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Chapter 0: Problem 121

In Problems 121-128, write each number in scientific notation. 454.2

Short Answer

Expert verified
4.542 \times 10^2

Step by step solution

01

Identify the significant figures

The number is 454.2. Identify the significant figures: 4, 5, 4, and 2.
02

Place the decimal point

Move the decimal point in the number 454.2 so that it is after the first significant figure. The number becomes 4.542.
03

Count the decimal places moved

Count how many places the decimal point was moved. It was moved 2 places to the left.
04

Express the number in the form of scientific notation

Combine the coefficient 4.542 with the exponent indicating the decimal moves. The scientific notation is: \[ 4.542 \times 10^2 \]

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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 contribute to its precision. In the number 454.2, the significant figures are 4, 5, 4, and 2. These digits represent the accuracy and detail of the measurement. Any zeros before the first significant figure are not counted, but zeros between or after the significant figures do count. Identifying these significant figures is crucial because they help determine the precision of the number when converting it into scientific notation.
Decimal Point
The position of the decimal point is key in converting a number to scientific notation. In the number 454.2, the decimal point is initially between the digits 4 and 2. To convert this to scientific notation, we need to shift the decimal point so it is immediately after the first significant figure. Moving the decimal point to create 4.542 accomplishes this. The necessity to shift the decimal allows us to express large or small numbers efficiently without losing precision.
Exponents
Exponents are used in scientific notation to show how many places the decimal point has moved. This helps in representing large or small numbers compactly. In our example, the decimal point in 454.2 moves 2 places to the left, giving us the exponent of 2. Therefore, the number in scientific notation is written as \[ 4.542 \times 10^2 \]. The exponent of 2 signifies that the decimal point was moved two places to the left. Positive exponents indicate a move to the left, while negative exponents indicate a move to the right.
Coefficient
The coefficient in scientific notation is the number formed after placing the decimal point correctly. In the example, after adjusting 454.2, we get 4.542. This is known as the coefficient and it must always be greater than or equal to 1 and less than 10. The coefficient retains the significant figures from the original number, ensuring accuracy. Combining this with the exponent, we get the complete scientific notation. For 454.2, the scientific notation is \[ 4.542 \times 10^2 \]. The coefficient \`4.542\` retains the vital significant figures, making the notation both precise and manageable.

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