Problem 12 - Introductory Chemistry Atoms First

Question

Convert \(47.325 \times 10^{3}\) to standard notation.

Step-by-Step Solution

Answer

The given number \(47.325 \times 10^{3}\) in standard notation is \(47,325\).

Step 1: Identify the Decimal Point Location in the Original Number

In the given number \(47.325\), the decimal point is located between 4 and 7.

Step 2: Move the Decimal Point to the Right

We need to move the decimal point three places to the right because of the exponent 3 in \(10^{3}\). Thus, we have: \(47.325\) → \(473.25\) → \(4732.5\) → \(47325\)

Step 3: Write the Final Answer in Standard Notation

The given number \(47.325 \times 10^{3}\) has now been converted into standard notation as \(47,325\).

Key Concepts

Scientific Notation

Scientific notation is a method of writing very large or very small numbers in a compact form. It's particularly useful in science and engineering to express numbers with many zeros, either after the decimal point for small numbers or before the decimal point for large numbers. In scientific notation, a number is written as the product of two factors: a decimal number between 1 and 10, and an exponent of 10. The general form is \( a \times 10^{n} \), where \( a \) is the decimal number, and \( n \) is the integer exponent. For example, the distance between the Earth and the Sun, approximately 149,600,000 kilometers, can be written in scientific notation as \( 1.496 \times 10^{8} \) kilometers.

Using scientific notation makes calculations with very large or small numbers more manageable because it simplifies the multiplication and division of numbers that have the same basis of 10. Converting from scientific notation to standard notation involves moving the decimal point to the right if the exponent is positive and to the left if it's negative, which is the number of times equal to the absolute value of the exponent.

Decimal Point

The decimal point is a dot used to separate the whole number part from the fractional part of a numeral. In the context of scientific notation and standard notation, the decimal point's position is critical because it determines the power of ten. When converting a number from scientific notation to standard notation, like \(47.325 \times 10^{3}\), you move the decimal point to make the number bigger or smaller.

To understand this, one must recognize that each place value in a number represents a power of ten. For example, moving the decimal point one place to the right multiplies the number by ten (or by \(10^{1}\)); moving it two places multiplies the number by a hundred (or by \(10^{2}\)), and so on. This logic is reversed for moving the decimal point to the left. Mastery of moving the decimal point is essential for efficient calculations and understanding number scales.

Exponents

Exponents are a shorthand way to show how many times a number, known as the base, is multiplied by itself. In the expression \(10^{3}\), the number 10 is the base and 3 is the exponent, meaning that 10 is multiplied by itself three times (\(10 \times 10 \times 10\)). Exponents play a crucial role in scientific notation, as they allow us to express the magnitude of a number.

Understanding Positive and Negative Exponents

Positive exponents indicate how many zeros should follow the number if it's greater than one. Negative exponents, on the other hand, show the number of zeros before the number, when it is less than one and doesn't have a whole numeral part. For instance, \(10^{-3}\) means you have to move the decimal point three places to the left, giving \(0.001\). By mastering the concept of exponents, students can effectively work with scientific notation and easily perform mathematical operations on large or small numbers.