Question

The velocity of light is 186,282.397 miles per second. Round off this value to three significant digits.

Short answer

The velocity of light is 186,000 miles per second when rounded to three significant digits.

Step-by-step solution

Identify Significant Figures

Significant figures start from the non-zero number on the left. For 186,282.397, the first three significant figures are 186.

Check the Fourth Digit

Look at the fourth digit after the first three significant figures. If the fourth digit is 5 or greater, round up. In 186,282.397, the fourth digit is 2.

Apply Rounding Rules

Since the fourth digit, 2, is less than 5, you retain the third significant figure as is. Thus, you round down and 186,282.397 is rounded to 186,000 when considering three significant figures.

Express the Number

Finally, express the rounded value, considering only the first three significant figures. It becomes 186,000, showing a precision that retains the significant figures.

Key concepts

Rounding Numbers

Rounding numbers is a crucial skill in mathematics and science, especially when dealing with long or unwieldy numbers. It makes the numbers easier to work with and more concise when a high level of precision is not necessary. The general rule for rounding is simple: look at the number immediately following the last digit you want to keep. If that number is 5 or greater, the last digit you keep should increase by one. If it is less than 5, the last digit remains the same.
In our case, rounding the speed of light, 186,282.397, to three significant digits means focusing only on the first three digits — the '186'. To decide whether '186' should remain '186' or become '187', look at the next digit, '2'. Because '2' is less than '5', '186' stays as is, which means the rounded figure becomes 186,000. This process not only simplifies calculations but also makes it easier to interpret and communicate scientific data.

Scientific Notation

Scientific notation is a method of expressing very large or very small numbers in a compact form. It is expressed as the product of a number between 1 and 10 and a power of ten. This notation is particularly essential in fields like science and engineering where quantities can vary tremendously. Think of the velocity of light, 186,282.397 miles per second. After rounding it to three significant figures, it is easier to express this as 1.86 × 10^5.
To convert the number to scientific notation, place the decimal after the first significant figure, counting how many places the decimal has "moved" to form the new number. This count becomes the exponent of 10. Not only does scientific notation simplify handling large numbers, but it also helps in maintaining precision throughout calculations by focusing on the significant figures.

Precision in Measurement

Precision in measurement refers to how closely multiple measurements or estimations of the same thing are to each other. It is an indicator of the reliability and consistency of a measurement. When rounding numbers to a certain number of significant figures, it's important to understand that you are expressing a value within a specific precision level. With the speed of light example, rounding to three significant figures — 186,000 — implies we have chosen to maintain accuracy to the nearest thousand. This level of precision indicates that while the result captures the essence of the measurement, it acknowledges that there is an inherent degree of uncertainty in the exactness.
For practical purposes in scientific work, maintaining the balance between precision and usefulness is key. Over-calculating precision in results often does not reflect actual experimental accuracy, while under-calculating can obscure important details.