Problem 32 - Prealgebra

Question

Express each number in scientific notation. The flow rate of some Antarctic glaciers is 0.00031 mile per hour.

Step-by-Step Solution

Answer

0.00031 in scientific notation is \( 3.1 \times 10^{-4} \).

Step 1: Understand Scientific Notation

Scientific notation is a way to express very large or very small numbers in the format of a number between 1 and 10 multiplied by a power of 10. It is written as \( a \times 10^n \), where \( 1 \leq a < 10 \) and \( n \) is an integer.

Step 2: Identify the Significant Figures

Look at the given number: 0.00031. Identify the significant figures, which are the non-zero digits. Here, the significant figures are 3 and 1.

Step 3: Move the Decimal Point

Move the decimal point to the right until there is only one non-zero digit to the left of the decimal point. For 0.00031, you move the decimal point 4 places to the right, resulting in 3.1. This is the number \( a \) in scientific notation.

Step 4: Determine the Power of 10

Count how many places you moved the decimal point to determine the exponent \( n \). Since you moved the decimal 4 places to the right, \( n \) is -4. The negative sign indicates the original number is less than one.

Step 5: Write in Scientific Notation

Combine the significant number and the power of ten to write the number in scientific notation. The given number 0.00031 is written as \( 3.1 \times 10^{-4} \) in scientific notation.

Key Concepts

Significant Figures

When dealing with numbers, especially in scientific contexts, it's important to understand significant figures. These are the digits in a number that contribute to its precision. Generally, they include all non-zero digits, zeros between non-zero digits, and any trailing zeros in a decimal number. This rule helps us decide how many digits are worth noting.

In the given number, 0.00031, the significant figures are 3 and 1. Non-zero numbers are always considered significant, providing the scale of measurement. Applying this to scientific notation gives us the most accurate representation without unnecessary details. Recognizing significant figures ensures that our representation is correct and that we avoid misrepresenting the number's precise value.

Decimal Point

A decimal point is a crucial element in numbers, separating the whole part from the fractional part. When converting a number into scientific notation, moving the decimal point is a key step.

For scientific notation, you move the decimal to get a number between 1 and 10. In 0.00031, the decimal point moves 4 places to the right to become 3.1, situating the non-zero digit to the left of the new decimal point. This movement helps us arrive at the proper form of a scientific notation, balancing precision with simplicity.

• Move the decimal to create a number between 1 and 10.
• This new position determines the power of ten used.
• The number of moves equals the magnitude of power of ten.

By shifting the decimal correctly, we preserve the initial value while adjusting for scale.

Power of Ten

The power of ten in scientific notation represents how many times we have shifted the decimal point. It shows whether the original number is large or small compared to one.

In the case of 0.00031, the decimal point moved 4 places to the right, indicating a negative exponent of -4. This negative sign tells us the original number was less than 1. Understanding this can help clarify large and small numbers across various sciences.

• The power of ten indicates the movement of the decimal.
• A negative exponent highlights numbers smaller than one.
• In scientific notation, it helps succinctly express very small values.

This system allows us to convey both grand and minuscule numbers in a uniform, comprehensible manner.