Problem 65 - Algebra 1: Concepts and Skills

Question

Write the number in scientific notation. The mass of a carbon atom is 0.00000000000000000000002 gram.

Step-by-Step Solution

Answer

The mass of the carbon atom is \(2 \times 10^{-23}\) grams in scientific notation.

Step 1: Understanding Scientific Notation

Scientific notation is a method of writing very large or very small numbers as a product of a number between 1 and 10 and a power of 10. It is generally written as \(a \times 10^n\), where \(1 \leq a < 10\) and \(n\) is an integer.

Step 2: Convert the Mass to Scientific Notation

Given the mass of a carbon atom is 0.00000000000000000000002 gram. This number can be written in scientific notation by shifting the decimal to the right until we are left with a number between 1 and 10, which in this case is 2. Count the number of places you moved the decimal point to find the decimal exponent. The decimal was moved 23 places to the right, so the exponent is -23. Therefore, the mass of a carbon atom in scientific notation is \(2 \times 10^{-23}\) grams.

Key Concepts

Writing Numbers in Scientific Notation

Scientific notation is a convenient way to express very large or very small numbers. It simplifies reading and writing such numbers by using powers of 10, making it easier to understand and manage data in scientific and mathematical contexts. To write a number in scientific notation, you must identify the significant figures of the number and the decimal placement to represent the number as a product of a number between 1 and 10, and a power of 10.

For instance, taking the mass of a carbon atom, we find it cumbersome to deal with so many zeros: 0.00000000000000000000002 grams. By shifting the decimal point 23 places to the right, the number is now between 1 and 10, which is 2. In scientific notation, this is expressed as \(2 \times 10^{-23}\) grams. It's critical to ensure the starting number, now 2, is indeed between 1 and 10, and the exponent reflects the number of decimal places moved.

Powers of 10

Powers of 10 are integral to understanding scientific notation. When a number is raised to a power, it is being multiplied by itself a certain number of times. For example, \(10^2\) equals 100 because 10 is multiplied by itself once (10 x 10). In scientific notation, we use positive powers to represent large numbers and negative powers to represent small numbers.

The exponent, or the power of 10, indicates how many places the decimal needs to move. If the exponent is positive, the decimal moves to the right, making the number larger. Conversely, if the exponent is negative, like in our carbon atom example \(2 \times 10^{-23}\), we move the decimal to the left, indicating a very small number. This system provides a clear and precise way to denote the scale of numbers.

Representing Small Numbers

Small numbers, such as those in the realm of atomic measurements, are challenging to read and manage when written in standard decimal form due to the proliferation of leading zeros. The power of scientific notation shines brightest when representing such infinitesimal values. The procedure for a small number involves moving the decimal point to the right until you have a non-zero digit starting off the new number.

In our carbon atom example, the number is extremely small and requires us to count each zero as we move the decimal point to get to a significant figure so we can express it succinctly as \(2 \times 10^{-23}\). This placement of the decimal point and use of a negative exponent compactly signifies the original number's smallness, making it comprehensible and useful for calculations and comparisons.