Problem 50 - Contemporary Precalculus

Question

Express the given numbers (based on 2006 estimates) in scientific notation. Population of the United States: 298,400,000

Step-by-Step Solution

Answer

Question: Express the population of the United States, 298,400,000, in scientific notation. Answer: The population of the United States, expressed in scientific notation, is 2.984 × 10^8.

Step 1: Determine the significant digits

First, we need to identify the significant digits in the number. Significant digits are non-zero digits starting from the first non-zero digit on the left. In our case, the number is 298,400,000, and the significant digits are 2, 9, and 8.

Step 2: Convert the significant digits into a decimal

Now, we need to convert these significant digits into a decimal number. To do this, simply place a decimal point after the first significant digit and write the remaining digits after the decimal point. So, the decimal number for our significant digits will be 2.984.

Step 3: Determine the exponent of 10

Next, we need to determine the exponent by calculating how many places the decimal point has shifted to the left from its original position. In our given number, 298,400,000, the decimal point has been shifted to the left by 8 places (as there are 8 zeros). Thus, the exponent will be 8.

Step 4: Write the number in scientific notation

Finally, we can express the given number in scientific notation by writing the decimal number found in Step 2 multiplied by 10 raised to the exponent determined in Step 3. Therefore, the scientific notation for 298,400,000 is: 2.984 \times 10^8.

Key Concepts

Significant Digits

In the realm of scientific notation, significant digits play a crucial role. They help identify the precision of a number. When given a number like 298,400,000, it's important to identify its significant digits. These are the non-zero numbers you encounter when reading from the left. Here, 2, 9, and 8 serve as the significant digits.
To determine significant digits, follow these steps:

• Look for the first non-zero digit (from the left).
• Include all subsequent digits, regardless of being zero or non-zero, up to and including the last non-zero digit observed from the left.

Identifying these digits allows you to move on to other steps in expressing numbers in scientific notation.

Decimal Conversion

Once you've identified the significant digits, the next step is decimal conversion. This involves positioning these digits in a decimal format for simplicity and ease of use. In our example, we rearrange 2, 9, and 8 from 298,400,000 into a decimal form: 2.984.
Here's how to do it:

• Place a decimal point immediately after the first significant digit.
• Write the remaining significant digits after the decimal point.

This conversion helps understand the base number without considering the magnitude, which will be addressed in subsequent steps.

Exponent Calculation

Exponent calculation is vital for expressing numbers in scientific notation. This step identifies how many places the decimal point has shifted from its original position. For 298,400,000, count the spaces between the original decimal position and its new position after conversion to get the exponent.
In our case, since you have shifted 8 places from 298,400,000 to 2.984, the exponent is 8. This method allows you to express large or tiny numbers efficiently.
The calculation of this shift leads to a crucial part of scientific notation where you write the base number multiplied by 10 raised to this exponent.

Number Representation

The ultimate goal of learning scientific notation is efficient number representation. This method is especially useful for expressing very large numbers like populations or astronomical distances. With our previous steps, we write 298,400,000 as 2.984 multiplied by 10 raised to the 8, or in mathematical terms, 2.984 \times 10^8.
Scientific notation simplifies using and interpreting large numbers by focusing on two components:

• The significant decimal number, here 2.984.
• The power of ten represented by the calculated exponent, here \(10^8\).

This form of number representation not only standardizes large figures but also conveys accuracy and thoroughness in scientific reports and calculations.