Question

Write the following numbers in scientific notation. a. 0.000 673 0 b. 50 000.0 c. 0.000 003 010

Short answer

a. 6.730 × 10^{-4}, b. 5.0 × 10^4, c. 3.010 × 10^{-6}

Step-by-step solution

Understand Scientific Notation

Scientific notation expresses numbers as a product of a number between 1 and 10 and a power of 10. It has the form: \[ a \times 10^n \]where 1 <= a < 10 and n is an integer.

Convert 0.000 673 0 to Scientific Notation

1. Identify the significant figures: 6730. 2. Place a decimal point to create a number between 1 and 10: 6.730. 3. Count the number of places the decimal was moved: 4 places to the right. 4. Write the number in scientific notation: \[ 0.000 673 0 = 6.730 \times 10^{-4} \]

Convert 50 000.0 to Scientific Notation

1. Identify the significant figures: 5. 2. Place a decimal point to create a number between 1 and 10: 5.0. 3. Count the number of places the decimal was moved: 4 places to the left. 4. Write the number in scientific notation: \[ 50 000.0 = 5.0 \times 10^4 \]

Convert 0.000 003 010 to Scientific Notation

1. Identify the significant figures: 3.010. 2. Place a decimal point to create a number between 1 and 10: 3.010. 3. Count the number of places the decimal was moved: 6 places to the right. 4. Write the number in scientific notation: \[ 0.000 003 010 = 3.010 \times 10^{-6} \]

Key concepts

Significant Figures

Significant figures are the digits in a number that are known with certainty plus the first uncertain digit. They play a crucial role in scientific notation because they represent the accuracy and precision of a number.
When converting to scientific notation, it's essential to identify all the significant figures.
For example, in the number 0.000 673 0, the significant figures are 6730. The trailing zero is significant because it comes after a decimal point and a non-zero digit.
Another example is the number 50 000.0. Here, the significant figures are 5. The trailing zeros are not significant because they are placeholders and don't provide additional precision information. The decimal point, however, makes the zero after it significant.
Understanding significant figures ensures that scientific notation accurately represents the number's precision and avoids misleading results.

Powers of 10

Powers of 10 are used in scientific notation to simplify the representation of very large or very small numbers. In the equation \[ a \times 10^n \], 'a' is a number between 1 and 10, and 'n' indicates how many places the decimal point has been moved.

• If 'n' is positive, the decimal point moves to the right, making 'a' a large number. For example, 50 000.0 is written as 5.0 \[ \times 10^4 \].
• If 'n' is negative, the decimal point moves to the left, making 'a' a small number. For instance, 0.000 003 010 becomes 3.010 \[ \times 10^{-6} \].

Powers of 10 make it easier to read, write, and work with numbers that would otherwise be unwieldy due to their size.

Decimal Point

The decimal point is a critical component in converting to scientific notation. Its placement determines the power of 10 and affects the significant figures.
For example, in the number 0.000 673 0, moving the decimal point four places to the right turns it into 6.730. Hence, it is written as 6.730 \[ \times 10^{-4} \]. The movement to the right is represented by a negative power of 10.
Conversely, for the number 50 000.0, moving the decimal point four places to the left gives 5.0. This is written as 5.0 \[ \times 10^4 \], where the movement to the left is depicted by a positive power of 10.
Understanding how the decimal point affects the conversion process is vital for accurately writing numbers in scientific notation.