Question

The number of significant figures in \(\pi\) is (1) one (2) two (3) three (4) infinite

Short answer

Option (4) infinite

Step-by-step solution

Understand the Question

The exercise requires determining the number of significant figures in the value of \(\backslash pi\). This is a concept in mathematics related to the accuracy and precision of numerical values.

Review the Value of \( \pi \)

The value of \( \pi \) is approximately 3.14159, but it is known to have an infinite number of decimal places without repeating. This value is derived from the ratio of the circumference to the diameter of a circle.

Define Significant Figures

Significant figures in a number are the digits that carry meaning contributing to its precision. This includes all numbers except leading and trailing zeros where they serve as placeholders.

Relate \( \pi \) to Significant Figures

Since \( \pi \) has an infinite number of decimal places, it also has an infinite number of significant figures. The precision of \( \pi \) is not limited by any finite number of digits.

Choose the Correct Answer

Given that \( \pi \) has infinite decimal places and thus infinite significant figures, the correct choice is (4) infinite.

Key concepts

value of pi

The value of \( \pi \) is fundamental in mathematics, especially in geometry and trigonometry. \( \pi \) is approximately 3.14159, but its true value extends infinitely without repeating. It represents the ratio of a circle's circumference to its diameter. \( \pi \) is an irrational number, which means it cannot be exactly expressed as a simple fraction and its decimal representation never terminates or repeats. This infinite nature makes \( \pi \) unique and crucial for precise calculations. When using \( \pi \) in equations, precise values often depend on the specific context and the desired level of accuracy.

mathematical precision

Mathematical precision refers to the exactness of a numerical value. When dealing with measurements and calculations, having high precision is crucial for accuracy. Significant figures play a major role in this context, as they indicate the reliability of a number. For instance, \( \pi \) having infinite significant figures means it is infinitely precise. However, in practical applications, we often use rounded approximations of \( \pi \) such as 3.14 or 3.14159, depending on the required precision. The precision of \( \pi \) ensures that our mathematical models and calculations accurately reflect real-world phenomena.

decimal places

Decimal places indicate the position of digits to the right of the decimal point in a number. The more decimal places a number has, the more precise it is. For example, \( \pi \) can be written as 3.14 when rounded to 2 decimal places, or 3.14159 to 5 decimal places. The value of \( \pi \) extends infinitely, meaning it has an infinite number of decimal places. In calculations, the number of decimal places retained depends on the level of precision required. For certain advanced scientific calculations, \( \pi \) may be used to far more decimal places to ensure accuracy.

circumference and diameter ratio

The value of \( \pi \) arises from the unique and constant ratio of a circle's circumference to its diameter. This ratio, \( \pi = \frac{C}{D} \), holds true for any circle irrespective of its size. If you measure the circumference (C) and diameter (D) of a circle, dividing C by D will always yield \( \pi \). This relationship is fundamental to many areas of mathematics and practical applications. For example, engineers use \( \pi \) to design circular objects, and mathematicians use it to explore properties of circles and other related shapes.