Question

Use the method of direct proof to prove the following statements.. If \(x\) is an even integer, then \(x^{2}\) is even.

Short answer

The proof that if an integer \(x\) is even, its square \(x^{2}\) is also even, was achieved by expressing the even-number property \(x = 2k\), substituting into the square equation to get \(4k^{2}\), which also has the even-number property. Thus proving the given statement.

Step-by-step solution

Define Even Number

An even number is any integer that can be expressed in the form \(2k\) where \(k\) is also an integer. So let's assume \(x\) to be an even integer. Therefore, we can represent it as \(x = 2k\). Here, the integer \(k\) represents any integer.

Substitution

The next step is to substitute \(x\) in the expression \(x^{2}\). Thus, replacing \(x\) with \(2k\) in \(x^{2}\), we get \((2k)^2\).

Simplify Expression

Simplify \((2k)^2\) to \(4k^{2}\). This expression can also be written as \(2(2k^{2})\).

Conclude the Proof

As the expression \(2(2k^{2})\) fits the definition of an even number (a number that can be written as \(2m\), where \(m\) is an integer), we can say that \((2k)^2\) is an even number. Meaning, when \(x\) is an even integer, \(x^{2}\) is also an even integer. Therefore, the statement 'If \(x\) is an even integer, then \(x^{2}\) is even' is proven to be true.

Key concepts

Even Integer

An even integer is one of the most fundamental concepts in mathematics. It is defined as any integer that can be exactly divided by 2, resulting in another integer. You may think of even integers as those numbers you would count while skipping every other number: 0, 2, 4, 6, 8, and so on. Each one of these can be written in the form of (2k), where k is itself an integer.

Understanding this is crucial when we proceed to explore how evenness behaves under various operations, such as squaring the number. This concept attaches to the idea of parity, where numbers are either even or odd, and it is vital in proving certain properties about integers in algebra and number theory.

Mathematical Proof

A mathematical proof is a logical argument that establishes the truth of a mathematical statement. This process is the bedrock of mathematics and involves creating a series of steps, each logically derived from the previous one, leading to the inevitable conclusion that the statement in question is true. Serving as the backbone of certainty in mathematics, proofs can come in many forms – direct proofs, indirect proofs, proof by contradiction, and so on. Each of these methods has its own usage, and selecting the right technique is crucial in effectively establishing a statement's validity.

In our example, the direct proof technique logic follows from the assumption that we start with what we know – in this case, that we have an even integer – and demonstrate that the statement follows naturally from this by using algebra and logic.

Algebraic Expression

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (such as addition, subtraction, multiplication, and division). The power of algebraic expressions lies in their ability to generalize mathematical relationships and patterns. For instance, when we talk about even integers as 2k, the '2k' is an algebraic expression.

Manipulating these expressions allows mathematicians to explore and prove properties of numbers and equations. In the proof we're discussing, squaring the expression that represents an even integer is the central algebraic operation. In this case, the algebraic expression grows from x, to x^2, and finally to its simplified form 4k^2, revealing more about the nature of the number in question.

Proof Techniques

Proof techniques are the various strategies used to prove mathematical statements. Direct proof, the technique used in our example, is one of these strategies. This approach involves starting with known facts or axioms and applying logical steps to arrive at the conclusion that a particular statement is true.

Other proof techniques include but are not limited to, proof by induction (used for statements that involve a sequence or series), proof by contradiction (where you assume the opposite of what you're trying to prove and show a contradiction occurs), and proof by contrapositive (where you prove that if the conclusion is false, then the premise must also be false). Understanding when and how to use these techniques allows us to uncover and affirm new truths in mathematics effectively.