Problem 83 - Algebra and Trigonometry

Question

Write each number in scientific notation. $$ 0.0027 $$

Step-by-Step Solution

Answer

0.0027 written in scientific notation is \(2.7 \times 10^{-3}\)

Step 1: Identify the position of the decimal point

To write this number in scientific notation, first identify the current position of the decimal point and count the number of places to move it to make the number between 1 and 10. In this case, the decimal point needs to move three places to the right, making the number 2.7.

Step 2: Multiply by a power of 10

The number needs to be multiplied by a power of 10 matching the number of places the decimal point moved. As the original number was less than 1, this power of 10 will be negative, denoting a division. In this case, the number of decimal places moved was three, hence the power of 10 is -3.

Step 3: Write in scientific notation

Combine the number obtained in step 1 with the power of 10 obtained in step 2. This renders the given number in scientific notation. Thus, \(0.0027 = 2.7 \times 10^{-3}\).

Key Concepts

Decimal Point

The decimal point is essential in representing both small and large numbers in a readable way. In a number like 0.0027, the decimal point is what distinguishes where the whole number part ends and the fractional part begins. When converting a number to scientific notation, one of the first steps is moving the decimal point to create a new number between 1 and 10.

This means shifting the decimal point either to the left or to the right, depending on whether the original number is less than or greater than 1:

• If the number is less than 1, like our example 0.0027, move the decimal point to the right until you get a number between 1 and 10. For 0.0027, you move it three places, making it 2.7.
• If the number is greater than 10, you'd move the decimal point to the left instead.

After moving the decimal point, remember the number of places you've moved it, as this will help you with the next steps.

Powers of Ten

The concept of powers of ten is pivotal in scientific notation, which helps simplify numbers. It involves using ten raised to an exponent, which tells you how many times to multiply or divide by 10. After moving the decimal point, you multiply the result by a suitable power of ten.

Here’s how it works:

• The exponent tells you how many times the number 10 is to be used as a multiplier.
• If you moved the decimal to the right, you use a negative power of ten (we’ll learn why in the next section). For example, from 0.0027 to 2.7, we moved the decimal three places right. Thus, we use \(10^{-3}\).
• Where you move the decimal point left, you'd use a positive power of ten.

Understanding how to determine this power is a stepping stone towards mastering scientific notation, as it simplifies the representation of incredibly small or large numbers.

Negative Exponent

Negative exponents signal a division, which is crucial when dealing with numbers less than 1 in scientific notation. A negative exponent indicates that a number is being divided by a power of ten rather than multiplied.

To comprehend negative exponents:

• A negative exponent like \(10^{-3}\) means "divide by ten" three times, or equivalently \(\frac{1}{10^3}\).
• In our scientific notation example, we convert 0.0027 to 2.7 by moving the decimal three places right, reflecting it as \(2.7 \times 10^{-3}\).

This tells us that 2.7 should be divided by 1000 (since \(10^3 = 1000\)), confirming that it indeed represents the original number 0.0027. Grasping the role of negative exponents in scientific notation helps demystify how small numbers efficiently convert to more manageable expressions.