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Chapter 1: Problem 15

Change the numbers from ordinary notation to scientific notation. $$609,000,000$$

Short Answer

Expert verified
The scientific notation is \( 6.09 \times 10^8 \).

Step by step solution

01

Identify the Significant Figures

In scientific notation, we need to express the number in the form of \( a \times 10^n \), where \( 1 \leq a < 10 \). Start by identifying the significant digits of the number. Here, the significant digits in \( 609,000,000 \) are \( 6.09 \).
02

Count the Number of Digits After the First Digit

Count how many places you need to move the decimal point to place it after the first significant figure 6. In \( 609,000,000 \), you move the decimal point 8 places to the left, from \( 609,000,000 \) to \( 6.09000,000 \).
03

Express the Number in Scientific Notation

Since the decimal point was moved 8 places to the left, the exponent of 10 will be 8. Hence, the scientific notation of \( 609,000,000 \) is \( 6.09 \times 10^8 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Significant Figures
Understanding significant figures is crucial when working with scientific notation. Significant figures are the digits in a number that are reliable and provide meaningful information about the precision of that number. For instance, in the number 609,000,000, the significant figures are the digits 6, 0, and 9, which give us a way to express the number concisely.
  • These figures provide the foundation for representing a number in scientific notation, which is a more compact form.
  • It's important to note that only the first set of digits before trailing zeros in a number are considered significant when converting to scientific notation.

Through selecting the significant figures first, you ensure that the number maintains its accuracy but avoids unnecessary details.
Ordinary Notation
Ordinary notation, also known as standard or decimal notation, is the way we typically write numbers in everyday life. For example, using ordinary notation, we write numbers like 609,000,000 with all its digits laid out. While this form is straightforward for smaller numbers, it becomes cumbersome for very large or very small values.
  • Ordinary notation doesn鈥檛 efficiently convey the scale or order of magnitude of the number.
  • As numbers grow larger, counting the zeros to understand magnitude can become time-consuming and error-prone.

Converting this into scientific notation simplifies the number, making it easier to communicate and understand its real-world meaning.
Exponent of 10
In scientific notation, the exponent of 10 is crucial as it indicates how many places the decimal point has moved, thus illustrating the scale of the number. In the number 609,000,000, we move the decimal point 8 places to the left to transform it to 6.09. The movement of the decimal point determines the exponent.
  • If the decimal point is moved to the left, which usually happens with large numbers, the exponent is positive, as in: \( 6.09 \times 10^8 \).
  • Conversely, if the decimal moves right for smaller numbers, the exponent would be negative.

This exponent makes scientific notation highly practical and powerful, providing a clear indication of the number鈥檚 magnitude without listing all the digits.

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