Two or more things are distinct if no two of them are the same thing. In mathematics, two things are called distinct if they are not equal.
A quadratic equation over the complex numbers sometimes has two roots.
x2 − 3x + 2 = 0
(x − 1)(x − 2) = 0
and thus has as roots x = 1 and x = 2. Since 1 and 2 are not equal, these roots are distinct.
In contrast, the equation:
x2 − 2x + 1 = 0
(x − 1)(x − 1) = 0
and thus has as roots x = 1 and x = 1. Since 1 and 1 are (of course) equal, the roots are not distinct; they coincide.
In other words, the first equation has distinct roots, while the second does not. (In the general theory, the discriminant is introduced to explain this.)
In order to prove that two things x and y are distinct, it often helps to find some property that one has but not the other. For a simple example, if for some reason we had any doubt that the roots 1 and 2 in the above example were distinct, then we might prove this by noting that 1 is an odd number while 2 is even. This would prove that 1 and 2 are distinct.
Along the same lines, one can prove that x and y are distinct by finding some function f and proving that f(x) and f(y) are distinct. This may seem like a simple idea, and it is, but many deep results in mathematics concern when you can prove distinctness by particular methods. For example,
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