Problem 60 - College Algebra and Calculus: An Applied Approach

Question

Write the number in scientific notation.One Micron (millionth of a meter): \(0.00003937\) inch

Step-by-Step Solution

Answer

In scientific notation, one micron, or \(0.00003937\) inch, is represented as \(3.937 \times 10^{-5}\).

Step 1: Determine the Decimal Point Position

Move the decimal point in \(0.00003937\) inch to the right until there is only one non-zero digit to the left. This would be \(3.937\). Here the decimal point has been moved 5 places to the right.

Step 2: Write the Result as a Product of a Number and a Power of 10

Now, express the number as a product of the new value, which is \(3.937\), and a power of 10. For the power of 10, use the number of places moved as negative power for small numbers, as here. With 5 places moved, the right notation is: \(3.937 \times 10^{-5}\).

Key Concepts

Decimal Point Position

Understanding the position of the decimal point is crucial when converting standard numbers into scientific notation. The main rule is to arrange the number such that only one non-zero digit remains to the left of the decimal point. This is done by moving the decimal point either to the right or left, depending on whether the number is smaller or larger than 1. For instance, take the number 0.00003937; to convert this into scientific notation, shift the decimal point to the right, stopping directly after the first non-zero digit (3 in this case), resulting in 3.937. It is essential to keep track of how many places the decimal has been moved, as this will define the power of 10 in the final expression of the number in scientific notation.

One typical mistake students make is not moving the decimal point to create only one non-zero digit to the left. Ensuring you have a single non-zero digit is a key step to correctly formatting a number in scientific notation.

Power of 10

The 'power of 10' in scientific notation indicates how many times the decimal point has been moved to get that single non-zero digit to the left of the decimal point. It's vital to remember that for numbers smaller than one, the power of 10 will be negative because the decimal point is moved to the right, indicating division by a power of 10 (as opposed to multiplication for larger numbers where the decimal is moved to the left). For the number 0.00003937, we moved the decimal point 5 places to the right. Therefore, the power of 10 is \(10^{-5}\). The negative sign highlights that we are working with a small number. The complete scientific notation for 0.00003937 is \(3.937 \times 10^{-5}\).

It's common for students to confuse when to use a positive or negative exponent. Remembering that small numbers use negative exponents because you are 'compressing' the size of the number can help avoid this mistake.

Expressing Small Numbers

Expressing small numbers in scientific notation is a way to make them easier to work with, especially when dealing with extremely large or small scales, such as in chemistry or physics. When writing small numbers like 0.00003937, scientific notation allows us to display this number succinctly as \(3.937 \times 10^{-5}\). It is a more efficient way to represent values that would otherwise be cumbersome to read and write. In the scientific community, this notation is necessary because it provides a standardized method to convey very small numbers without the risk of misinterpretation due to a large number of leading zeros.

Students sometimes overlook the significance of scientific notation for small numbers, not realizing its purpose in avoiding errors in calculation and communication. By consistently using scientific notation, the potential for error in decimal placement is reduced, thereby increasing accuracy in scientific work.