A student claims that the equation has no solution, since the square root of a negative number does not exist. Why is this argument wrong?

Imaginary numbers.

Unreal versus real numbers.

The student's argument is incorrect because it is based on a misunderstanding of the operations involving imaginary numbers and complex numbers.

To explain why this argument is wrong, let's break it down step by step:

1. First, let's consider the equation:

x^2 = -9

2. The student claims that there is no solution to this equation because the square root of a negative number does not exist.

3. However, the student's understanding is based on the assumption that only real numbers are allowed. In reality, mathematics extends beyond real numbers to include complex numbers as well.

4. Complex numbers involve a combination of real numbers and an imaginary unit, denoted by the symbol "i." The imaginary unit is defined as the square root of -1, i.e., i = √(-1).

5. Using the concept of complex numbers, let's solve the equation x^2 = -9:

x^2 = -9
Taking the square root of both sides:
√(x^2) = ±√(-9)
Simplifying further:
x = ±√(9) * √(-1)
x = ±3i

6. Therefore, the solution to the equation x^2 = -9 is x = ±3i. These are complex numbers that include the imaginary unit i.

In summary, the student's claim that the equation has no solution due to the square root of a negative number being undefined is incorrect. Complex numbers allow us to handle such situations and find solutions to equations involving square roots of negative numbers.