Problem 11 - Algebra 1: Concepts and Skills

Question

Write the number in scientific notation. $$ 0.99 $$

Step-by-Step Solution

Answer

The number 0.99 in scientific notation is \(9.9 \times 10^{-1}\).

Step 1: Write the given number

The given number is 0.99. We need to express this number in a form such that there is only one digit before the decimal point.

Step 2: Express the number in scientific notation

To convert 0.99 into scientific notation, split the number into two parts: the digit part and the decimal part. The digit part is 0.99, which becomes 9.9 when we move the decimal place to the right once. The decimal part is the power of 10, which is -1 because we moved the decimal place to the right once. Thus, the number in scientific notation is \(9.9 \times 10^{-1}\).

Key Concepts

Expressing Numbers in Scientific Notation

Understanding scientific notation is key to dealing with very large or very small numbers efficiently. To express a number in scientific notation, you need one non-zero digit before the decimal point, followed by any other digits. The decimal point is then adjusted to reflect how the number has been scaled, and this is combined with a power of 10 that indicates how many places the decimal was moved.

For the example of 0.99, we adjust the decimal to obtain a new number between 1 and 10, which is 9.9. Now, to compensate for this shift, we multiply by 10 raised to a power. Since we only moved one place to the right to make the number 9.9, the power is negative one (showing that the original number was less than 1). This results in the scientific notation of 0.99 as \(9.9 \times 10^{-1}\).

In general, the process for expressing a number in scientific notation involves:

• Identifying the most significant digit (the first non-zero digit).
• Moving the decimal point so that it follows this digit, counting the number of places moved.
• Multiplying by 10 raised to the negative of this count if the number is less than 1, or to the positive of this count if the number is greater than 1.

Scientific Notation Conversion

Converting numbers to and from scientific notation is a fundamental skill in science and mathematics. It facilitates easier computation and comparison of vastly different scales of values. When converting a decimal to scientific notation, you are packaging the same quantity in a more compact form. The exercise demonstrating the scientific notation conversion proves this principle.

The number 0.99, when converted to scientific notation, is written as \(9.9 \times 10^{-1}\). The reverse process of conversion is equally simple; to convert a number from scientific notation to standard form, you take the digit in front of the multiplication symbol and move the decimal point to the right for a positive exponent of 10, or to the left for a negative exponent. Here, you would move the decimal one place to the left yielding the original 0.99.

Decimal Place Adjustment

Decimal place adjustment is at the heart of writing numbers in scientific notation. It's a skill that requires an understanding of the place value system of numbers, determining the significance of each digit's position. The rule of thumb is straightforward: When converting to scientific notation,

• If the original number is greater than 1, the decimal moves to the left, and the exponent on 10 is positive.
• If the original number is less than 1, as in the exercise with 0.99, the decimal moves to the right, making the exponent negative.

The exponent indicates how many places the decimal was moved. This ensures that the original number and its scientific notation represent the same quantity, just in a different format that is more manageable for various scientific and mathematical applications.