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Chapter 9: Problem 5

Write each number in scientific notation. $$11,000$$

Short Answer

Expert verified
\(11,000\) in scientific notation is \(1.1 \times 10^4\).

Step by step solution

01

Identify the Significant Figures

The number given is 11,000. Identify the significant figures in this number. In this case, the significant figures are '11' because the trailing zeros are not significant.
02

Define the Base of Scientific Notation

Scientific notation is expressed as a product of a number between 1 and 10 and a power of 10. The '11' will be rewritten to fall between 1 and 10. We can achieve this by moving the decimal point to the left to create the number 1.1.
03

Count the Decimal Places Moved

Determine how many places the decimal point was moved. In '11,000', the decimal point is effectively at the end, so it moved 4 places to the left (from the end of '11,000' to between '1' and '1').
04

Write in Scientific Notation

Combine the number between 1 and 10 (1.1) with the power of ten indicating how many times the decimal was moved. This gives us \(1.1 \times 10^4\).

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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 crucial in scientific notation since they indicate the precision of a number. For the number 11,000, only the digits '11' are significant. The trailing zeros, in this specific context, merely indicate the magnitude of the number and don't contribute to its precision. When writing in scientific notation, our goal is to reflect the precision accurately. Remember:
  • Non-zero digits are always significant.
  • Leading zeros are never significant.
  • Trailing zeros are only significant if there鈥檚 a decimal point shown.
Therefore, translating 11,000 into scientific notation requires focusing on '11' as the significant figures.
Decimal Place
Understanding decimal places is essential when converting numbers into scientific notation. In 11,000, the decimal point is not visibly present; however, it's assumed to be at the end of the number (11,000.). When converting this to 1.1 (a number between 1 and 10), the decimal was "moved" 4 places to the left. This move is critical because it determines the power of ten, reflecting the magnitude of the number. Here鈥檚 a breakdown:
  • Visualize the number as 11,000.0.
  • See how moving the decimal left transforms 11,000 to 1.1.
  • The count of moves made sets the power of ten.
This skill in identifying and counting decimal movements is essential for correctly applying powers of ten.
Powers of Ten
Powers of ten help express large or tiny numbers more concisely and readably. They simplify calculations and comparisons. For 11,000, once we've identified the significant figures and counted the decimal places moved (4 places), we now need to use this count to set the power of ten. Remember:
  • The number of decimal places moved becomes the power's exponent.
  • In 11,000, moving 4 places left means multiplying 1.1 by \(10^4\).
  • This exponent reflects how many times the base number (1.1) scales up or down in magnitude.
Utilizing powers of ten is the backbone of scientific notation, allowing us to write numbers such as 11,000 elegantly as \(1.1 \times 10^4\).

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