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Chapter 2: Problem 4

If \(a \in \mathbb{R}\) satisfies \(a \cdot a=a\), prove that either \(a=0\) or \(a=1\).

Short Answer

Expert verified
The real numbers \(a\) that satisfy \(a \cdot a = a\) are \(a = 0\) and \(a = 1\).

Step by step solution

01

Formulate the Equation

The given condition is that \(a \cdot a = a\) for some real number \(a\). This can be rearranged and written as a quadratic equation. Move all terms to one side of the equality sign: \(a^2 - a = 0\).
02

Factoring the Quadratic Equation

Factor the quadratic equation \(a^2 - a = 0\). A common factor of \(a\) can be taken out, resulting in: \(a(a - 1) = 0\). In this factored form, it becomes clear that if the product is zero, then either \(a = 0\) or \(a - 1 = 0\).
03

Solve for a

Now, simply solve the equations obtained in the previous step: \(a = 0\) or \(a - 1 = 0\). The latter gives \(a = 1\). Therefore, the solutions for \(a\) that satisfy the original equation \(a \cdot a = a\) are \(a = 0\) and \(a = 1\).

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