Problem 32 - College Algebra with Modeling and Visualization

Question

Write the number in scientific notation. 0.62 (Number of miles in 1 kilometer)

Step-by-Step Solution

Answer

0.62 in scientific notation is \(6.2 \times 10^{-1}\).

Step 1: Understand Scientific Notation

Scientific notation is a way to express numbers as a product of a number between 1 and 10, and a power of ten. It is useful for expressing very large or very small numbers in a compact form.

Step 2: Identify the Decimal Position

To convert 0.62 into scientific notation, first identify where the decimal point is currently located. In this number, the decimal is between the 0 and 6, making the number 0.62.

Step 3: Shift the Decimal to Create a Number Between 1 and 10

Shift the decimal point so that the number is between 1 and 10. We move the decimal point one place to the right turning 0.62 into 6.2. This results in the need for a multiplying factor of a power of 10.

Step 4: Determine the Power of Ten

Since the decimal was moved one place to the right, the exponent needed will be negative. Therefore, the power of ten is \[-1\].

Step 5: Write the Number in Scientific Notation

Combine the number derived from Step 3 (6.2) with the power of 10 from Step 4. This results in the scientific notation: \[6.2 \times 10^{-1}\].

Key Concepts

Expressing Numbers

Expressing numbers can sometimes be challenging, especially when dealing with very large or small values. Scientific notation comes in handy in these situations. This method simplifies numbers, making them manageable and easier to read. When you see a number like 0.62, it might not seem too complex to handle.

• First, scientific notation breaks it down into two parts: a coefficient and a power of ten.
• The coefficient is a number between, and including, 1 and 10.
• The power of ten adjusts the scale of the number, determining how big or small it is.

Let's take the example of 0.62. By expressing it in scientific notation, you start by adjusting it to a coefficient that falls within 1 and 10, such as 6.2. This method ensures clarity in terms of how large or small a number really is, while keeping it simple and efficient.

Decimal Point

The decimal point is crucial in understanding and converting numbers into scientific notation. It's the small dot that separates whole numbers from fractional parts. In 0.62, the decimal point is initially placed between the 0 and the 6.

• To convert a number to scientific notation, you often need to shift this decimal point.
• This shift ensures that the number becomes a coefficient between 1 and 10.
• For instance, when we move the decimal point in 0.62 one place to the right, it becomes 6.2.

This change in the decimal point drastically alters the number's representation, which requires an accompanying adjustment in the power of ten to keep the value unchanged.

Power of Ten

The power of ten plays a pivotal role in scientific notation. It tells you how to scale the number up or down. The shift of the decimal point directly affects this power. Moving the decimal point to the right decreases the power because you're essentially making the number appear larger, and vice versa.

• For 0.62, moving the decimal from 0.62 to 6.2 means shifting it one place to the right.
• This results in a negative exponent in the scientific notation.
• Thus, 0.62 becomes 6.2 multiplied by 10 to the power of \(-1\), which physically means dividing by 10.

The negative exponent here indicates that the actual value of the coefficient gets smaller in comparison to its placement in the ten's power scale. This relationship is what ensures that scientific notation accurately represents the original number.