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. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Interactive Mathematics Miscellany and Puzzles — A collection of articles on various math topics, with interactive Java... See also the disambiguation page title equality. ...

clearly marked or understood

Example

A quadratic equation over the complex numbers always has two roots. Graph of a quadratic function: y = x2 - x - 2 = (x+1)(x-2) The x-coordinates of the points where the graph crosses the x-axis, x = -1 and x = 2, are the roots of the quadratic equation: x2 - x - 2 = 0 In mathematics, a quadratic equation is a polynomial... In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for the square root of minus one (−1), which cannot be represented by any real number. ...

The equation

x² − 3x + 2 = 0

factors as In mathematics, factorization or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. ...

(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

factors as

(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 mathematics, a polynomial P(T) has a discriminant, which is a polynomial function of its coefficients, and discriminates the case of a multiple root (for which the graph of P(x) would touch the x-axis). ...

Proving distinctness

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. In mathematics, a proof is a demonstration that, given certain axioms, some statement of interest is necessarily true. ... // Use of the term The concept of property or ownership has no single or universally accepted definition. ... In mathematics, any integer (whole number) is either even or odd. ... In mathematics, any integer (whole number) is either even or odd. ...

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, In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique type of another (possibly the same) set (the codomain, not to be confused with the range). ...

• The Hahn-Banach theorem says (among other things) that distinct elements of a Banach space can be proved to be distinct using only linear functionals.
• In category theory, if f is a functor between categories C and D, then f always maps isomorphic objects to isomorphic objects. Thus, one way to show two objects of C are distinct (up to isomorphism) is to show that their images under f are distinct (up to isomorphism).