Question

Let \(f(x+y)=f(x)+f(y)\) for all \(x\) and \(y\). Prove that there is a number \(m\) such that \(f(t)=m t\) for all rational numbers t. Hint: First decide what \(m\) has to be. Then proceed in steps, starting with \(f(0)=0, f(p)=m p\) for a natural number \(p\), \(f(1 / p)=m / p\), and so on.

Short answer

For all rational numbers \(t\), \(f(t) = m t\) where \(m = f(1)\).

Step-by-step solution

Determine Basic Property of f

We are given that \( f(x + y) = f(x) + f(y) \) for all \(x\) and \(y\). If we set \(x = y = 0\), then \(f(0 + 0) = f(0) + f(0)\), which simplifies to \(f(0) = 0\).

Determine f for Natural Numbers

For a natural number \(n\), consider \(f(n) = f(1 + 1 + \ldots + 1)\) (n times). Using the given functional equation iteratively, \(f(n) = f(1) + f(1) + \ldots + f(1) = n \cdot f(1)\). Thus, \(f(n) = m \cdot n\) where \(m = f(1)\).

Determine f for Rational Fractions

Now consider a rational fraction of the form \( \frac{1}{n} \) for a natural number \(n\). Observe that \(n \cdot f\left(\frac{1}{n}\right) = f(n \cdot \frac{1}{n}) = f(1) = m\). Therefore, \(f\left(\frac{1}{n}\right) = \frac{m}{n}\).

Extend to All Rational Numbers

Now consider a general rational number \(t = \frac{p}{q}\), where \(p\) and \(q\) are integers with \(q eq 0\). We have \(f\left(\frac{p}{q}\right) = p \cdot f\left(\frac{1}{q}\right) = p \cdot \frac{m}{q} = m \cdot \frac{p}{q} = m \cdot t\). Therefore, \(f(t) = m t\) for any rational \(t\).

Key concepts

Additive Functions

Additive functions are a fascinating concept in mathematics where a function, say \( f(x) \), satisfies the condition that \( f(x + y) = f(x) + f(y) \) for every value of \( x \) and \( y \). This property is quite significant because it implies that the function maintains addition between inputs. Understanding this helps simplify complex equations because, essentially, it means the function scales or extends linearly when summing its inputs.

To prove properties of additive functions, such as linearity on rational numbers, mathematicians often begin by evaluating the function at known base cases, such as \( f(0) = 0 \). This is a critical step because it establishes the baseline behavior of the function. Subsequent conclusions about the function are inferred by treating each component in stages—progressively expanding from known integers to fractions and, eventually, to all rational numbers.

This type of function becomes particularly meaningful when handling problems involving series or sequences where preserving the structure of addition is imperative. Additive functions are also foundational to more advanced concepts such as vector spaces in linear algebra.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction \( \frac{p}{q} \) of two integers, where \( p \) is the numerator, \( q \) is the non-zero denominator, and this concept is pivotal in various branches of mathematics.

These numbers include integers (since any integer \( n \) can be expressed as \( \frac{n}{1} \)) as well as fractions like \( \frac{1}{2} \), \( \frac{3}{4} \), and negative values like \( -\frac{7}{8} \). Rational numbers have a dense nature in mathematics; between any two rational numbers, another rational number exists. This density is what makes them so useful in approaching continuous values and their behavior under functional constraints, such as those in additive functions.

In solving functional equations, understanding how they apply to rational numbers allows one to extend proven properties from clear cases—like integers—to more complex cases. This is achieved using properties like the standard arithmetic and multiplicative inverses, enabling equations involving these numbers to be solved universally within their defined field without loss of generality.

Mathematical Proofs

Mathematical proofs are a logical process used to establish the truth of a statement or theorem, and they are an essential component in validating mathematical concepts, such as those involving functional equations.

Proofs can be structured in various forms, including direct proof, proof by contradiction, and proof by induction, each appropriate for different kinds of problems. For example, in proving that a function is additive across rational numbers, a direct proof is typically used. By starting from a given hypothesis (like \( f(x + y) = f(x) + f(y) \)) and logically deducing the necessary outcome (such as showing \( f(t) = mt \)), mathematicians can convince others of a functional relationship's correctness.

Additionally, constructing proofs involves a deep understanding of properties and behaviors of functions over different domains, like rational numbers, to systematically and incrementally build an argument. Mathematical proofs not only translate precise theoretical claims into credible assertions but also enrich our comprehension of how mathematical structures operate and interrelate within the number systems.