Problem 21 - Algebra for College Students

Question

For Problems \(19-32\), write each of the following in ordinary decimal notation. For example, \((3.18)(10)^{2}=318\). \((4.19)(10)^{3}\)

Step-by-Step Solution

Answer

4190

Step 1: Identify the Scientific Notation

The given expression is \((4.19)(10)^{3}\), which is in scientific notation.

Step 2: Understand the Power of 10

The expression \((10)^{3}\) indicates that the decimal point in 4.19 needs to be moved 3 places to the right.

Step 3: Move the Decimal

Starting with 4.19, move the decimal point 3 places to the right. This results in the number 4190.

Step 4: Write the Number in Decimal Notation

After moving the decimal, you obtain 4190, which is the ordinary decimal notation of the expression.

Key Concepts

Understanding Decimal Notation

Decimal notation is a way of expressing numbers using the base 10 system, which is the most common numerical system used in everyday life. This system relies on the position of digits to represent values. Each position in a number is a power of ten, with the rightmost digit representing the units (or ones) place, then the tens, hundreds, thousands, and so on as you move to the left.
For example, in the number 319, the 3 is in the hundreds place, the 1 is in the tens place, and the 9 is in the ones place. In essence, it is written as:

• 3 - hundreds place ( 300 or 3×10^2)
• 1 - tens place ( 10 or 1×10^1)
• 9 - ones place ( 9 or 9×10^0)

The value of each digit is determined by multiplying with powers of ten, which is why understanding powers of ten is crucial when converting scientific notation to decimal notation.

Exploring Powers of Ten

The power of ten refers to a mathematical expression that indicates how many times the number ten is used in a multiplication. Essentially, it's a way of writing and calculating large or small numbers in a compact form.
In scientific notation, powers of ten are important for expressing numbers that are either very large or very small. The exponent, written as a small number to the upper right of the base (which is 10), tells you how many times to multiply the base.

• An exponent of 1 means you multiply by ten once: 10^1 = 10
• An exponent of 2 means you multiply it twice: 10^2 = 10 × 10 = 100
• An exponent of 3 means you multiply it three times: 10^3 = 10 × 10 × 10 = 1000

In our example, (10)^{3}, the exponent is 3. This indicates we need to multiply the base number, 10, three times, leading to moving the decimal place three spaces to the right.

Moving the Decimal Point

Moving the decimal point is a key step in converting scientific notation into decimal notation. It determines the size of the number in base 10, influencing whether it is larger or smaller.
Let's look at the process:
With (4.19)(10)^3, the 10^3 tells us to move the decimal point three places to the right. Begin at the initial position of the decimal point in 4.19. Each number 0 you move across as you move the point right creates a higher value in the tens places bracket.

• Start with 4.19: the decimal is immediately after the 4.
• First move: 41.9.
• Second move: 419.
• Third move: 4190.

After moving the decimal point three places to the right, the number is now 4190, completing the conversion from scientific to ordinary decimal notation.