Problem 29 - Algebra for College Students

Question

Write each of the following in ordinary decimal notation. For example, \((3.18)(10)^{2}=318\). \((9.14)(10)^{-4}\)

Step-by-Step Solution

Answer

0.000914

Step 1: Understanding the Scientific Notation

The expression given is \((9.14)(10)^{-4}\). This means that the decimal point in the number 9.14 has to be moved 4 places to the left, because the exponent is negative.

Step 2: Moving the Decimal Point

Start with the number 9.14. Move the decimal point 4 places to the left. Since there aren't enough digits to the left of the decimal point, add zeros as placeholders.

Step 3: Achieving Ordinary Decimal Notation

When moving the decimal point 4 places to the left in 9.14, you end up with 0.000914. This is because each move requires placing a zero to fill in the gap.

Key Concepts

Decimal Notation

Decimal notation is a way of writing numbers that are standard in everyday mathematics. It's what most people think of as regular numbers. In decimal notation, values are expressed as whole numbers and fractions separated by a decimal point. For instance, the number 9.14 consists of a whole number part, 9, and a decimal part, 0.14. Decimal notation is used when you want to write numbers in their most common form without relying on scientific notation or fractions.

Decimal notation is especially useful in various practical scenarios:

• In financial transactions where precision is key.
• In displaying measurements like 2.75 inches or 3.56 liters.
• When solving mathematical problems requiring accuracy to several decimal places.

Understanding decimal notation is crucial as it simplifies complex values and communicates them in an accessible format. This forms the backbone of more advanced mathematical concepts that build on basic number representation.

Negative Exponents

Negative exponents indicate a reciprocal relationship with powers of ten. In simple terms, a negative exponent tells us to divide by a power of ten instead of multiplying. For instance, 10^{-4} means you divide by 10 four times, which is the same as multiplying by the reciprocal, \(\frac{1}{10^4}\).
In our example, when we have \((9.14)(10)^{-4}\), the negative exponent indicates that the number is smaller than 9.14, specifically by a factor of 10, four times:

• \(10^{-1} = \frac{1}{10} = 0.1\)
• \(10^{-2} = \frac{1}{100} = 0.01\)
• \(10^{-3} = \frac{1}{1000} = 0.001\)
• \(10^{-4} = \frac{1}{10000} = 0.0001\)

When dealing with negative exponents, remember to move the decimal point to the left. This adjustment is key to converting numbers into their decimal equivalents accurately.

Multiplication by Powers of Ten

Multiplication by powers of ten is a foundational concept in mathematics that makes it easier to convert and manipulate numbers. When you multiply a number by attractive powers of ten, such as 10, 100, or 1000, the number simply becomes larger. Conversely, with negative exponents, it becomes smaller.
Here's how it works:

• For every positive exponent, move the decimal point to the right.
• For every negative exponent, move the decimal point to the left.

In our case of \((9.14)(10)^{-4}\), since the exponent is -4, we move the decimal point four places to the left. This process either adds or removes zeros dependent on the exponent's sign and size.
Multiplication by powers of ten is straightforward but powerful, enabling the easy transformation and interpretation of large and small numbers alike.