Problem 4 - Physics for Scientists and Engineers with Modern Physics

Question

(1) Write out the following numbers in full with the correct number of zeros: (a) \(8.69 \times 10^{4}\), (b) \(9.1 \times 10^{3}\), (c) \(8.8 \times 10^{-1},\) (d) \(4.76 \times 10^{2}\), and (e) \(3.62 \times 10^{-5}\)

Step-by-Step Solution

Answer

(a) 86900, (b) 9100, (c) 0.88, (d) 476, (e) 0.0000362.

Step 1: Understanding Scientific Notation

Scientific notation expresses numbers as a product of a coefficient and a power of ten. For example, in the expression \(8.69 \times 10^4\), 8.69 is the coefficient, and \(10^4\) indicates the number will be multiplied by 10 raised to the fourth power, moving the decimal point four places to the right.

Step 1: Convert (a) \(8.69 \times 10^{4}\)

To convert \(8.69 \times 10^{4}\), move the decimal point 4 places to the right: \(86900\). This transformation is done by multiplying 8.69 by 10,000.

Step 2: Convert (b) \(9.1 \times 10^{3}\)

To convert \(9.1 \times 10^{3}\), move the decimal point 3 places to the right: \(9100\). This transformation is achieved by multiplying 9.1 by 1,000.

Step 3: Convert (c) \(8.8 \times 10^{-1}\)

To convert \(8.8 \times 10^{-1}\), move the decimal point 1 place to the left: \(0.88\). This is the same as dividing 8.8 by 10.

Step 4: Convert (d) \(4.76 \times 10^{2}\)

To convert \(4.76 \times 10^{2}\), move the decimal point 2 places to the right: \(476\). This transformation is done by multiplying 4.76 by 100.

Step 5: Convert (e) \(3.62 \times 10^{-5}\)

To convert \(3.62 \times 10^{-5}\), move the decimal point 5 places to the left, resulting in \(0.0000362\). This is the same as dividing 3.62 by 100,000.

Key Concepts

Decimal Point Conversion

The concept of decimal point conversion is fundamental when working with scientific notation. This process involves shifting the decimal point in a number based on the exponent of ten associated with it. For numbers with a positive exponent, move the decimal to the right.
Conversely, for numbers with a negative exponent, move it to the left.

• For example, take the number \(8.69 \times 10^{4}\). With an exponent of 4, shift the decimal point four places to the right, turning 8.69 into 86900.
• If the exponent is negative, such as in \(8.8 \times 10^{-1}\), move the decimal to the left by one place to transform it into 0.88.

The key to mastering decimal point conversion is to count the required places very carefully, as each shift dramatically changes the number's value.

Coefficient and Power of Ten

Scientific notation pairs a coefficient—a number between 1 and 10—with a power of ten. This format efficiently represents very large or small numbers.

• The coefficient indicates the significant figures of the number. For \(4.76 \times 10^{2}\), 4.76 is the coefficient showing the significant digits.
• The power of ten deals with the scaling factor. It tells us how many places to move the decimal point—right for positive and left for negative powers.

This representation is particularly handy in science and engineering where precision and brevity are crucial. When you understand both components, converting between scientific notation and standard notation becomes straightforward.

Multiplying by Ten Powers

When numbers are multiplied by powers of ten, it involves simplifying the process of moving the decimal point. This makes mathematical operations with very large or very small numbers easier.

• Consider \(9.1 \times 10^{3}\). To simplify, shift the decimal right by three places, converting 9.1 into 9100.
• In another example, \(3.62 \times 10^{-5}\), move the decimal left by five places, changing 3.62 into 0.0000362.

Multiplication by ten powers also aids in quickly comparing the scale of different values without tackling complex arithmetic. It's the exponent that plays a pivotal role in determining the final value's magnitude by dictating the number of decimal shifts needed.