Problem 51 - Elementary and Intermediate Algebra

Question

Write number in scientific notation. 0.0000000000000123

Step-by-Step Solution

Answer

The scientific notation is \( 1.23 \times 10^{-14} \).

Step 1: Identify the Significant Figures

Determine the non-zero numbers in the original number, which in this case is 123, as they represent the significant figures.

Step 2: Place the Decimal After the First Figure

Place the decimal point after the first digit of the significant figures to form a number between 1 and 10. For 123, this becomes 1.23.

Step 3: Count the Decimal Movements

Count how many places you move the decimal to get from the original number to this new form. Move the decimal from 0.0000000000000123 to 1.23, which requires 14 places to the right.

Step 4: Write in Scientific Notation Form

Express the number as a product of the new form and 10 raised to the power of the number of decimal places moved. This results in the scientific notation: \( 1.23 \times 10^{-14} \).

Key Concepts

Significant Figures

Significant figures are crucial in expressing a number's precision. They reflect the digits that carry meaningful information about the number’s precision. In the number 0.0000000000000123, the only significant figures are 123. Leading zeros are not considered significant because they do not contribute to the number’s value but only indicate scale or magnitude.
To identify significant figures:

• Ignore leading zeros as they are not significant.
• Include all non-zero digits (e.g., in 123, each digit is significant).
• Include zeros between non-zero digits (e.g., in 1002, the zeros are significant).

Correct identification of significant figures is the first step in converting a number to scientific notation.

Decimal Point

The placement of the decimal point is vital when working with scientific notation. You need to adjust the decimal point to create a new number between 1 and 10, which makes it easier to express very large or very small numbers.
In scientific notation, you move the decimal point:

• After the initial significant figure, creating a new decimal number between 1 and 10. In 123, placing it after the '1' results in 1.23.
• The number of decimal places shifted during this step will determine the exponent of 10 in scientific notation.

This step ensures that the number follows the proper format for scientific notation, simplifying it for practical use.

Powers of Ten

The concept of powers of ten is essential to scientific notation as it indicates the scale of the number, either making it larger or smaller. Once you have adjusted all significant figures to the proper position, you multiply by 10 raised to a power. This power is the number of places the decimal point was moved.
In our example:

• We moved the decimal 14 places right to convert 0.0000000000000123 to 1.23.
• The number is thus expressed as a product: 1.23 times 10 raised to the power of -14, written as \( 1.23 \times 10^{-14} \).

Powers of ten allow for clear communication and understanding of the number's size without extra zeros, which could be cumbersome to write.