Question

Write this quantity in scientific notation: one ten-millionth of a centimeter.

Short answer

Answer: \(1 \times 10^{-7}\) cm

Step-by-step solution

Convert the given quantity into a number

First, we need to convert one ten-millionth of a centimeter into a number. To do this, we can think of it as a fraction, where the numerator is 1 (one) and the denominator is the number ten-million (10,000,000). So, one ten-millionth of a centimeter can be written as a fraction: \(\frac{1}{10,000,000}\) centimeters.

Convert the fraction into a decimal

Now, we'll convert the fraction \(\frac{1}{10,000,000}\) into a decimal by dividing 1 by 10,000,000: \(\frac{1}{10,000,000} = 0.0000001\) centimeters.

Write the decimal in scientific notation

To write the decimal in scientific notation, we need to find the exponent that we'd need to multiply by 10 to get the decimal. In this case, since the decimal point is moved 7 places to the right to get the number 1, the exponent is -7. Now, we can write the decimal in scientific notation: \(1 \times 10^{-7}\) centimeters. So, one ten-millionth of a centimeter in scientific notation is \(1 \times 10^{-7}\) cm.

Key concepts

Converting Fractions to Decimals

Understanding how to convert fractions to decimals is a fundamental skill in math that allows us to work with numbers in different formats. To convert a fraction to a decimal, you simply divide the numerator (the top number) by the denominator (the bottom number). For example, to convert the fraction \(\frac{1}{10,000,000}\), we divide 1 by 10,000,000, resulting in 0.0000001. This process transforms the value represented by the fraction into a decimal that can be more easily understood and used in further calculations.

For longer denominators or complex fractions, a calculator can speed up this process. However, understanding the fundamental method of division is crucial for grasping more advanced mathematical concepts and for those times when a calculator isn't at hand. Being able to visualize the fraction as a division problem helps to reinforce the relationship between these two representations of numbers.

Scientific Notation in Physics

Scientific notation is an essential tool in physics for dealing with very large or very small numbers, which are common when measuring physical constants, distances in space, or subatomic particles. It expresses numbers as a product of two parts: a coefficient between 1 and 10, and a power of 10. This not only simplifies the numbers but also makes calculations easier.

In physics, quantities like the charge of an electron or the mass of a proton are incredibly small and using scientific notation allows physicists to communicate these measurements without the confusion of many decimal places. For example, the mass of a proton is approximately \(1.67 \times 10^{-27}\) kilograms in scientific notation, which is much more convenient than writing out 0.00000000000000000000000000167 kilograms.

The use of scientific notation also helps to quickly compare the magnitude of different physical quantities and simplifies mathematical operations such as multiplication and division that are frequently used in physics calculations.

Decimal to Scientific Notation Conversion

When converting a decimal to scientific notation, the goal is to rewrite the number so that it is easier to read and work with, especially if it contains a lot of decimal places. The process involves moving the decimal point until we have a number between 1 and 10, and then recording how many places the decimal was moved as an exponent of 10.

For the decimal 0.0000001, which was derived from the fraction \(\frac{1}{10,000,000}\), we move the decimal point 7 places to the right to get the number 1. Since we are moving the decimal point to the right, the exponent will be negative, reflecting the small magnitude of the original number. Therefore, this decimal converts to \(1 \times 10^{-7}\) in scientific notation. This method is not only common in physics, but is widely used in all science and engineering fields because it clearly represents how large or small a number is and reduces the potential for errors when dealing with extreme values.