Problem 47 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \begin{array}{l} \text { Find a rational number between } 3.14159 \text { and } \pi . \text { Note that }\\\ \pi=3.141592 \ldots \end{array} $$

Step-by-Step Solution

Answer

A rational number between 3.14159 and π is 3.141591.

Step 1: Understand the Question

We have two numbers given: 3.14159 and \( \pi \) (approximately 3.141592). We need to find a rational number (a number that can be expressed as a fraction) between these two.

Step 2: Identify the Range

Convert \( \pi \approx 3.141592 \). So our range is from 3.14159 to 3.141592.

Step 3: Calculate Midpoint

To find a rational number, calculate the midpoint between 3.14159 and 3.141592: \[ \text{Midpoint} = \frac{3.14159 + 3.141592}{2} \] \[ = \frac{6.283182}{2} \] \[ = 3.141591 \]

Step 4: Verify Midpoint

Check that the midpoint 3.141591 is indeed between 3.14159 and 3.141592: - 3.14159 < 3.141591 < 3.141592 This confirms that 3.141591 is correctly within the range and is rational.

Key Concepts

Pi

Pi, symbolized as \( \pi \), is a very special number in mathematics. It represents the ratio of a circle's circumference to its diameter. Although Pi is commonly approximated as 3.14159, or rounded to 3.14, it is in fact an irrational number. This means it has an infinite number of decimal places without a repeating pattern, unlike rational numbers that end or repeat.

Pi is significant in many areas of mathematics and science, especially in geometry and trigonometry.

• It is used to calculate areas and volumes related to circles and spheres.
• Pi also appears in many formulas beyond those purely geometric, showing its wide-reaching importance.

Understanding \( \pi \) can enhance your ability to work with problems involving circular shapes and is a building block for higher mathematics.

Midpoint

Finding the midpoint between two numbers is a straightforward way to identify a number that lies exactly between them on the number line. The midpoint is essentially the average of two numbers.

To find the midpoint of numbers \( a \) and \( b \), use the formula:

• Midpoint = \( \frac{a + b}{2} \).

This formula calculates the center of the two numbers by adding them together and dividing by 2, ensuring the midpoint lies exactly halfway between them. Remember:

- This number is especially useful for finding a rational number between two decimal numbers if it results in a terminating decimal.

Number Range

When looking for a number within a specific number range, you are working between two points on a number line. A range defines the limits within which a number must fall.

In this context, we are looking between 3.14159 and \( \pi \) (approximated as 3.141592). These two values are very close, so finding a number between them involves understanding precise limits and comparisons.

• Ranges can be small or large depending on the values.
• Finding a rational number within any range involves recognizing decimal equivalents or fractional forms.

It helps to be comfortable working with decimal places and understanding how they relate to fractions to navigate between points of a numerical range.

Fractions

A fraction represents a part of a whole and is composed of two integers: a numerator and a denominator. Rational numbers are those that can be expressed as fractions.

For example, 1/2, 3/4, and 5 are all rational numbers because they can be written as fractions. The number 5 can be written as 5/1 to demonstrate its fraction form.

Fractions denote a ratio or a division between two integers and are particularly useful when identifying rational numbers between other numbers.

• A rational number like the midpoint found between 3.14159 and \( \pi \)—3.141591—can be represented in fractional form (for further decimal accuracy).
• Understanding fractions is key to comparing and finding rational numbers within any given range.

While fractions represent quantities less than or equal to whole numbers, they are foundational in rational number discussions and problem-solving.