Adding Numbers In Scientific Notation Calculating (7.2 × 10⁻⁹) + (6.5 × 10⁻¹⁰)

Find the sum of (7.2 × 10⁻⁹) and (6.5 × 10⁻¹⁰). Write the final answer in scientific notation.

In the realm of mathematics and scientific calculations, scientific notation serves as a powerful tool for expressing extremely large or small numbers in a concise and manageable format. This article delves into the process of adding two numbers expressed in scientific notation, specifically (7.2 × 10⁻⁹) and (6.5 × 10⁻¹⁰). We will explore the underlying principles, step-by-step procedures, and practical considerations involved in arriving at the final answer, ensuring clarity and accuracy in our calculations.

Understanding Scientific Notation

Before embarking on the addition process, it is crucial to grasp the essence of scientific notation. Scientific notation represents a number as the product of two components: a coefficient and a power of 10. The coefficient, typically a decimal number between 1 and 10, embodies the significant digits of the number. The power of 10, expressed as 10 raised to an integer exponent, dictates the magnitude or scale of the number.

For instance, the number 3,000,000 can be expressed in scientific notation as 3 × 10⁶. Here, 3 is the coefficient, and 10⁶ signifies 10 raised to the power of 6, which equals 1,000,000. Similarly, the number 0.0000025 can be written as 2.5 × 10⁻⁶, where 2.5 is the coefficient, and 10⁻⁶ represents 10 raised to the power of -6.

Scientific notation proves particularly advantageous when dealing with numbers that span several orders of magnitude. It streamlines calculations, enhances readability, and minimizes the risk of errors associated with handling long strings of digits. By expressing numbers in scientific notation, we can readily compare their magnitudes and perform arithmetic operations with greater ease.

Step-by-Step Addition of (7.2 × 10⁻⁹) and (6.5 × 10⁻¹⁰)

Now, let's embark on the process of adding the two numbers in question: (7.2 × 10⁻⁹) and (6.5 × 10⁻¹⁰). The following steps outline the procedure:

1. Align the Exponents

The cornerstone of adding numbers in scientific notation lies in ensuring that the powers of 10 are identical. To achieve this, we must manipulate one of the numbers to match the exponent of the other. In our case, we can either convert (7.2 × 10⁻⁹) to have an exponent of -10 or transform (6.5 × 10⁻¹⁰) to have an exponent of -9. For convenience, let's opt for the latter approach.

To convert (6.5 × 10⁻¹⁰) to have an exponent of -9, we must effectively multiply the coefficient by 10 and reduce the exponent by 1. This yields:

6. 5 × 10⁻¹⁰ = 0.65 × 10⁻⁹

2. Add the Coefficients

With the exponents aligned, we can now proceed to add the coefficients. This step involves simply adding the decimal numbers while preserving the common power of 10:

7. 2 × 10⁻⁹ + 0.65 × 10⁻⁹ = (7.2 + 0.65) × 10⁻⁹

Performing the addition, we get:

7. 2 + 0.65 = 7.85

Therefore, the sum of the coefficients is 7.85.

3. Express the Result in Scientific Notation

The final step entails expressing the result in standard scientific notation. This necessitates ensuring that the coefficient falls within the range of 1 to 10. In our case, the coefficient 7.85 already meets this criterion. Consequently, the sum of (7.2 × 10⁻⁹) and (6.5 × 10⁻¹⁰) in scientific notation is:

7. 85 × 10⁻⁹

Alternative Approach: Converting Both Numbers

As an alternative approach, we could have converted both numbers to have the same exponent before adding. Let's illustrate this by converting both numbers to have an exponent of -10:

• 7.2 × 10⁻⁹ = 72 × 10⁻¹⁰
• 6.5 × 10⁻¹⁰ remains the same.

Now, we add the coefficients:

72 × 10⁻¹⁰ + 6.5 × 10⁻¹⁰ = (72 + 6.5) × 10⁻¹⁰ = 78.5 × 10⁻¹⁰

To express this in standard scientific notation, we adjust the coefficient:

8. 5 × 10⁻¹⁰ = 7.85 × 10⁻⁹

As we can see, this approach yields the same final result as the previous method.

Practical Considerations and Potential Pitfalls

While the process of adding numbers in scientific notation is relatively straightforward, it's essential to be mindful of certain practical considerations and potential pitfalls:

• Exponent Alignment: The cardinal rule of adding numbers in scientific notation is to ensure that the exponents are identical. Neglecting this step will lead to erroneous results.
• Coefficient Range: The coefficient in scientific notation must fall within the range of 1 to 10. If the sum of the coefficients exceeds 10, you must adjust the coefficient and exponent accordingly.
• Significant Digits: When dealing with experimental data or measurements, it's crucial to maintain the correct number of significant digits throughout the calculation. The final answer should be rounded to the same number of significant digits as the least precise number in the calculation.
• Calculator Usage: Modern calculators often have built-in functions for handling scientific notation. Familiarize yourself with these functions to streamline calculations and minimize the risk of errors.

Real-World Applications of Scientific Notation

Scientific notation finds extensive application in various scientific and engineering disciplines. Its ability to represent extremely large and small numbers with ease makes it indispensable in fields such as:

• Astronomy: Astronomers use scientific notation to express distances between celestial objects, the masses of stars and planets, and the ages of galaxies.
• Chemistry: Chemists employ scientific notation to represent the number of atoms or molecules in a sample, the concentrations of solutions, and the rates of chemical reactions.
• Physics: Physicists utilize scientific notation to express fundamental constants, such as the speed of light and the gravitational constant, as well as the sizes of atoms and subatomic particles.
• Computer Science: Computer scientists leverage scientific notation to represent memory sizes, processing speeds, and data storage capacities.

By mastering the art of working with scientific notation, you equip yourself with a valuable tool for tackling complex numerical problems in diverse fields.

Conclusion

In conclusion, adding numbers expressed in scientific notation involves aligning the exponents, adding the coefficients, and expressing the result in standard scientific notation. By adhering to these steps and keeping in mind the practical considerations discussed, you can confidently perform these calculations with accuracy and efficiency. Scientific notation stands as a fundamental tool in the scientific community, enabling us to express and manipulate numbers across a vast range of magnitudes. By mastering its principles, we unlock a deeper understanding of the numerical world around us.