Problem 15 - College Physics

Question

\(\cdot\) Express each of the following numbers to three, five, and eight significant figures: (a) \(\pi=3.141592654 \ldots,\) (b) \(e=\) \(2.718281828 \ldots,(\mathrm{c}) \sqrt{13}=3.605551275 \ldots\)

Step-by-Step Solution

Answer

Three figures: 3.14, 2.72, 3.61. Five figures: 3.1416, 2.7183, 3.6056. Eight figures: 3.1415927, 2.7182818, 3.6055513.

Step 1: Significant Figures for Pi (3.141592654...)

Identify the digits in \(\pi = 3.141592654\). For three significant figures, only the first three digits are considered: \(3.14\). For five significant figures, the first five digits are considered: \(3.1416\). For eight significant figures, use the first eight digits: \(3.1415927\). The digits are rounded accordingly, keeping in mind to round up if the next digit is 5 or more.

Step 2: Significant Figures for e (2.718281828...)

For the number \(e = 2.718281828\), identify the significant figures. For three significant figures, it's \(2.72\); for five significant figures, it's \(2.7183\); and for eight significant figures, it's \(2.7182818\). Like before, round the last digit you're keeping based on the next digit.

Step 3: Significant Figures for Square Root of 13 (3.605551275...)

For \sqrt{13} = 3.605551275\, three significant figures result in \(3.61\), five significant figures result in \(3.6056\), and eight significant figures result in \(3.6055513\). Round the digits based on standard rounding rules, considering the digit following the last one kept.

Key Concepts

The Art of Rounding Numbers

Rounding numbers is an essential skill in mathematics and everyday life. It involves adjusting a number to make it simpler or to focus on its most significant digits. This is particularly useful when dealing with lengthy decimals or to ensure consistency in measurements.

Here are some basic rules to help understand rounding better:

• Identify which digit is to be rounded based on specified significant figures.
• Look at the digit immediately after the one to be rounded. If it is 5 or more, round up the identified digit.
• If the next digit is less than 5, the identified digit remains unchanged.
• Drop all the digits following the rounded digit.

For example, in the number \(3.141592654\), to round to three significant figures, we focus on the first three digits and check the fourth. Since the fourth is a '1', it is less than 5, so our rounded number is \(3.14\). Similarly, this is done for five and eight significant figures accordingly.

Unveiling Mathematical Constants

Mathematical constants are fixed numbers that are widely recognized for their fundamental properties in mathematics. The most notable ones include \(\pi\), \(e\), and other irrational numbers.

• \(\pi = 3.1415926535...\) represents the ratio of the circumference of a circle to its diameter, a critical constant in geometry.
• \(e = 2.718281828...\) is the base of the natural logarithm and surfaces in fields like calculus and compound interest calculations.

These constants often appear with many decimal places, leading us to treat them with specific techniques like rounding. By doing so, they can be more easily used in calculations without losing substantial precision. It's vital to maintain a balance between accuracy and simplicity in numerical representation.

Understanding Numerical Approximation

Numerical approximation is a mathematical process used to represent a complex number with a simpler version, often involving fewer digits. This approach enhances computational efficiency and simplifies work without significantly compromising accuracy.

For instance, calculating square roots or powers can yield long decimals, which are rounded or truncated to a practical number of significant figures. Consider the square root of 13, \(\sqrt{13} = 3.605551275...\). Approximating this with three significant figures gives us \(3.61\), balancing precision and manageability. Approximations allow us to use these numbers efficiently in mathematical operations, scientific experiments, and engineering calculations, where full accuracy to many decimal places might not contribute significant extra value.