Question

Write each of the following as an "ordinary" decimal number. a. \(6.235 \times 10^{-2}\) b. \(7.229 \times 10^{3}\) c. \(5.001 \times 10^{-6}\) d. \(8.621 \times 10^{4}\)

Short answer

a. \(0.06235\) b. \(7,229\) c. \(0.000005001\) d. \(86,210\)

Step-by-step solution

a. Convert 6.235 × 10^{-2} to an "ordinary" decimal number.

To convert \(6.235 \times 10^{-2}\) to a decimal number, we move the decimal point 2 places to the left because the exponent is -2: 1. Move the decimal point 2 places to the left: \(6.235 \Rightarrow 0.6235\) There's our answer: \(6.235 \times 10^{-2} = 0.06235\)

b. Convert 7.229 × 10^{3} to an "ordinary" decimal number.

To convert \(7.229 \times 10^{3}\) to a decimal number, we move the decimal point 3 places to the right because the exponent is +3: 1. Move the decimal point 3 places to the right: \(7.229 \Rightarrow 7229\) There's our answer: \(7.229 \times 10^{3} = 7,229\)

c. Convert 5.001 × 10^{-6} to an "ordinary" decimal number.

To convert \(5.001 \times 10^{-6}\) to a decimal number, we move the decimal point 6 places to the left because the exponent is -6: 1. Move the decimal point 6 places to the left: \(5.001 \Rightarrow 0.000005001\) There's our answer: \(5.001 \times 10^{-6} = 0.000005001\)

d. Convert 8.621 × 10^{4} to an "ordinary" decimal number.

To convert \(8.621 \times 10^{4}\) to a decimal number, we move the decimal point 4 places to the right because the exponent is +4: 1. Move the decimal point 4 places to the right: \(8.621 \Rightarrow 86210\) There's our answer: \(8.621 \times 10^{4} = 86,210\)

Key concepts

Exponentiation

Exponentiation is a fundamental mathematical operation used to calculate repeated multiplication of a number by itself. When you see a number raised to the power of another, it means that the base number should be multiplied by itself as many times as the power suggests. For instance, in the expression \(10^3\), the base is 10, and the exponent is 3, which means 10 is multiplied by itself three times: \(10 \times 10 \times 10 = 1000\).

In scientific notation, exponentiation is used to express very large or very small numbers in a compact form. By aligning numbers with powers of 10, we simplify the way we work with these extreme values. For example, expressing 1000 as \(1 \times 10^3\) is a more straightforward and manageable form, especially when dealing with calculations.

When converting numbers in scientific notation to ordinary decimals, it's crucial to interpret these exponents correctly. A positive exponent indicates that the decimal point should be moved to the right, making the number larger, whereas a negative exponent means moving the decimal point to the left, creating a smaller number.

Decimal Conversion

Decimal conversion is all about changing the format of a number for easier understanding or application. In the steps provided, we saw how numbers written in scientific notation could be converted to their standard decimal form. This involves shifting the decimal point as instructed by the exponent.

Here's how you can achieve decimal conversion easily:

• Identify the exponent: It tells you how many places to move the decimal point.
• If the exponent is positive, move the decimal point to the right.
• If the exponent is negative, move the decimal point to the left.
• If you run out of numbers, fill in with zeros to complete the move.

For example, with \(7.229 \times 10^3\), move the decimal three places to the right to get 7,229. On the other hand, \(5.001 \times 10^{-6}\) requires moving the decimal six places left, resulting in 0.000005001. Performing these conversions allows us to handle both grandiose and minuscule figures effortlessly.

Mathematical Notation

Mathematical notation serves as a language to succinctly and universally describe mathematical concepts and operations. Scientific notation is one such tool, especially valuable when dealing with very large or small numbers. It not only saves space but also simplifies computations.

Let's break down a typical scientific notation: \(A \times 10^n\). Here, \(A\) is known as the significand or mantissa and represents the base number, while \(n\) refers to the power of 10, or the exponent. Whether \(n\) is positive or negative dictates the direction in which the decimal shifts:

• Positive \(n\): Decimal point moves right; number increases.
• Negative \(n\): Decimal point moves left; number decreases.

This notation is instrumental not just in mathematical chores but across scientific fields, including physics and engineering, where precision and ease of communication are paramount. By mastering this notation, we can seamlessly transition between representing tiny bacteria dimensions to astronomical distances in the cosmos.