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Chapter 5: Problem 29

Write number in scientific notation. \(23,000\)

Short Answer

Expert verified
\(23,000\) in scientific notation is \(2.3 \times 10^4\).

Step by step solution

01

Identify the Base Number

Identify the significant figures that form the base number of the scientific notation. Here, \(23,000\) has the significant figures \(23\), ignoring the trailing zeros.
02

Count the Decimal Places to Shift

Determine how many places you need to shift the decimal point to convert the original number into the scientific notation. For \(23,000\), the decimal point moves 4 places to the left (i.e., from \(23000.0\) to \(2.3\)).
03

Construct the Scientific Notation

Combine the base number and the power of ten derived from the shift. Since the decimal moved 4 places, the scientific notation is \(2.3 \times 10^4\).

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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 are crucial because they reflect the precision of a number. They include all non-zero digits, any zeros between them, and any trailing zeros in a decimal part. For the number 23,000, we consider only the digits 2 and 3 as the significant figures. All the trailing zeros are not considered significant since they don't affect the precision in this whole number. By identifying significant figures, we can clearly express numbers in a compact form, especially when using scientific notation.
Shifting the Decimal Point
Shifting the decimal point is a key step in converting a number to scientific notation. In scientific notation, every number is represented as a number between 1 and 10 multiplied by a power of ten. This requires moving the decimal point to a position just after the first significant figure. For 23,000, we need to move the decimal point 4 places to the left, which effectively shrinks its size to between 1 and 10. Thus, 23,000 becomes 2.3, which provides a compact and precise way of expressing large numbers. Remember, by shifting the decimal point, we are keeping the number's value but changing its format.
The Power of Ten
The power of ten compensates for the decimal shift when expressing a number in scientific notation. When you move the decimal point to the left, you increase the exponent of ten. Conversely, if you move it to the right, you decrease the exponent. The number of places the decimal shifts translates directly to the exponent of ten. In our case of 23,000, the shift of 4 places to the left results in the scientific notation of \(2.3 \times 10^4\). This multiplication by a power of ten maintains the original value of the number, allowing scientists and mathematicians to manage and work with very large or very small numbers more efficiently.

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