Convert $7.5 \times 10^{-2}$ To Ordinary Number

by Alex Johnson 50 views

Converting numbers from scientific notation to ordinary numbers can seem tricky at first, but once you understand the process, it becomes quite straightforward. Scientific notation is a way of expressing numbers that are either very large or very small in a compact form. It's written as a×10ba \times 10^b, where aa is a number between 1 and 10 (but not including 10), and bb is an integer. In our case, we have 7.5×10−27.5 \times 10^{-2}, and we want to express this as an ordinary number.

Understanding Scientific Notation

Before we dive into the conversion, let's ensure we grasp the concept of scientific notation. The number 7.5×10−27.5 \times 10^{-2} tells us that we're dealing with the number 7.5, which is then multiplied by 10 raised to the power of -2. The exponent -2 is crucial because it dictates how we move the decimal point to convert it to an ordinary number. A negative exponent means we're dividing by a power of 10, while a positive exponent would mean we're multiplying by a power of 10.

When you encounter scientific notation, always pay close attention to the exponent. It determines the magnitude of the number. For instance, 10310^3 means 1000, so if you had 2.5×1032.5 \times 10^3, it would be 2.5 multiplied by 1000, resulting in 2500. Conversely, 10−310^{-3} means 1/1000 or 0.001. Therefore, 2.5×10−32.5 \times 10^{-3} would be 2.5 multiplied by 0.001, giving you 0.0025. Understanding this relationship is key to correctly converting between scientific and ordinary notation.

Moreover, consider the significance of the number before the power of 10. In our example, it's 7.5. This number provides the significant digits of the original number. When converting, you're essentially adjusting the decimal place of this number based on the exponent. Keeping this in mind helps maintain the accuracy of your conversion.

Step-by-Step Conversion

Now, let's convert 7.5×10−27.5 \times 10^{-2} into an ordinary number. The exponent is -2, which means we need to move the decimal point two places to the left. Here's how we do it:

  1. Start with the number 7.5.
  2. Move the decimal point one place to the left: 0.75
  3. Since we need to move it two places, add a zero before the 7: 0.075

So, 7.5×10−27.5 \times 10^{-2} as an ordinary number is 0.075. It’s that simple!

When you're moving the decimal point, it's crucial to keep track of how many places you've moved it. If the exponent is a larger negative number, you might need to add several zeros to the left of the number. For example, if you were converting 3.2×10−53.2 \times 10^{-5}, you would need to move the decimal point five places to the left, resulting in 0.000032.

On the other hand, if the exponent is positive, you move the decimal point to the right. If you run out of digits, you add zeros to the right of the number. For instance, converting 1.8×1041.8 \times 10^4 would require moving the decimal point four places to the right, resulting in 18000.

Always double-check your work, especially when dealing with larger exponents. A small mistake can lead to a significant difference in the final result. Practice makes perfect, so the more you convert numbers between scientific and ordinary notation, the more comfortable and accurate you'll become.

Examples and Practice

Let's do a few more examples to solidify our understanding. Consider the number 2.3×10−32.3 \times 10^{-3}. To convert this to an ordinary number, we move the decimal point three places to the left:

  1. Start with 2.3
  2. Move one place: 0.23
  3. Move two places: 0.023
  4. Move three places: 0.0023

Thus, 2.3×10−32.3 \times 10^{-3} is 0.0023 as an ordinary number.

Now, let's convert 9.1×1029.1 \times 10^2. Since the exponent is positive, we move the decimal point two places to the right:

  1. Start with 9.1
  2. Move one place: 91
  3. Move two places: 910

Therefore, 9.1×1029.1 \times 10^2 is 910 as an ordinary number.

To further enhance your skills, try converting the following numbers on your own:

  • 5.6×10−45.6 \times 10^{-4}
  • 1.2×1051.2 \times 10^5
  • 8.7×10−18.7 \times 10^{-1}

Check your answers by converting them back to scientific notation. If you arrive at the original number, you've done it correctly!

Common Mistakes to Avoid

When converting between scientific and ordinary notation, several common mistakes can occur. Being aware of these pitfalls can help you avoid them.

One common mistake is moving the decimal point in the wrong direction. Always remember that a negative exponent means you move the decimal point to the left (making the number smaller), while a positive exponent means you move it to the right (making the number larger).

Another frequent error is miscounting the number of places to move the decimal point. Double-check the exponent and ensure you move the decimal point the correct number of places. It's easy to lose track, especially with larger exponents.

Forgetting to add zeros is also a common mistake. If you run out of digits when moving the decimal point, you need to add zeros as placeholders. Make sure you add enough zeros to move the decimal point the required number of places.

Finally, some people struggle with understanding the basic concept of scientific notation. Remember that scientific notation is simply a way to express very large or very small numbers in a more manageable form. If you're unsure about the concept, review the definition and examples before attempting to convert numbers.

By being mindful of these common mistakes, you can improve your accuracy and confidence in converting between scientific and ordinary notation.

Real-World Applications

Understanding how to convert between scientific and ordinary notation isn't just an academic exercise; it has numerous real-world applications. Scientific notation is widely used in various fields, including science, engineering, and finance.

In science, particularly in fields like astronomy and chemistry, scientists often deal with extremely large and small numbers. For example, the distance to a star might be expressed in light-years, which is a very large number. Similarly, the size of an atom is incredibly small. Scientific notation allows scientists to express these numbers in a compact and manageable way.

Engineers also use scientific notation extensively. When designing structures or circuits, engineers often work with very large or small values. Using scientific notation helps them avoid errors and simplify calculations.

In finance, scientific notation can be used to represent large sums of money or very small interest rates. It provides a convenient way to express these values without having to write out many zeros.

Furthermore, many calculators and computer programs use scientific notation to display numbers that are too large or too small to fit on the screen. Understanding scientific notation allows you to interpret these results correctly.

By mastering the conversion between scientific and ordinary notation, you'll be better equipped to understand and work with numbers in a variety of real-world contexts.

In conclusion, converting 7.5×10−27.5 \times 10^{-2} to an ordinary number involves understanding the principles of scientific notation and applying a simple step-by-step process. By moving the decimal point two places to the left, we find that 7.5×10−27.5 \times 10^{-2} is equal to 0.075. Remember to practice and be mindful of common mistakes to improve your accuracy. Understanding scientific notation and its conversion to ordinary numbers is a valuable skill in various fields, making it a worthwhile endeavor.

For further reading and more detailed explanations, you can visit Khan Academy's scientific notation section.