Question

Prove that \(n\) is odd if and only if \(n^{2}\) is odd. (See Problem \(73 .\) )

Short answer

An integer \( n \) is odd if and only if \( n^2 \) is odd, as shown by proving both directions of the statement.

Step-by-step solution

Understand the Problem Statement

We need to prove that an integer \( n \) is odd if and only if \( n^2 \) is odd. This is a biconditional statement, meaning we have to show two directions: (1) If \( n \) is odd, then \( n^2 \) is odd, and (2) If \( n^2 \) is odd, then \( n \) is odd.

Prove that if n is odd, n^2 is odd

Assume \( n \) is an odd integer. By definition of odd numbers, \( n = 2k + 1 \) for some integer \( k \). Calculate \( n^2 \):\[n^2 = (2k + 1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1.\]Since \( 2(2k^2 + 2k) \) is clearly an even number (as it is a multiple of 2), adding 1 makes \( n^2 \) odd. Therefore, if \( n \) is odd, then \( n^2 \) is also odd.

Prove that if n^2 is odd, n is odd

Assume \( n^2 \) is odd. We need to show that \( n \) must be odd. Suppose for contradiction that \( n \) is even, then \( n = 2m \) for some integer \( m \). Calculate \( n^2 \):\[n^2 = (2m)^2 = 4m^2 = 2(2m^2).\]This expression shows that \( n^2 \) would be even if \( n \) is even. But this contradicts the assumption that \( n^2 \) is odd, hence \( n \) cannot be even and must be odd.

Conclusion

We have shown both directions of the biconditional statement: If \( n \) is odd, then \( n^2 \) is odd, and if \( n^2 \) is odd, then \( n \) is odd. Therefore, \( n \) is odd if and only if \( n^2 \) is odd.

Key concepts

Odd and Even Numbers

Numbers can be categorized as odd or even based on their divisibility by 2. An even number is one that can be divided by 2 without leaving any remainder. For instance, numbers like 2, 4, 6 are even. Odd numbers, however, leave a remainder of 1 when divided by 2, examples include 1, 3, 5. Understanding this difference is essential because it determines how these numbers interact in mathematical operations.

When you square an odd number, the result is always odd. This happens because of the inherent properties of these numbers. For example, if you start with any odd number, you can express it in the form of \( n = 2k + 1 \), where \( k \) is an integer. Its square will always end up as \( n^2 = 2m + 1 \) for some integer \( m \). This formula shows that the square of an odd number remains odd because it retains the form where 1 is added to a multiple of 2, making it odd.
Understanding these basic properties helps in recognizing patterns or proving theorems involving odd and even numbers.

Biconditional Statements

Biconditional statements are logical constructs used in mathematics to express that two statements are both true or both false. They are often written using "if and only if" or the symbol \( \iff \). In this particular exercise, the statement "\( n \) is odd if and only if \( n^2 \) is odd" is a biconditional statement.

To prove a biconditional statement, you need to demonstrate that both conditions imply each other. This involves showing two separate proofs:

• The first proof (\( A \) implies \( B \)) shows that if the first condition is true, then the second must also be true.
• The second proof (\( B \) implies \( A \)) demonstrates that if the second condition is true, then the first one must also be true.

Successfully proving both directions is what establishes the biconditional relationship.
In terms of logic, biconditional statements create a very strong connection between the concepts involved, which is crucial for solid theorem proving in mathematics.

Integer Properties

Integers are whole numbers that can be positive, negative, or zero. They form a fundamental part of number theory, and understanding their properties is key in solving many mathematical problems.

Odd and even numbers discussed previously are both types of integers defined by their divisibility by 2. But there is much more to integer properties than just divisibility. For proving mathematical concepts involving integers, knowing their basic operations and their properties helps immensely.

• Closure: Integers are closed under addition, subtraction, and multiplication. This means that if you add, subtract, or multiply any two integers, the result is always an integer.
• Associative and Commutative properties: These are crucial for simplifying expressions and proving statements.
• Identity and Inverse Elements: For integers, 0 is the additive identity because adding 0 to any integer doesn’t change its value. Similarly, 1 is the multiplicative identity.

Understanding these properties is important when working with integers, particularly when trying to prove theorems about their behavior, such as the exercise we discussed concerning the parity (oddness or evenness) of integers and their squares.