Problem 73 - Calculus Early Transcendentals: Pearson New International Edition

Question

. Prove the following statements. (a) If \(n\) is odd, then \(n^{2}\) is odd. (Hint: If \(n\) is odd, then there exists an integer \(k\) such that \(n=2 k+1 .\) ) (b) If \(n^{2}\) is odd, then \(n\) is odd. (Hint: Prove the contrapositive.)

Step-by-Step Solution

Answer

(a) If \( n \) is odd, then \( n^2 \) is odd. (b) If \( n^2 \) is odd, then \( n \) is odd.

Step 1: Understanding the Problem for (a)

We need to prove that if an integer \( n \) is odd, then \( n^2 \) is also odd. An odd integer can be represented in the form \( n = 2k + 1 \) where \( k \) is an integer.

Step 2: Substituting the Odd Expression

Substitute the expression for odd \( n \) into \( n^2 \): \[ n = 2k + 1 \quad \Rightarrow \quad n^2 = (2k + 1)^2. \]

Step 3: Expanding the Expression

Expand \( (2k + 1)^2 \): \[ n^2 = (2k + 1)(2k + 1) = 4k^2 + 4k + 1. \]

Step 4: Simplifying

The expanded form \( 4k^2 + 4k + 1 \) can be written as \( 2(2k^2 + 2k) + 1 \), which is clearly in the form of an odd integer (i.e., \( 2m + 1 \), where \( m \) is an integer). Hence, \( n^2 \) is odd.

Step 5: Understanding the Problem for (b)

We need to prove that if \( n^2 \) is odd, then \( n \) is odd. We will prove this by proving the contrapositive: If \( n \) is even, then \( n^2 \) is even.

Step 6: Expression for Even Integer

An even integer can be expressed as \( n = 2k \) for some integer \( k \). Now, compute \( n^2 \): \[ n^2 = (2k)^2 = 4k^2. \]

Step 7: Conclusion of Even Result

Since \( 4k^2 = 2(2k^2) \), \( n^2 \) is clearly even as it is divisible by 2. Thus, by proving the contrapositive, if \( n^2 \) is odd, then \( n \) must be odd.

Key Concepts

Odd and Even Numbers

In number theory, an integer is categorized as either odd or even based on its divisibility by 2. A number is even if it can be divided by 2 without leaving a remainder, such as 2, 4, 6, etc. These numbers take the form of \(n = 2k\), where \(k\) is an integer.

On the other hand, an odd number divides by 2 leaving a remainder of 1. Examples include 1, 3, 5, etc., and they can be expressed as \(n = 2k + 1\). This expression helps in many mathematical proofs because it provides a standard way to write any odd integer. By knowing these basics, we can delve into mathematical proofs concerning these numbers.

• Odd numbers: \(n = 2k + 1\)
• Even numbers: \(n = 2k\)

Selecting the correct form helps in proving more complex properties about numbers.

Contrapositive Proof

Contrapositive is a fundamental logical method in mathematics used to prove conditional statements. It involves proving that if \(Q\) is false, then \(P\) is false when the statement is originally \("If P, then Q"\).

Thus, a contrapositive transforms "If \(n^2\) is odd, then \(n\) is odd" into "If \(n\) is even, then \(n^2\) is even." This approach leverages the idea that proving the contrapositive is logically equivalent to proving the original statement.

• Original statement: "If \(P\), then \(Q\)"
• Contrapositive: "If not \(Q\), then not \(P\)"

By leveraging the symmetry in logic, contrapositive proofs can simplify the problem and make it easier to handle.

Integer Representation

In mathematical proofs, representations like \(n = 2k\) for even numbers and \(n = 2k + 1\) for odd numbers are crucial. These expressions simplify the formulation of proofs and allow us to describe properties like parity (odd or even status) of integers.

Representing integers in these forms helps to clearly identify their behavior under mathematical operations. For example, the square of an odd integer can be expressed as:

• \((2k + 1)^2 = 4k^2 + 4k + 1\)

which is structured to reveal its odd nature readily.
This method of representation provides clarity and is a cornerstone technique within proofs involving integer arithmetic. It's important because it connects abstract concepts with tangible structures in mathematics.

Square of a Number

The square of a number is a fundamental concept, especially in proofs about integers. For any integer \(n\), its square \(n^2\) involves the number multiplying itself. This operation significantly affects whether the result is odd or even.

Squaring an even integer (e.g., \(n = 2k\)) results in an even square, \(n^2 = 4k^2\), which is always divisible by 2.
However, squaring an odd integer, represented as \(n = 2k + 1\), results in \(n^2 = 4k^2 + 4k + 1\). Here, the expression is of the form \(2m + 1\), verifying the square is odd.

• Square of an even number: \(2k\) leads to \(4k^2\)
• Square of an odd number: \(2k + 1\) leads to \(4k^2 + 4k + 1\)

This understanding is vital in constructing mathematical proofs concerning parity and other properties.