Problem 65 - Calculus Single Variable

Question

$$ \alpha=1728148040-140634693 \sqrt{151} $$ and $$ \beta=1728148040+140634693 \sqrt{151} $$ A direct computation (with integers) shows that \(\alpha \cdot \beta=1\). Calculate decimal representations of \(\alpha\) and \(\beta .\) Use them to compute the product \(\alpha \cdot \beta .\) Note: To twenty significant digits, \(140634693 \sqrt{151}\) equals \(1728148039.9999999997 .\) Calculators that do not record enough significant digits will give 0 for the value of \(\alpha\).

Step-by-Step Solution

Answer

The product \( \alpha \cdot \beta \) using decimal representations is approximately 1.

Step 1: Calculate Decimal Representation of \( \alpha \)

To find the decimal representation of \( \alpha \), we substitute the given value for \( 140634693 \sqrt{151} \) into \( \alpha = 1728148040 - 140634693 \sqrt{151} \). This gives us \( \alpha = 1728148040 - 1728148039.9999999997 \).Simplifying this, we have \( \alpha \approx 0.0000000003 \).

Step 2: Calculate Decimal Representation of \( \beta \)

Similarly, to find \( \beta \), we substitute the same value into \( \beta = 1728148040 + 140634693 \sqrt{151} \). This gives us \( \beta = 1728148040 + 1728148039.9999999997 \).Simplifying this, \( \beta \approx 3456296079.9999999997 \).

Step 3: Compute the Product \( \alpha \cdot \beta \)

Now we compute \( \alpha \cdot \beta \) using their decimal representations. We have:\[0.0000000003 \times 3456296079.9999999997 \approx 1.0368888239 \approx 1\]This confirms the provided direct computation that \( \alpha \cdot \beta = 1 \).

Key Concepts

Decimal Representation

Decimal representation is a way to express numbers in their base 10 form, which is most commonly used in everyday arithmetic. When we translate numbers that involve roots or irrational components into their decimal forms, we are approximating the values with as many digits as needed to provide a clear and accurate representation. This is crucial, especially when working with very small or very large numbers, to understand their scale and adjust calculations accordingly.

In the given exercise, \( \alpha = 1728148040 - 140634693 \sqrt{151} \)becomes approximately \( \alpha \approx 0.0000000003 \).Similarly,\( \beta = 1728148040 + 140634693 \sqrt{151} \)translates to \( \beta \approx 3456296079.9999999997 \).Both these expressions use subtraction and addition to incorporate the decimal approximation of\( 140634693 \times \sqrt{151} \) which is given as \( 1728148039.9999999997 \).When numbers are dense with digits, decimal representation helps make them more manageable.

Significant Digits

Understanding significant digits is essential for precision in calculations. They refer to the number of meaningful digits in a number, contributing to its accuracy. More significant digits imply higher precision.

In calculations, maintaining the correct number of significant digits is crucial when numbers are added, subtracted, multiplied, or divided, especially in scientific and engineering contexts.

In the exercise, knowing that\( 140634693 \sqrt{151} = 1728148039.9999999997\)with twenty significant digits, we notice the importance of these digits in the subtraction involving\( \alpha \).The value showcases that\( 0.0000000003 \)is not simply zero, emphasizing the precision required to see the true value of\( \alpha \).If fewer significant digits were used, the result could misleadingly appear as zero, highlighting why precision is paramount in such operations.

Algebraic Calculation

Algebraic calculations are foundational in resolving mathematical problems by manipulating and simplifying expressions using operations and symbols. In our exercise, to find the product \( \alpha \cdot \beta \), we apply the calculations directly corrected to align with algebraic principles.

By substituting the accurate decimal representations of\( \alpha \approx 0.0000000003 \)and\( \beta \approx 3456296079.9999999997 \),we can perform the multiplication reliably:\[0.0000000003 \times 3456296079.9999999997 \approx 1.0368888239\].This approximates to 1, which is already known through integer computation.This serves as a practical example of how accurate algebraic calculations affirm the result through decimal approximations.