Problem 21 - Intermediate Algebra

Question

For Problems \(19-32\), write each of the following in ordinary decimal notation. $$ (4.19)\left(10^{3}\right) $$

Step-by-Step Solution

Answer

4190

Step 1: Understand the Problem

We are given an expression in scientific notation, \[(4.19) \times 10^{3},\] and we need to write this number in ordinary decimal notation.

Step 2: Identify the Power of Ten

In scientific notation, multiplying by a power of ten shifts the decimal point. The expression involves \(10^3\) which means we need to move the decimal point 3 places to the right.

Step 3: Move the Decimal Point

Start with the number 4.19. To convert it to ordinary decimal notation, move the decimal point 3 places to the right. This changes 4.19 to 4190.

Step 4: Write the Result

After moving the decimal point 3 places to the right, we obtain the ordinary decimal notation \[4190.\] This is the number represented by \(4.19 \times 10^3.\)

Key Concepts

Ordinary Decimal Notation

Ordinary decimal notation is the everyday way of writing numbers that you are already familiar with. It represents numbers using the base-ten system, where the position of each digit signifies its value relative to powers of ten. For instance, in the number 4190, each digit has its own place value:

• 4 is in the thousands place (4 x 1000)
• 1 is in the hundreds place (1 x 100)
• 9 is in the tens place (9 x 10)
• 0 is in the ones place (0 x 1)

Using ordinary decimal notation allows us to express numbers in an understandable and straightforward format. You don't need special symbols or operations to read these numbers. They immediately convey their size and value.

Powers of Ten

The concept of powers of ten is fundamental in understanding scientific notation and how numbers scale. A power of ten is expressed as 10 raised to an exponent, indicating how many times to multiply 10 by itself. For example:

• \(10^1 = 10\) (ten multiplied once)
• \(10^2 = 100\) (ten multiplied twice)
• \(10^3 = 1000\) (ten multiplied three times)

Using powers of ten makes it easy to work with very large or very small numbers by simplifying how many places the decimal point will move. For example, in the expression \(4.19 \times 10^3\), the power \(10^3\) signifies moving the decimal 3 places, enlarging 4.19 to 4190. This multiplication by a power of ten doesn’t change the digits themselves but allows for scaling up or down, converting the number into ordinary notation when applicable.

Decimal Point Movement

Decimal point movement is a straightforward mechanism when converting scientific notation into ordinary decimal notation. It involves shifting the decimal point either to the right or left based on the power of ten.The number of moves corresponds directly to the power. For instance, if you have \(4.19 \times 10^3\):

• Start with 4.19.
• Move the decimal 3 places to the right because of \(10^3\).
• This results in the number 4190.

If the exponent is positive, as in \(10^3\), the shift is to the right, enlarging the number. Conversely, a negative exponent, such as \(10^{-3}\), signifies moving the decimal point to the left, making the number smaller. Understanding how to move the decimal point correctly is key to transforming various notations and comprehending the size of different numbers.