Problem 30 - College Algebra with Modeling and Visualization

Question

Write the number in scientific notation. \(29,285,000\) (People worldwide living with HTV)

Step-by-Step Solution

Answer

The number in scientific notation is \(2.9285 \times 10^7\).

Step 1: Identify the Significant Figures

First, we determine the significant figures of the number. In this case, the significant figures are 29285, which is the digit portion of 29285000, ignoring the trailing zeros.

Step 2: Determine the Power of Ten

Next, count the number of decimal places the decimal point needs to move to the left, to position it immediately after the first single digit. For 29285000, you move the decimal place 7 places to the left (29285000 -> 2.9285).

Step 3: Assemble in Scientific Notation

Now, write the number in scientific notation by placing the significant figures identified in Step 1 before a multiplication sign, and expressing the power of ten determined in Step 2 as an exponent. This yields: \(2.9285 \times 10^7\).

Key Concepts

Significant Figures

Significant figures are crucial in both scientific and mathematical calculations because they indicate the precision of a number. When dealing with large numbers, like in our exercise where we have 29,285,000, identifying significant figures helps simplify the number without losing essential information. In this specific example, **significant figures** refer to all non-zero digits and zeros that occur between them or to the right of them. This means that the number 29,285,000 has the significant figures 29285. Here are some helpful points to remember:

• The first non-zero digit is the beginning of our count of significant figures.
• Trailing zeros in a whole number without a decimal point are not considered significant.
• Our example has five significant figures, which form the essential part of the scientific notation.

Understanding the concept of significant figures allows you to accurately reflect the precision of the data you are working with, which is essential for any scientific calculation.

Power of Ten Calculation

The power of ten calculation is a vital step in forming scientific notation. It is used to signify how many times we need to multiply or divide by ten to get the original number from its scientific form. For the number 29,285,000, we need to convert it so that it falls between 1 and 10, achieving this by moving the decimal point.

• In our exercise, the decimal point moves from the end of the number to just after the first digit, the '2'.
• Counting these moves reveals that the decimal point has shifted 7 places to the left.

Therefore, we express this movement as a power of ten: specifically, an exponent of 7. So, when you have moved the decimal point the correct number of steps, your power of ten accurately reflects the magnitude of the original number.

Decimal Place Movement

Decimal place movement is a visual method to help write a number in scientific notation. This technique involves shifting the decimal point to create a new number between 1 and 10. Imagine the number written with a decimal at the very end, 29,285,000.0000000.

• To express the number in scientific notation, move the decimal point 7 places to the left until it is just after the '2', making the number 2.9285.
• Each movement is fundamental because it forms the power of ten you'll need to raise in scientific notation.

Knowing how to manipulate the decimal place is crucial for accurate scientific notation. This ensures that once the number has been simplified, it can easily be converted back to its original form using multiplication.