Problem 59 - Prealgebra

Question

Write each number in scientific notation. $$42,240$$

Step-by-Step Solution

Answer

The number 42,240 in scientific notation is \( 4.224 \times 10^4 \).

Step 1: Identify the Significant Digits

First, determine which digits are significant in the number 42,240. The significant figures here are 4224, as the trailing zero does not impact the number's precision in scientific notation.

Step 2: Place the Decimal

Position the decimal point to create a number between 1 and 10. In this case, moving the decimal after the first 4 in 42240 gives 4.224.

Step 3: Determine the Exponent

Calculate the power of 10 needed by counting how many places the decimal has been moved from its original position. For 42,240, the decimal is moved 4 places to the left, yielding an exponent of 4.

Step 4: Write in Scientific Notation

Combine the result from Step 2 and the exponent from Step 3 to write the number in scientific notation: \[ 4.224 \times 10^4 \]

Key Concepts

Understanding Significant Figures

Significant figures refer to the digits in a number that contribute to its precision. They are important because they help us understand the accuracy of a measurement or calculation. In the number 42,240, the significant figures are 4224. The trailing zero here is not considered significant as it doesn't add to the precision of the number when expressed in scientific notation.

• Any non-zero digit is always significant.
• Zeros between non-zero digits are significant.
• Leading zeros, those at the beginning of a number, are not significant.
• Trailing zeros in a decimal number are significant.

Understanding which digits are significant ensures that when we convert numbers into scientific notation, we maintain their accuracy.

Decimal Point Placement

When converting a number to scientific notation, placing the decimal point correctly is crucial. Scientific notation requires the decimal to be positioned so that the resultant figure is between 1 and 10. In the case of 42,240, we move the decimal point to follow the first digit 4, resulting in the number 4.224. This placement ensures that the value stays true to its original size while simplifying its expression in scientific form. The movement of the decimal point also helps in deciding the power of 10 that complements the scientific notation. Correct placement highlights the number's significant figures and makes comparison with other quantities much easier in mathematical operations.

Role of Power of 10

The power of 10 in scientific notation represents how many times we multiply or divide a number by 10 to return to its original size. For 42,240, moving the decimal point four places to the left gives us a power of 4. This is expressed in scientific notation as 10 raised to the power of 4, written as \( 10^4 \).The exponent indicates the scale of the number. If the decimal is moved to the left, the power is positive, while moving it to the right results in a negative power. Comprehending the power of 10 helps in encoding large or small numbers in a more manageable and readable form, simplifying computations and analyses.

Significance in Mathematics Education

Teaching scientific notation is a cornerstone of mathematics education because it streamlines complex calculations and provides a clear method for working with very large or small numbers.

• Builds a foundation for higher-level math, like calculus and physics.
• Promotes understanding of number size and significance in different contexts.
• Facilitates easier comparison and computation of differing magnitudes.

Students who learn scientific notation gain valuable skills in precision and accuracy. It encourages logical thinking, fosters numerical literacy, and instills confidence when tackling sophisticated scientific and mathematical problems.