Converting Scientific Notation To Decimal Notation Rewrite 3.647 X 10^-8

by ADMIN 73 views

In the realm of mathematics, expressing numbers in different forms is a fundamental skill. Decimal notation and scientific notation are two common ways to represent numerical values. While decimal notation presents numbers in their standard form, scientific notation provides a compact way to express very large or very small numbers. This article delves into the process of converting a number from scientific notation to decimal notation, specifically focusing on the number 3.647 x 10^-8.

What is Decimal Notation?

Decimal notation, often referred to as the standard form, is the way we commonly write numbers. It uses a base-10 system, where each digit's position represents a power of 10. For instance, the number 123.45 can be broken down as follows:

  • 1 x 10^2 (hundreds place)
  • 2 x 10^1 (tens place)
  • 3 x 10^0 (ones place)
  • 4 x 10^-1 (tenths place)
  • 5 x 10^-2 (hundredths place)

In decimal notation, the decimal point separates the whole number part from the fractional part. The digits to the left of the decimal point represent positive powers of 10, while the digits to the right represent negative powers of 10.

Demystifying Scientific Notation

Scientific notation is a convenient method for expressing extremely large or small numbers. It follows the format:

a x 10^b

where:

  • a is a number between 1 and 10 (the coefficient or significand)
  • 10 is the base
  • b is an integer (the exponent)

For example, the number 3,000,000,000 can be written in scientific notation as 3 x 10^9, and the number 0.000000005 can be expressed as 5 x 10^-9.

Understanding the Exponent: The exponent indicates how many places the decimal point needs to be moved to convert the number back to decimal notation. A positive exponent signifies a large number, and the decimal point is moved to the right. A negative exponent indicates a small number, and the decimal point is moved to the left.

Converting from Scientific Notation to Decimal Notation: A Step-by-Step Guide

Now, let's focus on the primary task: converting the number 3.647 x 10^-8 from scientific notation to decimal notation. Here's a step-by-step approach:

1. Identify the Components: In the number 3.647 x 10^-8, we have:

  • Coefficient (a): 3.647
  • Base: 10
  • Exponent (b): -8

2. Interpret the Exponent: The exponent is -8, which is negative. This tells us that the number is less than 1, and we need to move the decimal point 8 places to the left.

3. Move the Decimal Point: Start with the coefficient, 3.647. Since the exponent is -8, we move the decimal point 8 places to the left. To do this, we may need to add leading zeros as placeholders:

  • Original: 3. 647
  • Move 1 place: 0. 3647
  • Move 2 places: 0. 03647
  • Move 3 places: 0. 003647
  • Move 4 places: 0. 0003647
  • Move 5 places: 0. 00003647
  • Move 6 places: 0. 000003647
  • Move 7 places: 0. 0000003647
  • Move 8 places: 0. 00000003647

4. Write the Result in Decimal Notation: After moving the decimal point 8 places to the left, we get the number 0.00000003647. This is the decimal notation of 3.647 x 10^-8.

Therefore, 3.647 x 10^-8 in decimal notation is 0.00000003647.

Practical Examples and Applications

Understanding how to convert between scientific and decimal notation is not just an academic exercise; it has practical applications in various fields, including:

  • Science: Expressing measurements in chemistry, physics, and astronomy, such as the size of atoms or the distance between stars.
  • Engineering: Representing very small tolerances or very large forces.
  • Computer Science: Dealing with memory sizes or processing speeds.
  • Finance: Handling large sums of money or very small interest rates.

For example, the wavelength of visible light is often expressed in nanometers (1 nm = 1 x 10^-9 meters). So, a wavelength of 500 nm can be written as 500 x 10^-9 meters or 5 x 10^-7 meters in scientific notation, which is 0.0000005 meters in decimal notation.

Another example is Avogadro's number, which represents the number of atoms or molecules in a mole (approximately 6.022 x 10^23). This massive number is much more manageable in scientific notation than writing it out in decimal notation.

Common Pitfalls and How to Avoid Them

While the conversion process is straightforward, there are some common pitfalls to watch out for:

  • Incorrect Direction: Moving the decimal point in the wrong direction. Remember, a negative exponent means moving the decimal point to the left (making the number smaller), and a positive exponent means moving it to the right (making the number larger).

  • Miscounting Places: Not moving the decimal point the correct number of places. Double-check the exponent and ensure you've moved the decimal point accordingly.

  • Forgetting Leading Zeros: Not adding enough leading zeros when dealing with small numbers (negative exponents). Make sure to include enough zeros to accurately represent the magnitude of the number.

To avoid these pitfalls, always double-check your work and consider using a calculator or online converter for verification, especially when dealing with complex numbers.

Practice Problems

To solidify your understanding, try converting the following numbers from scientific notation to decimal notation:

    1. 5 x 10^-5
    1. 98 x 10^6
    1. 123 x 10^-3

Solutions:

  1. 5 x 10^-5 = 0.000025
  2. 98 x 10^6 = 3,980,000
  3. 123 x 10^-3 = 0.004123

Conclusion

Converting numbers from scientific notation to decimal notation is a crucial skill in mathematics and various scientific and technical fields. By understanding the underlying principles of exponents and decimal places, you can confidently rewrite numbers in the desired format. In this article, we have demonstrated a step-by-step approach to converting 3.647 x 10^-8 into decimal notation, which is 0.00000003647. By following the guidelines and avoiding common pitfalls, you can master this skill and apply it effectively in your studies and professional endeavors. Remember to practice regularly and use real-world examples to reinforce your understanding. With consistent effort, you'll become proficient in handling both scientific and decimal notation, enhancing your mathematical fluency and problem-solving abilities. The ability to switch between these notations not only simplifies calculations but also aids in comprehending the magnitude of numbers, whether they represent the vastness of the cosmos or the minuscule world of atoms. This versatility is indispensable for anyone venturing into scientific research, engineering design, or advanced mathematical studies. Embrace the challenge, and you'll find that the world of numbers becomes much more accessible and intriguing.