Question

Fix \(b>1\). (a) If \(m, n, p, q\) are integers, \(n>0, q>0\), and \(r=m / n=p / q\), prove that $$\left(b^{m}\right)^{1 / n}=\left(b^{p}\right)^{1 / q} .$$ Hence it makes sense to define \(b^{\prime}=\left(b^{m}\right)^{1 / n}\). (b) Prove that \(b^{\prime+1}=b^{\prime} b^{\prime}\) if \(r\) and \(s\) are rational. (c) If \(x\) is real, define \(B(x)\) to be the set of all numbers \(b^{t}\), where \(t\) is rational and \(t \leq x\). Prove that $$b^{\prime}=\sup B(r)$$ when \(r\) is rational. Hence it makes sense to define $$b^{x}=\sup B(x)$$ for every real \(x\). (d) Prove that \(b^{x+y}=b^{x} b^{y}\) for all real \(x\) and \(y\).

Short answer

Short Answer: By proving that \(\left(b^{m}\right)^{1 / n}=\left(b^{p}\right)^{1 / q}\) given \(m/n = p/q\) and defining \(b^{\prime}\) accordingly, we can then show that \(b^{\prime+1}=b^{\prime} b^{\prime}\) when \(r\) and \(s\) are rational. By defining \(B(x)\) as the set of all numbers \(b^{t}\) where \(t\) is rational and \(t \leq x\), we establish the property that \(b^{\prime}=\sup B(r)\) for rational \(r\). We proceed to define \(b^{x}=\sup B(x)\) for every real \(x\). Lastly, we proved that \(b^{x+y}=b^{x} b^{y}\) for all real \(x\) and \(y\).

Step-by-step solution

Part (a): Proving \(\left(b^{m}\right)^{1 / n}=\left(b^{p}\right)^{1 / q}\)

To prove this, first recall the properties of exponents. We are given that \(m/n = p/q\), which means that \(mq = np\). Now, since \(mq = np\), we have: $$b^{m / n} = b^{mq / nq} = \left(b^{pq}\right)^{1 / nq} = \left(b^{np}\right)^{1 / nq} = \left(b^{p}\right)^{1 / q}$$ Thus, it makes sense to define \(b^{\prime}=\left(b^{m}\right)^{1 / n}\).

Part (b): Proving \(b^{\prime+1}=b^{\prime} b^{\prime}\) if \(r\) and \(s\) are rational

If \(r\) and \(s\) are rational, then using the definition of \(b^{\prime}\) from part (a), we have: $$b^{\prime+1} = \left(b^m\right)^{\frac{1}{n}(r+s)} = \left(b^m\right)^{\frac{1}{n}(r)} \left(b^m\right)^{\frac{1}{n}(s)} = b^{\prime r} b^{\prime s} = b^{\prime} b^{\prime}$$

Part (c): Defining \(B(x)\) and proving \(b^{\prime}=\sup B(r)\) for rational \(r\)

To define \(B(x)\), let \(B(x)\) be the set of all numbers \(b^{t}\), where \(t\) is rational and \(t \leq x\). Now, we need to prove that \(b^{\prime}=\sup B(r)\) when \(r\) is rational. Consider an arbitrary \(t_1\) and \(t_2\) such that \(t_1, t_2 \in B(r)\) and \(t_1 < t_2\). Since \(t_1 \leq r\) and \(t_2 \leq r\), we have \(b^{t_1} \leq b^{\prime}\) and \(b^{t_2} \leq b^{\prime}\), from the definition of \(b^{\prime}\). Moreover, since \(r\) is rational, there are no other rational numbers between \(t_1\) and \(t_2\). Therefore, \(b^{t_1} \leq b^{\prime} \leq b^{t_2}\), implying that \(b^{\prime}\) is an upper bound for \(B(r)\). As \(b^{\prime}\) is the smallest such upper bound, we have \(b^{\prime}=\sup B(r)\). Thus, it makes sense to define \(b^{x}=\sup B(x)\) for every real \(x\).

Part (d): Proving \(b^{x+y}=b^{x} b^{y}\) for all real \(x\) and \(y\)

We want to prove that \(b^{x+y}=b^{x} b^{y}\) for all real \(x\) and \(y\). Using the definition of \(b^{x}\) from part (c), we have: $$b^{x+y} = \sup B(x+y)$$ Now, consider the sets \(B(x)\) and \(B(y)\). For every \(t_1 \in B(x)\) and \(t_2 \in B(y)\), we have \(b^{t_1} \in B(x)\) and \(b^{t_2} \in B(y)\). Therefore, by the properties of exponents, we have \(b^{t_1} b^{t_2} = b^{t_1 + t_2} \in B(x+y)\). Thus, the set \(\{b^{t_1} b^{t_2} : t_1 \in B(x), t_2 \in B(y)\}\) is a subset of \(B(x+y)\). Furthermore, for any \(t_3 \in B(x+y)\), we have \(t_3 \leq x+y\). Thus, there must exist \(t_1 \in B(x)\) and \(t_2 \in B(y)\) such that \(t_3 = b^{t_1 + t_2}\). Now, since \(t_1 \leq x\) and \(t_2 \leq y\), we have \(b^{t_1} \leq b^{x}\) and \(b^{t_2} \leq b^{y}\). Therefore, \(b^{t_3} = b^{t_1 + t_2} \leq b^{x} b^{y}\). Hence, the set \(B(x+y)\) is a subset of \(\{b^{t_1} b^{t_2} : t_1 \in B(x), t_2 \in B(y)\}\), and we have \(b^{x+y} = b^{x} b^{y}\) for all real \(x\) and \(y\) as required.

Key concepts

Real Analysis

Real Analysis is a branch of mathematics dealing with the set of real numbers and the functions of real variables. It provides a rigorous framework for discussing limits, continuity, differentiation, and integration. The core focus of real analysis is to study how real numbers behave and how they are used to describe real-world phenomena through precise mathematical formulations.
The exercise we are exploring uses real analysis to establish how exponential functions work with real numbers. The fundamental properties and definitions are grounded in real analysis, especially when extending concepts from rational to real numbers.
In this context, understanding limits and the notion of the supremum (which we will explore further) is crucial as it supports why we can define exponentiation even for irrational and real numbers beyond simple rational arithmetic.

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction of two integers, specifically in the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b eq 0\). They play a significant role in this exercise as the stepping stone for defining exponentiation for real numbers.
In the provided problem, rational numbers are used to construct sequences that approach real numbers. By leveraging the density of rational numbers in real numbers, we extend exponentiation from the rational domain to the whole set of real numbers. This technique involves using a rational approximation to define exponentiation rigorously.
Thus, rational numbers provide a bridge between simple integer exponentiation and the more complex real number exponentiation, using fraction-based powers to gradually define real number powers.

Properties of Exponents

The properties of exponents are essential rules that govern operations involving exponential expressions. They allow us to simplify expressions involving powers with various bases and exponents. The fundamental properties include:

• Product of powers: \(a^m \cdot a^n = a^{m+n}\)
• Power of a power: \((a^m)^n = a^{m\cdot n}\)
• Power of a product: \((ab)^m = a^m \cdot b^m\)
• Zero exponent: \(a^0 = 1\) for any non-zero \(a\)

These properties allow us to manipulate expressions in this exercise, simplifying the proofs and extending results from rational to real exponents. For instance, in proving \(b^{x+y} = b^x b^y\), the properties of exponents are applied through rational approximations to the sum \(x+y\), ensuring consistent results for real numbers.

Supremum

Supremum is a concept from real analysis that represents the least upper bound of a set. If you have a set of numbers, the supremum is the smallest number that is greater than or equals all numbers in the set, even if it is not an element of this set. This is essential when we try to define limits or understand boundedness within the context of real numbers.
In this exercise, the concept of supremum is used to define the exponentiation of non-rational real numbers through the notion of the set \(B(x)\), as it captures the idea of approaching but never exceeding \(b^x\) for real \(x\). The definition of \(b^x = \sup B(x)\) ensures that the exponential function is well-defined for irrational and real numbers, leveraging the completeness property of real numbers.

Exponentiation of Real Numbers

Exponentiation of real numbers extends the idea of multiplying a base number by itself a certain number of times, even when this number isn't whole. This includes irrational and complex expressions. It involves using limits and the concept of supremum to ensure that powers of numbers extend consistently beyond simple integer or rational exponents.
In this exercise, defining \(b^x\) as the supremum of \(B(x)\) for any real number \(x\), where \(B(x)\) consists of powers \(b^t\) with \(t \leq x\), guarantees continuity and consistency of exponential expressions. This approach allows exponential functions to accurately model growth and decay processes in nature and science, where inputs aren't limited to neat fractions but exist in the realm of real numbers.
Real number exponentiation is thus a pivotal concept that builds on simpler exponent rules and rational approximations, expanding them to fully encompass the complexity and richness of the real number line.