Problem 70 - Chemistry Matter and Change

Question

How does scientific notation differ from ordinary notation?

Step-by-Step Solution

Answer

Scientific notation expresses numbers as a product of two factors: a decimal number (between 1 and 10) and a power of 10, making it easier to represent very large or small numbers. In contrast, ordinary notation uses digits and place values, often requiring more digits and zeros. For example, the ordinary notation number 800,000,000 can be written as \(8.0 \times 10^8\) in scientific notation.

Step 1: Convert the number to decimal form

First, we need to convert the given number to a decimal number between 1 and 10 (but not equal to 10). To do this, simply place a decimal point after the first non-zero digit:\(8.0\).

Step 2: Count the number of places moved

Now, count the number of decimal places you moved from the original number to reach the decimal form. Here, we moved 8 places to the left, so the power of 10 will be 8.

Step 3: Write the number in scientific notation

Now, multiply the decimal number obtained in step 1 by the power of 10 obtained in step 2. In this case, the scientific notation will be: \(8.0 \times 10^8\). In conclusion, the given number (800,000,000) in ordinary notation can be represented as \(8.0 \times 10^8\) in scientific notation.

Key Concepts

Ordinary Notation

Ordinary notation is the standard way of writing numbers, just as you would see them in everyday life. It's how numbers are typically represented and understood, without any special formatting. For instance, the number 800,000,000 is an example of ordinary notation. Here, the number is expressed in a straightforward manner, with all zeros written out explicitly. This method of writing numbers is intuitive and clear, but can become cumbersome for very large or very small numbers because you have to write numerous zeros, which can be impractical and error-prone. As a result, this led to the development of a more convenient approach for handling such magnitudes.

Decimal Form

Decimal form is an essential part of converting numbers to scientific notation. Decimal form entails expressing a number as a decimal between 1 and 10, but not equal to 10. For instance, when you have a large number like 800,000,000 and you want to write it in decimal form, you find the first non-zero digit and place a decimal point after it. In our example, the non-zero number is 8. Therefore, the decimal form of 800,000,000 is 8.0. This step simplifies the number, making it easier to use in scientific notation. Remember, the key is to adjust the position of the decimal until the number matches this criterion, lending itself to a more manageable and compact format.

Power of 10

The power of 10 is a crucial concept that supports expressing numbers in scientific notation. It represents how many times you must multiply the base (which is 10) to get the actual figure back from its decimal form. In scientific notation, determining the power of 10 involves counting how many places you moved the decimal point to convert the original number into decimal form. For the number 800,000,000, you move the decimal point 8 places to the left to reach 8.0. Thus, the power of 10 here is 8, denoting that 8.0 should be multiplied by 10 raised to the eighth power to return to its original value. This component of scientific notation makes it particularly powerful and efficient for expressing very large or small numbers succinctly.

Convert Numbers

Converting numbers between ordinary notation and scientific notation involves a straightforward process, breaking down numbers into their component parts.

• First, find the decimal form by identifying the first non-zero digit and placing a decimal immediately after it.
• Next, count how many places the decimal has shifted in the transition from ordinary notation.
• This count becomes your power of 10.
• Finally, write the number in scientific notation by multiplying the decimal form by 10 raised to the power of the shift count.

By following these steps, numbers are simplified and captured in a format that is less prone to errors in reading and writing, especially useful in scientific calculations and reports.