Problem 45 - College Algebra with Corequisite Support

Question

A dime is the thinnest coin in U.S. currency. A dime's thickness measures \(1.35 \times 10^{-3}\) \(\mathrm{m}\). Rewrite the number in standard notation.

Step-by-Step Solution

Answer

The number in standard notation is 0.00135.

Step 1: Understanding Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is expressed as \(a \times 10^n\) where \(1 \leq |a| < 10\). "Standard notation" means writing the number without using exponents.

Step 2: Identifying the Components

In the expression \(1.35 \times 10^{-3}\), the coefficient is 1.35 and the exponent is -3. This indicates a positive number as we are multiplying by a power of 10.

Step 3: Moving the Decimal Point

Since the exponent is -3, we need to move the decimal point in 1.35 three places to the left to express it in standard notation. The decimal point moves from between the '1' and the '3', moving past three zeros.

Step 4: Writing the Number in Standard Notation

After moving the decimal point three places to the left, the number is 0.00135. The original decimal point moves left and fills in with zeros to reflect the multiplication by \(10^{-3}\).

Key Concepts

Standard Notation

Standard notation is essential because it allows us to write numbers in their traditional, easily recognizable form, without the use of exponents. The key here is to translate numbers given in scientific notation, like \(1.35 \times 10^{-3}\), into a format where we can see all decimal places clearly. In this context, standard notation means writing the number as it would appear on a calculator or when writing it longhand on paper. For example, while \(1.35 \times 10^{-3}\) might seem abstract due to the exponent, it transforms into the more familiar 0.00135 in standard notation. This method helps us better understand the magnitude of the number and connect it to real-world values, such as a dime's thickness in the exercise.

Exponents

Exponents are a powerful mathematical tool that indicate how many times we multiply a number by itself. In scientific notation, exponents are used in the form of powers of ten to make very large or very small numbers more manageable. Let's delve deeper into this: - An exponent of \(-3\) in situations like \(10^{-3}\) means we divide 1 by 10 three times in succession, resulting in a number that's one-thousandth of the original.- Therefore, multiplying by \(10^{-3}\) shifts the decimal three places to the left, effectively making the resulting number smaller. Understanding exponents thus helps in comprehending the scale and size of various measurements, such as the extremely thin 0.00135-meter thickness of a dime.

Decimal Point Movement

Decimal point movement is a key operation when converting numbers from scientific to standard notation. This movement easily reflects which direction and how far the decimal point goes. In the case of the number \(1.35 \times 10^{-3}\), the exponent \(-3\) tells us something crucial:

• A negative exponent moves the decimal to the left.
• Thus, starting from 1.35, we move the decimal three places to the left.
• Every time we shift, we place a zero in the newly created space.

This results in 0.00135, making the number smaller and demonstrating how the multiplication by \(10^{-3}\) affects size. Grasping this concept is vital for tackling any scientific notation problem successfully.