Write The Numbers In Scientific Notation. 673.5

Mastering Scientific Notation: A Comprehensive Guide

Scientific notation, also known as standard form or exponential notation, is a way of expressing numbers that are too large or too small to be conveniently written in standard decimal form. It's particularly useful in scientific fields like physics, chemistry, and astronomy, where dealing with extremely large or minuscule values is commonplace. This article provides a detailed explanation of how to write numbers in scientific notation, specifically focusing on converting the number 673.5 into its scientific notation equivalent.

Why Use Scientific Notation?

Before diving into the mechanics, it's crucial to understand the benefits of using scientific notation:

• Conciseness: Scientific notation allows us to express very large and very small numbers in a compact and manageable format. Imagine writing the distance to the nearest star in kilometers versus using scientific notation – the latter is far more efficient.
• Clarity: It eliminates the ambiguity associated with long strings of zeros, making it easier to compare and manipulate numbers.
• Ease of Calculation: Scientific notation simplifies calculations involving very large or small numbers, particularly when combined with exponents.
• Standardization: It provides a standardized way of representing numerical values, facilitating communication and collaboration within the scientific community.

The Anatomy of Scientific Notation

A number expressed in scientific notation takes the following form:

a × 10^b

Where:

• a is the coefficient, also known as the significand or mantissa. It's a decimal number greater than or equal to 1 and less than 10 (1 ≤ |a| < 10).
• 10 is the base. It's always 10 in scientific notation.
• b is the exponent or power of 10. It's an integer (positive, negative, or zero) that indicates how many places the decimal point must be shifted to convert the number back to standard decimal form.

Converting 673.5 to Scientific Notation: A Step-by-Step Guide

Let's apply this knowledge to convert the number 673.5 into scientific notation.

Step 1: Identify the Decimal Point

In the number 673.5, the decimal point is located between the 3 and the 5.

Step 2: Move the Decimal Point to Obtain a Coefficient Between 1 and 10

We need to move the decimal point to the left until we have a number between 1 and 10. In this case, we move the decimal point two places to the left:

6. 735

Now, 6.735 is our coefficient (a). It satisfies the condition 1 ≤ |a| < 10.

Step 3: Determine the Exponent (b)

The exponent (b) represents the number of places we moved the decimal point. Since we moved the decimal point two places to the left, the exponent is +2. If we had moved the decimal point to the right, the exponent would be negative.

Step 4: Write the Number in Scientific Notation

Now we can write 673.5 in scientific notation:

6. 735 × 10²

Therefore, 673.5 expressed in scientific notation is 6.735 × 10².

Examples of Converting Other Numbers to Scientific Notation

Let's reinforce the process with a few more examples:

• Example 1: Convert 12,500 to Scientific Notation

  • Move the decimal point four places to the left: 1.2500
  • Exponent: +4
  • Scientific Notation: 1.25 × 10⁴ (Trailing zeros after the decimal point are usually dropped.)

• Example 2: Convert 0.00048 to Scientific Notation

  • Move the decimal point four places to the right: 4.8
  • Exponent: -4
  • Scientific Notation: 4.8 × 10^-4

• Example 3: Convert 900,000,000 to Scientific Notation

  • Move the decimal point eight places to the left: 9.0
  • Exponent: +8
  • Scientific Notation: 9 × 10⁸

• Example 4: Convert 0.0000000256 to Scientific Notation

  • Move the decimal point eight places to the right: 2.56
  • Exponent: -8
  • Scientific Notation: 2.56 × 10^-8

Understanding Positive and Negative Exponents

The sign of the exponent is crucial and indicates the magnitude of the number:

• Positive Exponent: A positive exponent indicates a number greater than 1. It tells you how many places to move the decimal point to the right to convert from scientific notation to standard form. For example, 3.0 × 10⁵ = 300,000.
• Negative Exponent: A negative exponent indicates a number less than 1 (a decimal fraction). It tells you how many places to move the decimal point to the left to convert from scientific notation to standard form. For example, 2.0 × 10^-3 = 0.002.
• Zero Exponent: An exponent of zero indicates that the number is between 1 and 10. Any number (except zero) raised to the power of 0 equals 1 (x⁰ = 1). For example, 5.0 × 10⁰ = 5.

Performing Arithmetic Operations with Scientific Notation

Scientific notation is particularly useful when performing arithmetic operations on very large or small numbers. Here's a brief overview of how to handle these operations:

1. Multiplication:

• Multiply the coefficients.

• Add the exponents.

• Adjust the coefficient and exponent if necessary to ensure the coefficient is between 1 and 10.

  Example: (2 × 10³) × (3 × 10⁴) = (2 × 3) × 10⁽³⁺⁴⁾ = 6 × 10⁷

2. Division:

• Divide the coefficients.

• Subtract the exponents.

• Adjust the coefficient and exponent if necessary.

  Example: (8 × 10⁵) / (2 × 10²) = (8 / 2) × 10^(5-2) = 4 × 10³

3. Addition and Subtraction:

• The numbers must have the same exponent before you can add or subtract them. If they don't, adjust one of the numbers to match the exponent of the other.

• Add or subtract the coefficients.

• Keep the same exponent.

• Adjust the coefficient and exponent if necessary.

  Example: (3 × 10⁴) + (5 × 10³) = (3 × 10⁴) + (0.5 × 10⁴) = (3 + 0.5) × 10⁴ = 3.5 × 10⁴

Scientific Notation and Significant Figures

When using scientific notation, it's important to maintain the correct number of significant figures. Significant figures represent the digits in a number that are known with certainty plus one uncertain digit. Here's how significant figures relate to scientific notation:

• The coefficient (a) in scientific notation should have the same number of significant figures as the original number.
• The exponent (b) does not affect the number of significant figures.

For example, if you're converting 673.5 (which has four significant figures) to scientific notation, the coefficient should also have four significant figures: 6.735 × 10². If you rounded the coefficient to 6.74 × 10², you would be implying that the original number had only three significant figures.

Common Mistakes to Avoid

• Incorrect Coefficient: Ensure the coefficient is always between 1 and 10.
• Incorrect Exponent Sign: Pay close attention to whether the decimal point was moved left (positive exponent) or right (negative exponent).
• Forgetting the Base: Always include the base 10 (× 10^b).
• Incorrect Significant Figures: Maintain the correct number of significant figures in the coefficient.
• Not Adjusting After Operations: After multiplication, division, addition, or subtraction, double-check that the coefficient is still between 1 and 10 and adjust the exponent accordingly.

Tools and Calculators for Scientific Notation

While understanding the principles of scientific notation is essential, there are many tools and calculators available online that can assist with conversions and calculations:

• Online Scientific Notation Converters: These tools allow you to input a number and automatically convert it to scientific notation.
• Scientific Calculators: Most scientific calculators have a dedicated mode for scientific notation, making it easy to perform calculations with large or small numbers.
• Spreadsheet Software: Programs like Microsoft Excel and Google Sheets offer functions for displaying numbers in scientific notation and performing calculations.

These tools can be helpful for checking your work and performing complex calculations, but remember that understanding the underlying principles is crucial for applying scientific notation correctly.

Real-World Applications of Scientific Notation

Scientific notation is widely used in various fields:

• Astronomy: Expressing distances between stars and galaxies (e.g., the distance to the Andromeda Galaxy is approximately 2.5 × 10¹⁹ meters).
• Physics: Representing the mass of subatomic particles (e.g., the mass of an electron is approximately 9.11 × 10^-31 kilograms).
• Chemistry: Expressing the number of molecules in a mole (Avogadro's number: 6.022 × 10²³).
• Computer Science: Representing storage capacities (e.g., a terabyte is 1 × 10¹² bytes).
• Geology: Describing the age of the Earth (approximately 4.543 × 10⁹ years).
• Engineering: Calculating extremely small tolerances or extremely large loads.

Conclusion

Mastering scientific notation is an essential skill for anyone working with numbers in science, engineering, or mathematics. It provides a convenient and standardized way to represent very large and very small numbers, simplifies calculations, and facilitates communication. By understanding the principles outlined in this guide and practicing with examples, you can confidently convert numbers to and from scientific notation and apply it effectively in your field of study or work. Remember to pay attention to the coefficient, exponent, significant figures, and the rules for arithmetic operations to ensure accuracy. Applying these principles to the initial number 673.5, we confidently arrive at its scientific notation representation: 6.735 × 10².