Question

The speed of light to five significant figures is \(2.9979 \times 10^{8} \mathrm{~m} / \mathrm{s}\). What is the speed of light to three significant figures?

Short answer

The speed of light to three significant figures is \(3.00 \times 10^8 \mathrm{~m/s}\).

Step-by-step solution

Understand Significant Figures

Significant figures are the digits in a number that are known with certainty plus one estimated digit. When reducing the number of significant figures, we maintain the most significant digits based on the original information provided.

Identify the First Three Significant Figures

The given speed of light is \(2.9979 \times 10^8\). The first three significant figures are '2', '9', and '9'.

Round to Three Significant Figures

After identifying '2', '9', and '9', the next digit is '7'. According to rounding rules, if the next digit is 5 or greater, increment the last significant digit by 1. Thus, '9' becomes '10', rolling over to make '2.99' into '3.00'.

Express in Scientific Notation

The value now is '3.00', which can be expressed in scientific notation as \(3.00 \times 10^8\), ensuring we maintain three significant figures in our answer.

Key concepts

Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a compact form. It's particularly useful in sciences, like physics, where we often deal with extreme quantities. In scientific notation, a number is written as the product of two parts:

• A coefficient that is between 1 and 10.
• An exponent of 10.

For example, the speed of light can be written as \(2.9979 \times 10^8\, \mathrm{m/s}\). Here, '2.9979' is the coefficient, and '\(10^8\)' denotes how many times to multiply the coefficient by 10.
This method ensures that we concisely represent and work with numbers, maintaining clarity and precision. It's exceptionally useful for maintaining consistency when calculating or conveying scientific measurements.

Rounding Rules

Rounding is the process of simplifying a number to make it easier to work with, while maintaining as much accuracy as necessary. The rules of rounding are essential when dealing with significant figures, which dictate how many digits should be kept for precision.
To round to a certain number of significant figures, you:

• Identify the last digit you need to keep.
• Look at the next digit (the one you're considering dropping).
• If that digit is 5 or higher, increase the last kept digit by 1.
• If that digit is less than 5, leave the last kept digit unchanged.

For instance, rounding \(2.9979\) to three significant figures involves looking at '2.997'. Since the next digit is 9, you round the second 9 up to make it '3.00'.
These simple rules allow us to express values succinctly, while avoiding unnecessary complexity.

Speed of Light

The speed of light is a fundamental constant in physics, denoted by the symbol \(c\). It's known to be approximately \(3 \times 10^8\, \mathrm{m/s}\) in a vacuum. This constant is not just a measure of speed, but it's instrumental in various scientific laws and equations.
For instance, Einstein's theory of relativity is based on the principle that nothing can travel faster than the speed of light. It serves as a "speed limit" for the universe. Besides its scientific implications, the speed of light is used to define the meter, the base unit of length in the International System of Units (SI). A meter is defined as the distance light travels in a vacuum in \(1/299,792,458\) of a second.
Understanding the speed of light helps us grasp the vast distances in space and the principles that govern electromagnetic radiation. It’s fascinating how such a constant can be crucial in both everyday technologies, like GPS, and in understanding the cosmos.