Question

The first several digits of \(\pi\) are known to be \(\pi=3.14159265358979 \ldots\). What is \(\pi\) to (a) three significant figures and (b) five significant figures?

Short answer

π is 3.14 to three significant figures and 3.1416 to five significant figures.

Step-by-step solution

Understanding Significant Figures

Significant figures are the digits in a number that contribute to its measurement resolution, including the first non-zero digit. In this exercise, we need to express \(\pi\) to both three and five significant figures.

Applying Three Significant Figures for π

To express \(\pi\) to three significant figures, count the first three digits from the left, starting with the first non-zero digit. For \(\pi = 3.14159265358979\ldots\), the first three digits are 3, 1, and 4. Therefore, \(\pi\) to three significant figures is 3.14.

Determining Five Significant Figures for π

To determine \(\pi\) to five significant figures, count the first five digits from the left starting with the first non-zero digit. From \(\pi = 3.14159265358979\ldots\), these digits are 3, 1, 4, 1, and 5. Therefore, \(\pi\) to five significant figures is 3.1416, as we need to round the fifth digit because the sixth digit is 9.

Key concepts

Rounding Numbers

Rounding numbers is a key skill in mathematics that helps simplify values to make them easier to work with, without significantly losing precision. Let's say you have a number with many decimal places and you need a simpler, shorter version of it. That's where rounding comes in.

When rounding numbers, you look at the digit immediately following the last digit you want to keep. If that digit is 5 or greater, you increase the last digit you keep by one. If it’s less than 5, you keep the last digit the same, trimming off the extra digits.

• Example: Round 3.146 to two decimal places. Look at the third decimal place (6). Since it's greater than 5, you round up: 3.15.
• Example: Round 2.433 to two decimal places. Look at the third decimal place (3). Since it's less than 5, you leave it as: 2.43.

Rounding simplifies numbers which is crucial in many fields, especially when dealing with large datasets or when presenting clear, concise reports.

Mathematical Constants

Mathematical constants are numbers with fixed values that appear universally in mathematics and science. These constants are significant because they help express fundamental truths about the universe. Recognizing and understanding these constants can illuminate various aspects of mathematical theory and its application.

One of the most famous mathematical constants is \(\pi\), known for its role in circle geometry. It represents the ratio of the circumference of any circle to its diameter. This value is unique and does not change, no matter the size of the circle.

• Popular constants include: \(\pi \approx 3.14159\), Euler's number \( e \approx 2.71828\), and the golden ratio \(\phi \approx 1.61803\).
• Constants are used to create formulas and expressions that remain true under specified conditions.

These constants not only aid in solving equations but also in understanding deeper scientific laws.

Pi Approximation

Approximating \(\pi\) is an ancient mathematical practice dating back thousands of years. Since \(\pi\) is an irrational number, its decimal representation goes on infinitely without repeating, making it impossible to express exactly as a finite decimal or fraction.

To work with \(\pi\) practically, we often approximate it to a set number of significant figures or decimal places. This method's effectiveness is determined by the level of precision needed.

• Common approximations include 3.14 (two decimal places) and 22/7, a fraction used historically which equals roughly 3.142857.
• For three significant figures, \(\pi\) is represented as 3.14. For five, it’s 3.1416.

Each approximation allows for practical use in measurements and calculations, like understanding perimeter in real-world applications, while maintaining an acceptable degree of accuracy.