Question

Use the method of direct proof to prove the following statements. If \(x\) is an odd integer, then \(x^{3}\) is odd.

Short answer

By substituting \(x = 2n+1\) (odd number form) and expanding \(x^{3}\) (odd number form), it can be seen that \(x^{3}\) is also in the form of an odd number, \(2m + 1\), with \(m = 4n^{3} + 6n^{2} + 3n\). Thus, when \(x\) is odd, \(x^{3}\) is also odd.

Step-by-step solution

Defining the Knowns

First, let's define the known conditions. An odd integer \(x\) can be expressed in the format \(2n+1\), where \(n\) is an integer.

Substitution

Next, substitute \(x\) with \(2n+1\) in \(x^{3}\). That gives \((2n+1)^{3}\).

Expanding the Cube

Expand \((2n+1)^{3}\) using the binomial theorem which is \(a^{3} + 3a^{2}b + 3ab^{2} + b^{3}\). Therefore, \((2n+1)^{3}\) becomes \(8n^{3} + 12n^{2} + 6n + 1\).

Rearranging Terms

Rearrange the terms to factor out 2, thus obtaining the equation as \(2(4n^{3} + 6n^{2} + 3n) + 1\).

Iron-clad Conclusion

If \(2(4n^{3} + 6n^{2} + 3n) + 1\) is of the form \(2m + 1\), where \(m\) is an integer, then it can be concluded that \(x^{3}\) is an odd number whenever \(x\) is odd. In this case, \(m = 4n^{3} + 6n^{2} + 3n\), which is an integer because \(n\) is an integer.

Key concepts

Understanding Odd and Even Integers

Knowing the properties of odd and even integers is fundamental when working with proofs in mathematics. An even integer is any number that can be divided by two without resulting in a remainder, making it possible to be written as 2 times another integer, expressed as \(2n\) where \(n\) is an integer. An odd integer, on the other hand, has a remainder of one when divided by two. This leads to the representation of odd numbers in the form \(2n+1\).
Understanding this concept is essential because it enables us to manipulate and work with integers in proofs effectively, allowing us to demonstrate certain properties of numbers as done in our example exercise.

The Binomial Theorem Unpacked

The binomial theorem is a tool that helps us expand expressions that are raised to a power, such as \( (a + b)^n \) where \(a\) and \(b\) are any numbers and \(n\) is a non-negative integer. The theorem provides a formula for expanding these expressions into a sum involving terms of the form \(a^{n-k}b^k\), and it's a powerful ally in simplifying mathematical proofs.
In the context of our exercise, where we expand \( (2n+1)^3 \), the binomial theorem allows us to break down and restructure the cube of an odd integer, proving that the result is still an odd integer. This theorem shows its usefulness in proofs by giving a clear pathway to follow and providing the structure of our desired expression.

Proof by Substitution Technique

In mathematical proofs, proof by substitution is a common method where we replace variables with specific numbers or expressions to prove the validity of a statement. This strategy is often used when we have known information about an equation or an expression that can be replaced with another, which simplifies our task.

Application in Direct Proofs

In direct proofs like our exercise example, substitution helps us directly demonstrate a fact by transforming the original condition into a form that clearly satisfies the property we are testing. When we substitute \(x\) with \(2n+1\) in \(x^3\), it becomes a tangible expression, making the subsequent proof steps much more manageable and straightforward.

Building Blocks of Mathematical Proofs

Mathematical proofs are the cornerstone of mathematics, providing the process to validate truths within the language of mathematics. A proof is a logical argument that demonstrates the truth of a given statement, and it involves a series of logical steps supported by definitions, axioms, propositions, and theorems.
Proofs can come in various forms, such as direct proofs, indirect proofs (by contradiction), inductive proofs, and proof by counterexample. Understanding the nature and structure of proofs is crucial for anyone delving into more advanced mathematics. Direct proofs, like the one used in our exercise, are generally the most straightforward approach as they create a sequence of logical statements that directly lead from the premises to the desired conclusion, which—in our case—is the odd nature of the cube of an odd integer.