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Chapter 2: Problem 40

Fix an integer \(N\), and call a real number \(x\) representable if \(x=q 2^{n}\), where \(1 / 2 \leq q<\) 1 and \(|n| \leq N\). Prove that if \(x_{1}, x_{2}, \ldots, x_{k}\) are representable and if their product is representable, then so is \(u v\), where \(u=\max \left(x_{i}\right)\) and \(v=\min \left(x_{i}\right)\).

Short Answer

Expert verified
If the product of representable numbers is representable, the product of their max and min is also representable.

Step by step solution

01

Understanding Representability

A real number \(x\) is representable if it can be expressed in the form \(x = q \cdot 2^n\), where \( \frac{1}{2} \leq q < 1\) and \(|n| \leq N\). This definition implies that \(x\) is a dyadic rational number between \(2^{-N}\) and \(2^N\), scaled by powers of 2.
02

Recognizing Product Conditions

Assume that \(x_1, x_2, \ldots, x_k\) are representable numbers and their product \(x_1x_2\cdots x_k\) is also representable. Then there exist integers \(m_i\) and rational numbers \(q_i\) for each \(x_i = q_i \cdot 2^{m_i}\) such that their collective product is also representable.
03

Define Maximum and Minimum

Let \(u = \max(x_1, x_2, \ldots, x_k)\) and \(v = \min(x_1, x_2, \ldots, x_k)\). By definition, there exist \(i, j\) such that \(u = x_i = q_i \cdot 2^{m_i}\) and \(v = x_j = q_j \cdot 2^{m_j}\).
04

Product of Maximum and Minimum

Calculate the product \(uv = (q_i \cdot 2^{m_i})(q_j \cdot 2^{m_j}) = (q_i q_j) \cdot 2^{m_i + m_j}\). Here, \(q_i q_j\) is a rational number since both \(q_i\) and \(q_j\) are rational and \(m_i + m_j\) is an integer.
05

Ensuring q_i q_j in Range

The product \(q_i q_j\) must satisfy \(\frac{1}{2} \leq q_i q_j < 1\). Since \(q_i\) and \(q_j\) are both at least \(\frac{1}{2}\), their product \(q_i q_j\) will be at least \(\frac{1}{4}\), meeting half of the condition. Depending on further constraints given by specific \(N\) values, additional checks ensure \(q_i q_j < 1\). If it doesn't naturally fall into this range, adjust using normalization factors or scaling adjustments which comply with conditions.
06

Final Step: Concluding Representability

Because adjustments or natural conditions (like minimums being above \(\frac{1}{2}\)) ensure \(\frac{1}{2} \leq q_i q_j < 1\) and \(|m_i + m_j| \leq N\), the product \(uv = (q_i q_j) \cdot 2^{m_i + m_j}\) satisfies the criteria for representability.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dyadic Rational Numbers
Dyadic rational numbers are a special type of rational number that can be expressed as a fraction, where the denominator is a power of two. In our context, a number is representable if it can be expressed in the form \( x = q \cdot 2^n \), where \( \frac{1}{2} \leq q < 1 \) and \( |n| \leq N \). These numbers lie between \( 2^{-N} \) and \( 2^N \), making them quite useful for certain kinds of digital computations and representations.
  • They are easy to manipulate computationally because binary systems naturally align with powers of two.
  • All dyadic rationals within a certain range are finite decimal fractions in binary, simplifying arithmetic operations.
  • Because these numbers are dense, many types of values can be neatly approximated within a given range.
Understanding dyadic rational numbers is integral to solving our problem, as it underlies the concept of representability, describing how these numbers can be scaled and combined while remaining within a specific bounds.
Mathematical Proof Strategies
Mathematical proof strategies are essential tools for verifying whether certain conditions hold true. In this problem, we engage with a structured proof to show that if a set of numbers is representable, then the maximum and minimum of that set must also be representable when multiplied. Here鈥檚 a breakdown of the approach:
  • Assumption: Begin by assuming what is already known鈥攅ach number \( x_i \) within the set is expressible in the form of \( x_i = q_i \cdot 2^{m_i} \).
  • Finding Extremes: Determine the maximum \( u \) and minimum \( v \) of these numbers, ensuring they adhere to the original assumption structure.
  • Calculate Product: Multiply these extremes to find \( uv = (q_i q_j) \cdot 2^{m_i + m_j} \).
  • Verify Conditions: Check that the product satisfies the same conditions: \( \frac{1}{2} \leq q_i q_j < 1 \) and \(|m_i + m_j| \leq N\).
By logically and sequentially verifying each step of the process, this strategy allows us to conclude with certainty about the representability of the product of maximum and minimum values. This systematic approach demonstrates the power and necessity of structured proof strategies in mathematics.
Scaling by Powers of Two
Scaling by powers of two is a fundamental concept in this exercise because it influences both the representation and manipulation of numbers. The structure \( x = q \cdot 2^n \) naturally allows the transformation of numbers by merely adjusting their exponent. Here's why this is so crucial:
  • Simplicity: Multiplying by powers of two simplifies the computation and helps maintain binary system accuracy and efficiency.
  • Range Management: The ability to scale up or down by powers of two provides great control over the numerical range within constraints like \( |n| \leq N \).
  • Representation Integrity: Maintaining a consistent framework for representation ensures that the deep mathematical properties are preserved even after operations like multiplication.
In our proof, when we compute the product of representable values, scaling by powers of two ensures the product maintains its representability form \((q_i q_j) \cdot 2^{m_i + m_j}\). This property not only helps in calculations but also assures the reliability of the results within the defined bounds, reinforcing both the understanding and application of scaling in powers of two.

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Most popular questions from this chapter

Consider the function \(f(x)=x^{-1}(1-\cos x)\). a. What is the correct definition of \(f(0) ;\) that is, the value that makes \(f\) continuous? b. Near what points is there a loss of significance if the given formula is used? c. How can we circumvent the difficulty in part \(\mathbf{b}\) ? Find a method that does not use the Taylor series. d. If the new formula that you gave in part \(\mathbf{c}\) involves subtractive cancellation at some other point, describe how to avoid that difficulty.

Suppose that \(x\) is a machine number in the range \(-\infty

Let \(f(x)=-e^{-2 x}+e^{x}\). Which of these formulas \(x, 3 x, 3 x(1-x / 2), 2-3 x\), or \(e^{x}\left(1-e^{-3 x}\right)\) for \(f\) is most accurate for small values of \(x\) ?

Discuss the calculation of \(e^{-x}\) for \(x>0\) from the series $$e^{-x}=1-x+\frac{x^{2}}{2 !}-\frac{x^{3}}{3 !}+\cdots$$ Suggest a better way, assuming that the system function for \(e^{x}\) is not available.

Find a suitable trigonometric identity so that \(1-\cos x\) can be accurately computed for small \(x\) with calls to the system functions for \(\sin x\) or \(\cos x .\) (There are two good answers.)
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