Problem 84 - Algebra and Trigonometry

Question

Write each number in scientific notation. $$ 0.0083 $$

Step-by-Step Solution

Answer

The number 0.0083 can be expressed in scientific notation as \(8.3 \times 10^{-3}\)

Step 1: Identify the First Non-Zero Digit

Looking at the initial number 0.0083, the first non-zero digit is 8. So the next step will revolve around shifting the decimal point in the position right after 8.

Step 2: Move the Decimal Point

To move the decimal point after the 8 to present it in the required form, it should be shifted three places to the right. This action will transform 0.0083 to 8.3.

Step 3: Express in Scientific Notation

The final step is to express this movement of the decimal point as a power of ten. Since the decimal point was moved to the right, the power of ten will be negative. That's how we transformed the decimal 0.0083 to scientific notation: \(8.3 \times 10^{-3}\)

Key Concepts

Understanding the Decimal Point

The decimal point is a critical component in writing numbers, particularly when dealing with sums like 0.0083. In its essence, the decimal point separates the whole number part from the fractional part. For the number 0.0083, the point comes shortly after the zero, indicating that the number is less than one but greater than zero.

Understanding how the decimal point operates becomes essential in scientific notation as well. It determines where counting for powers of ten begins. By shifting the decimal to the position needed, we reform how the original number is expressed.

In our example, moving the decimal point three places to the right will change 0.0083 into 8.3. This shift is crucial in simplifying how numbers are understood and manipulated in scientific notation.

Powers of Ten in Scientific Notation

Scientific notation relies heavily on the power of ten, which acts as a scalable factor that simplifies expressions of very large or very small numbers. Powers of ten in notation indicate how many places the decimal point has shifted from its original position.

In the instance of 0.0083, moving the decimal three places to the right means we employ a negative power, specifically, -3. Therefore, 0.0083 becomes expressed as 8.3 multiplied by ten raised to the power of -3: \(8.3 \times 10^{-3}\).

This reveals that the decimal was moved to make the number easier to handle and easier to understand within calculations.

The Role of Non-Zero Digits

Non-zero digits are the backbone of scientific notation. In any set of digits, the first non-zero one is essential as it indicates the beginning of significant numbers or the figure's essential value.

Take the example of 0.0083: the digit '8' is the first non-zero digit. This makes '8' significant in determining how the rest of the number is expressed.

When translating a number into scientific notation, the placement, and selection of non-zero digits determines how compact the expression becomes. Here, the presence of '8' amidst zeros leads us to standout 8.3 as the core expression of 0.0083 in scientific notation (coupled with a power of ten for accuracy).