Problem 18 Express each number in scientifi... [FREE SOLUTION]

Chapter 4: Problem 18

Express each number in scientific notation. $$2,000,000$$

Short Answer

Expert verified

The number 2,000,000 in scientific notation is $2.0 \times 10^6$.

Step by step solution

Identify the Significant Figures

For the number 2,000,000, identify the digits that are significant. Here, the number 2 alone is the significant figure, as it captures the essence of the number.

Place the Decimal Point

Place the decimal point immediately after the significant figure. This turns the number into 2.0 for scientific notation purposes.

Count the Number of Decimal Places Moved

From the original number 2,000,000, count how many places you move the decimal point to end up with 2.0. Here, you move the decimal point 6 places to the left.

Write in Scientific Notation

Express the number as the product of the significant figure and 10 raised to the power of the number of places moved. Here, it is expressed as $2.0 \times 10^6$.

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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. It's essential to recognize them, especially when converting a number into scientific notation. In the number 2,000,000, the only significant digit is 2, marking the essential value of the number. Typically, the significant figures include:

All non-zero numbers
Zeros between non-zero digits
Trailing zeros in a decimal number

For instance, in 2,000,000, zeros do not contribute to the significant figures, hence the focus on 2. Identifying significant figures helps simplify and neatly express large or small numbers.

Decimal Point Placement

In scientific notation, decimal point placement is crucial. It frames the number in a format where all significant figures precede the decimal point. For 2,000,000, the decimal point shifts right after the significant number to form 2.0. The logic behind this placement is:

Ensure a single non-zero digit is to the left of the decimal
Simplify understanding of large numbers, as seen in scientific notation

For 2,000,000, moving the decimal point 6 places left ensures the number transforms into 2.0. This strategy aids in attaining a clearer, compact representation, which is handy in scientific calculations.

Powers of Ten

Powers of ten signify how many times the decimal point shifts to represent a number in scientific notation. It denotes the scaling factor reflecting the size of a number. In 2,000,000, the decimal point is moved 6 places to the left, making the number more manageable while retaining its magnitude:

The number of shifts gives rise to the exponent
Thus, the expression becomes a combination of the significant figure and a power of ten

Essentially, 2,000,000 becomes $2.0 \times 10^6$. Here, 6 is the power of ten, illustrating the transformation from a large numeric form to a concise one. This exponential format is immensely helpful in sciences, ensuring computations are both efficient and precise.

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91影视

Express each number in scientific notation. $$2,000,000$$

Short Answer

Step by step solution

Identify the Significant Figures

Place the Decimal Point

Count the Number of Decimal Places Moved

Write in Scientific Notation

Key Concepts

Significant Figures

Decimal Point Placement

Powers of Ten

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