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Chapter 5: Problem 75

Write each number in scientific notation. See Example \(5 .\) $$ 1,160,000 $$

Short Answer

Expert verified
\(1.16 \times 10^6\)

Step by step solution

01

Identify the significant figures

Identify the non-zero digits in the number 1,160,000. These digits are 1.16. These form the significant figures in the number.
02

Determine the power of 10

Count how many places you need to move the decimal point to change 1,160,000 into 1.16. You need to move the decimal 6 places to the left.
03

Write in scientific notation

Now write the number in scientific notation using the significant figures and the power of 10 found in Step 2. This gives us \(1.16 \times 10^6\).

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Key Concepts

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

Understanding Significant Figures
Significant figures represent the digits in a number that contribute to its precision. When writing a number in scientific notation, you focus on the non-zero digits as these form the backbone or foundation of your figure. In the number 1,160,000, the significant figures are 1.16. These three digits are all non-zero, and they tell us the level of detail in the measurement.
  • Non-zero digits are always significant.
  • Any zeros between significant figures are also significant.
  • Leading zeros or trailing zeros in a large whole number are not considered significant unless specified by a decimal point.

This understanding helps maintain accuracy when converting numbers into scientific notation, ensuring we retain the true value and importance of the original measurement.
The Role of Power of 10 in Scientific Notation
The power of 10 is what gives scientific notation its ability to scale numbers up or down efficiently. This concept is vital because it represents how far the decimal point has moved in the number. In scientific notation, the power of 10 indicates multiplication by 10 raised to a certain exponent.
In the example of 1,160,000, we need to convert this number into 1.16 by moving the decimal point 6 places to the left. This movement tells us that our power of 10 will be 6, hence the expression is written as: \[1.16 \times 10^6\]
  • If you move the decimal to the left, the power is positive.
  • If you move the decimal to the right, the power is negative.

Correct use of powers of 10 allows us to express very large or very small numbers succinctly and conveniently.
Decimal Point Movement in Scientific Notation
Moving the decimal point is a crucial step in transforming any number into scientific notation. This action helps standardize the number into a manageable form, typically between 1 and 10. Each shift of the decimal impacts the power exponent you will use.
For the number 1,160,000, we move the decimal point six places to the left to arrive at 1.16. This shift not only changes the number's magnitude but also determines the exponent of the power of 10.
  • Moving the decimal left results in a positive exponent because the original number was large.
  • Each place moved corresponds to one count in the exponent of the power of 10.

Understanding decimal point movement ensures accurate conversion and vital consistency in scientific calculations.

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