Question

The velocity of light is 299,792,458 meters per second. Round off this value to three significant digits.

Short answer

The velocity of light rounded to three significant digits is 3.00 x 10^8 m/s.

Step-by-step solution

Identify Significant Digits

Significant digits are the digits in a number that carry meaningful information about its precision. For the velocity of light, given as 299,792,458, all digits are significant until we decide to round them off to a specific number of significant digits.

Determine Digits to Keep

We need to round off the value to three significant digits. This means we will keep the first three non-zero digits of the number, which are 2, 9, and 9.

Decide on Rounding

Look at the fourth digit, which is 7. If this digit is 5 or greater, we round up the last significant digit, which in this case is the second 9.

Perform the Rounding

Since the fourth digit (7) is greater than 5, increase the third significant digit (9) by 1. Add 1 to 9, which becomes 10. Thus, the number rounds to 300.

Apply Scientific Notation

To express 299,792,458 in three significant figures in scientific notation, we write it as 3.00 x 10^8, capturing the rounded value with three significant digits and the appropriate power of ten.

Key concepts

Velocity of Light

The velocity of light, often denoted as "c," is a fundamental constant in physics. It represents the speed at which light travels through a vacuum, and its value is approximately 299,792,458 meters per second.
This speed is not only a crucial component of Einstein's theory of relativity, but it also plays an essential role in many scientific calculations and experiments.
Here is why it is so important:

• Consistency: The speed of light remains constant regardless of the observer's motion or the source of the light. This property is key to the theory of special relativity.
• Measurement Standard: Since 1983, the meter has been defined based on the speed of light, making it a cornerstone in systems of measurement.
• Universal Limit: The velocity of light is considered the maximum speed at which all information, energy, and matter can travel in our universe.

Understanding and being able to work with the velocity of light is crucial for students dealing with physics, especially in areas focusing on electromagnetism and astronomy.

Scientific Notation

Scientific notation is a method to express very large or very small numbers in a concise form, making them easier to work with.
It is particularly useful in fields like science and engineering, where such numbers frequently occur.

• Format: A number is written as the product of two factors: a coefficient (a number typically between 1 and 10) and a power of ten. For example, the speed of light, when expressed in scientific notation, becomes 3.00 \( \times \) 10^8 meters per second.
• Purpose: This format simplifies calculations by reducing the number of zeros and making the magnitude of the number immediately apparent.
• Example: The number 4,500 can be written in scientific notation as 4.5 \( \times \) 10^3.

By using scientific notation, students can handle computations more easily and avoid potential errors related to miscounting digits or placing decimals incorrectly.

Rounding Numbers

Rounding numbers is the process of reducing the number of digits in a number while maintaining its value to a certain extent of accuracy.
It is a key skill in mathematics and science, allowing students to simplify complex numbers and focus on the most critical figures.

• Significant Figures: These are the digits in a number that contribute to its precision. When rounding to significant figures, you keep a select number of digits from the original number to reflect its precision after rounding.
• Rounding Rules: Generally, if the digit immediately right of your desired number of significant figures is 5 or greater, you round up the last kept digit. Conversely, if it is less than 5, you leave the last digit unchanged.
• Application: In the case of the velocity of light, we rounded 299,792,458 to three significant figures. We focused on 2, 9, and 9, then rounded based on the ensuing 7, resulting in a final value of 300, which can also be expressed as 3.00 \( \times \) 10^8 in scientific notation.

Understanding how to properly round numbers is essential for maintaining the integrity of calculations while making them manageable and easier to interpret.