Expressing 2,100,000,000 In Scientific Notation Pedro's Computer Operations

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In the realm of computer science and mathematics, dealing with extremely large numbers is commonplace. Scientific notation provides a concise and efficient way to represent such numbers. This article delves into the concept of scientific notation and applies it to a specific scenario involving the computational power of a computer microprocessor.

Understanding Scientific Notation

Scientific notation, also known as standard form, is a method of expressing numbers as a product of two factors: a coefficient and a power of 10. The coefficient is a number typically between 1 and 10, while the power of 10 indicates the number's magnitude. The general form of scientific notation is:

Coefficient × 10^Exponent

For instance, the number 3,000 can be expressed in scientific notation as 3 × 10^3, where 3 is the coefficient and 3 is the exponent. Similarly, the number 0.0025 can be written as 2.5 × 10^-3. Scientific notation offers several advantages, including:

  • Conciseness: It simplifies the representation of very large or very small numbers.
  • Ease of Comparison: It facilitates the comparison of numbers with vastly different magnitudes.
  • Computational Efficiency: It streamlines calculations involving large or small numbers.

Pedro's Computer Microprocessor: A Case Study

Consider Pedro's computer, whose microprocessor can perform 2,100,000,000 operations per second. This number is quite large and can be cumbersome to work with in its standard form. To express it in scientific notation, we need to identify the coefficient and the exponent. To convert 2,100,000,000 to scientific notation, move the decimal point to the left until there is only one non-zero digit to the left of the decimal point. Count the number of places you moved the decimal point. This number will be the exponent of 10. So, to express 2,100,000,000 in scientific notation, we follow these steps:

  1. Identify the Coefficient: Move the decimal point nine places to the left, resulting in the number 2.1. This number, 2.1, falls between 1 and 10, fulfilling the requirement for the coefficient in scientific notation. It represents the significant digits of the original number, providing the precision of the value without being encumbered by the scale.
  2. Determine the Exponent: We moved the decimal point nine places to the left. This movement signifies a scaling down of the original number by a factor of 10 nine times. Consequently, the exponent is 9, indicating the power of 10 by which the coefficient must be multiplied to obtain the original number. The exponent essentially captures the magnitude or scale of the number. A positive exponent indicates a number greater than 1, while a negative exponent indicates a number between 0 and 1.

Therefore, the number 2,100,000,000 expressed in scientific notation is 2.1 × 10^9. This representation is much more compact and easier to comprehend than the original number. In the context of Pedro's computer, expressing the microprocessor's operations per second in scientific notation provides a clear understanding of its immense computational capabilities without the visual clutter of numerous zeros. The scientific notation not only simplifies the number but also allows for an easier comparison with other computational speeds, highlighting the efficiency and power of the microprocessor. This method is especially useful in fields like computer science and engineering, where dealing with very large or very small numbers is common, and the clarity offered by scientific notation can significantly aid in both communication and calculation.

Analyzing the Options

Now, let's examine the given options and determine which one correctly expresses 2,100,000,000 in scientific notation:

a. 2.1 × 10^8 b. 2.1 × 10^10 c. 2.1 × 10^9 d. 21 × 10^10

  • Option a (2.1 × 10^8): This option represents 210,000,000, which is significantly smaller than 2,100,000,000. The exponent 8 indicates that the decimal point should be moved eight places to the right, resulting in a number that is one-tenth of the actual value. This misrepresentation underscores the importance of correctly identifying the magnitude of the number when converting to scientific notation.
  • Option b (2.1 × 10^10): This option represents 21,000,000,000, which is ten times larger than 2,100,000,000. The exponent 10 suggests a scale that is an order of magnitude greater than the actual number. Such a difference highlights the critical role the exponent plays in accurately conveying the size of a number in scientific notation.
  • Option c (2.1 × 10^9): As we determined earlier, this is the correct representation of 2,100,000,000 in scientific notation. It accurately captures the scale of the number, with the coefficient 2.1 and the exponent 9 providing a concise and precise expression of the microprocessor's computational speed. This option aligns with the methodology of scientific notation, where the coefficient is between 1 and 10, and the exponent correctly positions the decimal point to reflect the number's actual size.
  • Option d (21 × 10^10): While this option does represent the correct magnitude (21,000,000,000), it violates the convention of scientific notation that the coefficient should be between 1 and 10. The coefficient 21 is outside this range, making this representation technically incorrect, even though it conveys the same numerical value. This example illustrates the importance of adhering to the standard form in scientific notation to maintain clarity and consistency in mathematical and scientific communication.

Therefore, the correct answer is c. 2.1 × 10^9.

Conclusion

In conclusion, scientific notation is a valuable tool for representing large and small numbers concisely and efficiently. By expressing 2,100,000,000 in scientific notation as 2.1 × 10^9, we gain a clearer understanding of the computational power of Pedro's computer microprocessor. This representation not only simplifies the number but also allows for easier comparisons and calculations. Mastering scientific notation is essential for anyone working with numerical data, particularly in fields like science, technology, engineering, and mathematics, where precision and clarity are paramount.

Practice Questions

To solidify your understanding of scientific notation, try converting the following numbers into scientific notation:

  1. 5,280
  2. 0.000067
  3. 1,000,000,000,000
  4. 0.000000001

These exercises will help reinforce the principles of scientific notation and enhance your ability to work with large and small numbers effectively.