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Chapter 1: Problem 18

Convert to scientific notation. 3,090,000,000,000

Short Answer

Expert verified
3.09 脳 10^{12}

Step by step solution

01

- Identify the significant figures

Locate the significant figures in the number. For 3,090,000,000,000, the significant figures are 3.09.
02

- Count the decimal places to shift

Count how many places the decimal point must move to the left to position it after the first non-zero digit. Here, the decimal point moves 12 places to the left.
03

- Write in scientific notation

Express the number as a product of the significant figures and a power of 10. This can be written as 3.09 脳 10^{12}.

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Key Concepts

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

Significant Figures
Significant figures are the digits in a number that carry meaningful information about its precision. When converting large numbers to scientific notation, you first identify the significant figures. For the number 3,090,000,000,000, the significant figures are 3.09. These figures are derived by focusing on digits that are not zero, along with zeros that are between significant figures or after a decimal point and to the right of a non-zero digit.

To illustrate this, in the number 3.09:
  • '3' is a non-zero digit and significant.
  • '0' is between two significant figures and hence significant.
  • '9' is a non-zero digit and significant.
Remember, trailing zeros in a large number are not considered significant unless they are after a decimal point.
Decimal Places
Decimal places refer to the number of digits to the right of the decimal point in a number and play a crucial role in placing numbers in scientific notation. When converting 3,090,000,000,000, we count how many decimal places the point needs to shift to position after the first non-zero digit. In this case, the decimal starts at '3' and must move 12 places to the left.

Thus, we reformulate the number to place the decimal after 3.09, obtaining 3.09 脳 10^{12}. By doing so, we ensure that the number is easier to read and use in scientific calculations. This step emphasizes that understanding how far to move the decimal is key to the correct transformation of the number.
Powers of 10
The notation for large numbers often involves powers of 10, which helps to simplify and convey the size of these numbers efficiently. In scientific notation, the power of 10 represents how many places the decimal has been shifted. For the number 3,090,000,000,000, moving the decimal point 12 places to the left translates to a power of 10, hence 10^{12}.

Here's how to understand it:
  • The original number is 3,090,000,000,000.
  • After shifting the decimal point 12 places left, it becomes 3.09.
  • This change is represented mathematically as multiplying by 10^{12}.
Therefore, 3.09 脳 10^{12} efficiently expresses the large number using scientific notation, making it easier for calculations and comprehension.

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Most popular questions from this chapter

Use roster notation to write each set. The set of all natural numbers that are multiples of 5

Write an equivalent expression using an associative law. $$ (3 x) y $$

Which numbers in the list provided are (a) whole numbers? (b) integers? (c) rational numbers? (d) irrational numbers? (e) real numbers?. $$ -17,-0.01,0, \frac{5}{4}, 8, \sqrt{77} $$

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Translate to an algebraic expression. \(\\{x | x=3 n, n \text { is a natural number }\\}\)
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