Problem 96 - Algebra 1

Question

SCIENTIFIC NOTATION Rewrite the number in scientific notation. $$0.0012$$

Step-by-Step Solution

Answer

The number \(0.0012\) in scientific notation is \(1.2 \times 10^{-3}\)

Step 1: Identify the Decimal Point

Locate the decimal point in the number \(0.0012\) . The goal is to rewrite this number such that it's between 1 and 10 and multiply it by a power of 10.

Step 2: Move the Decimal Point

Move the decimal point on \(0.0012\) three places to the right.

Step 3: Write the Number in Scientific Notation

Now the number appears as 1.2 after moving decimal point. We have moved the decimal point in Step 2 three places to the right, so that corresponds to a power of 10 to the \(-3\). Thus, we have \(1.2 \times 10^{-3}\) as the number in scientific notation.

Key Concepts

Decimal Point Movement

By understanding decimal point movement, we can convert seemingly complex numbers into a more manageable form. When dealing with numbers like 0.0012, our primary objective is to modify the number so it's between 1 and 10. We achieve this by moving the decimal point.

• Locate the decimal point in the original number. In this case, it's at the beginning of 0.0012.
• Move the decimal point towards the first non-zero digit, which is '1'. For 0.0012, this means shifting it three places to the right, creating 1.2.

Each movement to the right increases the power of 10 by a negative integer, as you're essentially multiplying the number by fractions of ten (such as 1/10, 1/100, etc.). This is crucial for rewriting the number in scientific notation.

Power of Ten

Once the decimal point has been adjusted, the next step involves determining the power of ten, which is an essential part of scientific notation. The key here is to count how many places the decimal point has been moved.

• For each movement to the right, we assign a negative power of ten, as this represents division by ten.
• In the example of 0.0012, moving the decimal three places to the right is noted as multiplying by \(10^{-3}\).

By expressing the number this way, you can clearly see the scale of the number compared to powers of ten. It helps in understanding the number's size quickly, especially when dealing with very large or very small numbers.

Expressing Numbers Between 1 and 10

To express a number in scientific notation, it must be adjusted to fit between 1 and 10. This range is chosen as it simplifies handling of figures significantly in mathematical operations and comparisons.

• The original number, 0.0012, was reformed to 1.2 after adjusting the decimal point. This brings the value within the required range.
• Once adjusted, pair it with the appropriate power of ten. In our example, 0.0012 becomes \(1.2 \times 10^{-3}\) in scientific notation.

This process helps in standardizing numbers and makes it easier to compare and compute large sets of data across scientific fields.