Problem 67 - College Algebra

Question

Write each number in decimal notation. $$ 4 \times 10^{6} $$

Step-by-Step Solution

Answer

The decimal notation of \(4 \times 10^{6}\) is 4000000.

Step 1: Understand Scientific Notation

Scientific notation is a convenient way of expressing very large or very small numbers. It is represented as a number between 1 and 10 multiplied by a power of 10. In the given notation, \(4 \times 10^{6}\), 4 is the number and 6 is the power of 10.

Step 2: Conversion to Decimal Notation

To convert the number from scientific to decimal notation, move the decimal point in the number part to the right if the exponent of 10 is positive. You move it as many places as the exponent indicates. In this case, there is no decimal point in 4. So, consider 4 as 4.0 and move the decimal point 6 places to the right.

Step 3: Write the Decimal Notation

After moving the decimal point 6 places to the right, fill in zeroes in the gaps that have been created. The resulting number is 4000000.

Key Concepts

Decimal Notation

Decimal notation is the regular way of writing numbers that most people use every day. When exchanging amounts of money or measuring ingredients for a recipe, we naturally use decimal notation. It allows us to represent each digit using a base of 10, spanning positions such as ones, tens, hundreds, and so on. Each position in a number denotes a power of ten, increasing from right to left. For example:

• The number 5,647 means 5 thousands, 6 hundreds, 4 tens, and 7 ones.
• We write these numbers as they are, without special symbols or formatting.

To convert a number from scientific to decimal notation, place the decimal point where it needs to be, according to the given power of ten. This process makes handling large or small numbers more straightforward.

Large Numbers

Large numbers can be challenging to read, write, and especially comprehend. Scientific notation presents a useful solution by breaking them down into a manageable form, but ultimately, they often need conversion back to decimal form for day-to-day use. In decimal notation, large numbers can seem intimidating with their long strings of zeros.

• A number like 4,000,000 in scientific notation is simplified as \(4 \times 10^{6}\).
• When converting, you move the decimal to accommodate the power of ten, filling in with zeros as necessary.

Understanding large numbers in decimal notation assists greatly in real-world applications such as finance or science, where accuracy and ease of interpretation are critical.

Power of Ten

The power of ten is integral to both scientific and decimal notation. It indicates how many times the number 10 is used as a multiplier. In scientific notation, the power of ten tells us how many places to move the decimal point.

• If the power is positive, the decimal point moves to the right, indicating a larger number.
• Conversely, a negative power moves the decimal point to the left, indicating a small number.

For example, in \(4 \times 10^{6}\), the 6 tells us to move the decimal point 6 places to the right, making the number bigger. This system simplifies mathematical calculations and data representation.