Comparing Scientific Notation How Much Smaller Is 1.2 X 10^6 Than 1.47 X 10^7

by ADMIN 78 views

Understanding the magnitude of numbers expressed in scientific notation is a fundamental skill in mathematics and various scientific disciplines. This article aims to provide a comprehensive explanation of how to determine how many times smaller one number is compared to another, specifically focusing on the example of comparing 1.2 x 10^6 and 1.47 x 10^7. We will delve into the underlying principles of scientific notation, the process of division involving these numbers, and the practical implications of such calculations. Whether you are a student grappling with scientific notation for the first time or a professional seeking a refresher, this guide will equip you with the necessary knowledge and skills.

Understanding Scientific Notation

Before we dive into the calculation, it is crucial to understand what scientific notation represents. Scientific notation is a way of expressing numbers that are either very large or very small in a compact and standardized form. It consists of two parts: a coefficient (a number between 1 and 10) and a power of 10. For example, the number 1,200,000 can be written in scientific notation as 1.2 x 10^6, where 1.2 is the coefficient and 10^6 is the power of 10. Similarly, 0.00000147 can be expressed as 1.47 x 10^-6.

The power of 10 indicates how many places the decimal point needs to be moved to obtain the original number. A positive exponent means the decimal point is moved to the right, while a negative exponent means it is moved to the left. In the case of 1.2 x 10^6, the exponent 6 signifies that the decimal point in 1.2 should be moved six places to the right, resulting in 1,200,000. Conversely, in 1.47 x 10^-6, the exponent -6 indicates that the decimal point in 1.47 should be moved six places to the left, resulting in 0.00000147. Mastering scientific notation is essential for simplifying calculations and making comparisons between numbers of vastly different magnitudes.

Scientific notation not only simplifies the representation of very large and very small numbers but also streamlines mathematical operations involving them. When multiplying numbers in scientific notation, you multiply the coefficients and add the exponents of 10. For instance, (2 x 10^3) * (3 x 10^4) = (2 * 3) x 10^(3+4) = 6 x 10^7. Similarly, when dividing numbers in scientific notation, you divide the coefficients and subtract the exponents of 10. For example, (8 x 10^5) / (2 x 10^2) = (8 / 2) x 10^(5-2) = 4 x 10^3. These rules make calculations involving scientific notation much more manageable than working with the numbers in their standard decimal form. Understanding these principles is crucial for effectively using scientific notation in various mathematical and scientific contexts.

Comparing 1.2 x 10^6 and 1.47 x 10^7

Now, let's address the core question: how many times smaller is 1.2 x 10^6 than 1.47 x 10^7? To determine this, we need to divide the larger number by the smaller number. This will give us the factor by which the larger number is greater than the smaller number. In other words, it will tell us how many times the smaller number fits into the larger number. The mathematical operation we need to perform is (1.47 x 10^7) / (1.2 x 10^6).

To perform this division, we can separate the coefficients and the powers of 10. This gives us (1.47 / 1.2) x (10^7 / 10^6). First, let's divide the coefficients: 1.47 / 1.2 = 1.225. Next, let's divide the powers of 10. According to the rules of exponents, when dividing powers with the same base, we subtract the exponents: 10^7 / 10^6 = 10^(7-6) = 10^1 = 10. Therefore, the result of the division is 1.225 x 10. This means that 1.47 x 10^7 is 1.225 x 10 times larger than 1.2 x 10^6.

Converting 1.225 x 10 to its standard decimal form, we get 12.25. This implies that 1.2 x 10^6 is 12.25 times smaller than 1.47 x 10^7. This result provides a clear understanding of the relative magnitudes of the two numbers. By dividing the larger number by the smaller number, we have successfully quantified the difference between them. This method can be applied to compare any two numbers expressed in scientific notation, making it a valuable tool in various scientific and mathematical applications. The ability to accurately compare numbers in scientific notation is essential for interpreting data, making predictions, and solving problems in a wide range of fields.

Step-by-Step Calculation

To further illustrate the process, let's break down the calculation into a step-by-step guide:

  1. Identify the numbers: We have 1.2 x 10^6 and 1.47 x 10^7.
  2. Determine which number is larger: 1.47 x 10^7 is larger than 1.2 x 10^6 because it has a higher power of 10.
  3. Divide the larger number by the smaller number: (1.47 x 10^7) / (1.2 x 10^6)
  4. Separate the coefficients and the powers of 10: (1.47 / 1.2) x (10^7 / 10^6)
  5. Divide the coefficients: 1.47 / 1.2 = 1.225
  6. Divide the powers of 10: 10^7 / 10^6 = 10^(7-6) = 10^1 = 10
  7. Combine the results: 1.225 x 10
  8. Convert to standard decimal form (if needed): 1.225 x 10 = 12.25

This step-by-step approach provides a clear and organized method for comparing numbers in scientific notation. By following these steps, you can confidently determine how many times smaller one number is compared to another. The key is to break down the problem into smaller, more manageable steps, making the calculation less daunting and more accurate. This method is particularly useful when dealing with complex scientific or mathematical problems where precision and clarity are paramount. Practice applying this method to various examples to solidify your understanding and enhance your problem-solving skills.

Practical Applications

Understanding how to compare numbers in scientific notation has numerous practical applications across various fields. In science, it is essential for comparing the sizes of cells, the distances between stars, or the magnitudes of earthquakes. For example, if one cell has a diameter of 1.5 x 10^-5 meters and another has a diameter of 3.0 x 10^-6 meters, we can determine that the first cell is five times larger than the second cell by dividing 1.5 x 10^-5 by 3.0 x 10^-6. This type of comparison is crucial for biological research and understanding the relative scales of microscopic entities.

In astronomy, the vast distances between celestial objects are often expressed in scientific notation. Comparing these distances helps astronomers understand the structure of the universe and the relationships between different stars and galaxies. For instance, if the distance to one star is 4.5 x 10^16 meters and the distance to another star is 1.5 x 10^17 meters, we can calculate that the second star is approximately 3.33 times farther away than the first star. These calculations are fundamental for mapping the cosmos and understanding the scale of astronomical phenomena.

In engineering, scientific notation is used to represent and compare very large and very small quantities, such as the electrical resistance of a material or the speed of data transfer in a computer network. These comparisons are vital for designing efficient systems and ensuring the proper functioning of devices. In finance, large sums of money or market capitalization are often expressed in scientific notation, and the ability to compare these values is essential for making informed investment decisions. Whether it's comparing the sizes of nanoparticles, the energy outputs of different power plants, or the financial assets of multinational corporations, the ability to work with and compare numbers in scientific notation is a valuable skill in a wide array of disciplines.

Common Mistakes and How to Avoid Them

When working with scientific notation, several common mistakes can lead to incorrect results. One frequent error is misinterpreting the exponent. It's crucial to remember that the exponent indicates the number of places the decimal point should be moved, and a negative exponent signifies movement to the left, while a positive exponent signifies movement to the right. Confusing the direction of decimal point movement can result in drastically different values. For example, mistaking 10^-3 for 10^3 would lead to an error of a factor of one million.

Another common mistake occurs during division or multiplication when dealing with the exponents. Remember that when multiplying numbers in scientific notation, you add the exponents, and when dividing, you subtract them. Forgetting to apply these rules correctly can lead to significant errors in your calculations. For instance, incorrectly adding exponents during division or subtracting them during multiplication will produce a wrong result. Double-checking the exponent arithmetic is essential to avoid these mistakes.

Furthermore, errors can arise when performing the division or multiplication of the coefficients. It's important to perform these operations accurately, paying close attention to decimal placement. Using a calculator can help reduce the chances of making arithmetic errors. Additionally, ensuring that the final answer is expressed in proper scientific notation, with the coefficient between 1 and 10, is crucial for maintaining consistency and clarity. By being mindful of these common pitfalls and practicing careful calculation techniques, you can significantly improve your accuracy when working with scientific notation.

Conclusion

In conclusion, determining how many times smaller 1.2 x 10^6 is than 1.47 x 10^7 involves dividing the larger number by the smaller number. This calculation reveals that 1.2 x 10^6 is 12.25 times smaller than 1.47 x 10^7. This process highlights the importance of understanding scientific notation and the rules for performing mathematical operations with numbers expressed in this form. The ability to compare numbers in scientific notation is a valuable skill with practical applications in various fields, including science, astronomy, engineering, and finance. By mastering this skill, you can confidently tackle problems involving very large and very small numbers and gain a deeper understanding of the relative magnitudes of different quantities. Practice and attention to detail are key to avoiding common mistakes and ensuring accurate results in your calculations. Whether you are a student, a researcher, or a professional, a solid grasp of scientific notation will undoubtedly enhance your analytical and problem-solving abilities.