can be any fraction with an appropriately small numerator and a large denominator. The given example of \( \frac{1}{1000000} = 0.000001 \) is a positive rational number smaller than 0.00001. This fraction shows how a very small part of a large number still remains a positive rational number.
To determine or create more examples of positive rational numbers smaller than 0.00001, remember the essence of rational numbers being divideable into smaller fractional parts. Any fraction such as \( \frac{1}{1000001} \) or \( \frac{2}{2000000} \) would also fit this criteria, as long as the decimal result is less than 0.00001.

• A fraction is rational if it can be written as \( \frac{a}{b} \).
• Both \( a \) and \( b \) need to be integers, and \( b eq 0 \).

Positive Irrational Number

An irrational number is any number that cannot be expressed as a simple fraction of two integers. This means the number cannot be exactly represented as a repeating or terminating decimal.
Common examples of irrational numbers include \( \pi \) and \( \sqrt{2} \), among others. These numbers go on forever without repeating in any predictable pattern when written as decimals.
To achieve a positive irrational number smaller than 0.00001, take a known irrational number and multiply it by a small rational number. For instance, \( \sqrt{2} \) is inherently larger, but when multiplied by \( 0.000001 \), which is a small positive rational number, the result becomes an irrational number smaller than 0.00001. This product, approximately \( 0.00000141421 \), maintains the irrational characteristics due to the presence of \( \sqrt{2} \).

• Irrational numbers cannot be fully represented as fractions of two integers.
• They often involve non-repeating, non-terminating decimals.
• Multiplying an irrational number by a small positive rational number keeps it irrational but ensures it is extraordinarily small.

Number Comparison

When comparing numbers, especially looking to identify which are greater or smaller than a given threshold, it's important to remember that both rational and irrational numbers can be orderly compared when expressed in decimal form.
Understanding the place value of decimal digits is crucial in number comparison. When evaluating if a number is less than 0.00001, each digit's position informs its size relative to this threshold. Numbers are typically aligned by their decimal points and compared digit by digit.
For example, \( 0.000001 \) and \( 0.00000141421 \) both fall below the target number of 0.00001. Here, the leading zeroes dictate the small magnitude of these numbers, categorizing them properly as smaller.

• Compare numbers by aligning decimal points and assessing digit by digit from left to right.
• Smaller numbers have more leading zeroes in decimal places.
• Both rational and irrational numbers can be effectively compared when they are in decimal form.