Problem 67 - Thinking Mathematically

Question

Express each expanded form as a Hindu-Arabic numeral. \(\left(7 \times 10^{-1}\right)+\left(2 \times 10^{-4}\right)+\left(3 \times 10^{-6}\right)\)

Step-by-Step Solution

Answer

The Hindu-Arabic numeral of given expanded form is 0.700203.

Step 1: Understand the values of negative powers of ten

For any value of \( n \), \( 10^{-n} \) will be 1 divided by 10 raised to the power \( n \). Meaning \( 10^{-n} = \frac{1}{10^n} \). So the following conversions take place: \( 10^{-1} = 0.1 \), \( 10^{-4} = 0.0001 \), and \( 10^{-6} = 0.000001 \)

Step 2: Convert expanded form into numbers

To convert the expanded form into numbers, multiply each numeral by its respective power of 10. So:\( 7 \times 10^{-1} = 7 \times 0.1 = 0.7 \),\( 2 \times 10^{-4} = 2 \times 0.0001 = 0.0002 \),\( 3 \times 10^{-6} = 3 \times 0.000001 = 0.000003 \)

Step 3: Add up the numbers

Add the numbers from step 2 to get the final Hindu-Arabic number:\(0.7 + 0.0002 + 0.000003 = 0.700203 \)

Key Concepts

Negative Powers of Ten

When we talk about negative powers of ten, we're diving into the world of tiny numbers. In mathematics, any base number raised to a negative exponent represents the reciprocal of that base raised to the opposite positive exponent. Simply put, if you have a number like \(10^{-n}\), it's equal to \(\frac{1}{10^n}\). For example:\

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• \(10^{-1} = \frac{1}{10^1} = 0.1\)
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• \(10^{-4} = \frac{1}{10^4} = 0.0001\)
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• \(10^{-6} = \frac{1}{10^6} = 0.000001\)
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\This conversion process helps us break down expressions with negative powers into more manageable decimal numbers, making calculations simpler. Understanding this concept is crucial because it enables us to express numbers that are less than one in a precise and efficient way.

Expanded Form

The expanded form is a way of writing numbers that helps to see the value of each individual digit. It's like unpacking a number so that all its components are visible. Writing a number in an expanded form involves breaking it down into its separate parts, each represented by its digit multiplied by its power of ten.\
\Expanded notation is particularly useful in illustrating how small decimal numbers can be expressed in terms of powers of ten. For example, when dealing with \((7 \times 10^{-1}) + (2 \times 10^{-4}) + (3 \times 10^{-6})\), each term shows the digit multiplied by a specific negative power of ten.\
\This visualization helps you understand exactly how much each digit contributes to the total number. It emphasizes the scale and size of each amount, enabling a clearer comprehension of the number’s composition.

Mathematical Conversion

Mathematical conversion is the process of changing numbers from one form to another while preserving their value. This could involve converting expressions in expanded form into Hindu-Arabic numerals.\
\For instance, in the expression \((7 \times 10^{-1}) + (2 \times 10^{-4}) + (3 \times 10^{-6})\), each component must be transformed to a decimal number by calculating the value of each term. This involves multiplying the digit by the value of the negative power of ten like so:\

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• \(7 \times 10^{-1} = 0.7\)
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• \(2 \times 10^{-4} = 0.0002\)
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• \(3 \times 10^{-6} = 0.000003\)
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\Once each part is calculated, add them together: \(0.7 + 0.0002 + 0.000003 = 0.700203\). This sum is the original number in its concise Hindu-Arabic numeral form. Through this method, conversion clarifies complex mathematical expressions into readable numbers that are easier to understand.