Problem 17 - Intermediate Algebra

Question

For Problems \(1-18\), write each of the following in scientific notation. \(0.00000000194\)

Step-by-Step Solution

Answer

The scientific notation is \(1.94 \times 10^{-9}\).

Step 1: Identify the Base Number

To convert a number into scientific notation, start by identifying the base number, which is a number between 1 and 10. In the case of \(0.00000000194\), the base number will be \(1.94\) because we move the decimal 9 places to the right.

Step 2: Count the Decimal Places

Count how many places you move the decimal point to turn \(0.00000000194\) into \(1.94\). In this case, we move the decimal 9 places to the right.

Step 3: Write in Scientific Notation

Combine the base number and the exponent to express the original number in scientific notation. Since the decimal moved 9 places to the right, the exponent will be negative: \(1.94 \times 10^{-9}\).

Key Concepts

Understanding Base Number

In scientific notation, the base number is crucial and is always a number between 1 and 10. It represents the significant figures in the original number. By focusing on these numbers, we can simplify and accurately convey the information stored in very large or very small numbers.
To determine the base number in a given number like 0.00000000194, we need to adjust the decimal point so that the resulting number is comfortably within this range of 1 to 10.
This means shifting the decimal point to create the number 1.94, a value between 1 and 10 that retains the precision of the original digits.

The Role of the Exponent

Once the base number has been identified, the next step in scientific notation is determining the exponent. The exponent indicates the number of places the decimal point was moved to convert the original number to the base number.

• If you move the decimal point to the right, the exponent will be negative, signifying a small initial number.
• Moving the decimal point to the left results in a positive exponent, indicating the original number was large.

In the example 0.00000000194, we shifted the decimal point 9 places to the right, resulting in an exponent of -9. This means that the actual number is 1.94 times ten to the power of negative nine, as shown in the notation, 1.94 \( \times 10^{-9} \).

Decimal Point Movement Explained

The movement of the decimal point is a fundamental part of converting a number into scientific notation. This movement compresses the number into its significant digits while clearly indicating its scale or size.
For 0.00000000194, moving the decimal point 9 places to the right changes it to 1.94, transforming its original form to a more efficient format suitable for scientific analysis.

• The direction of movement determines the sign of the exponent: right for negative, left for positive.
• Counting each place the decimal is moved helps in correctly identifying the exponent, which plays a crucial role in scientifically expressing very large or small values.

In scientific notation, understanding the purpose and process of decimal point movement aids in grasping how vast quantities can be managed and communicated succinctly.