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Equal


Two quantities are said to be equal if they are, in some well-defined sense, equivalent. Equality of quantities a and b is written a=b. Equal is implemented in the Wolfram Language as Equal[A, B, ...], or A==B==....

Equality of multiple expressions is commonly denoted

a=b
(1)
=c,
(2)

which is equivalent to a=b=c. Equality is transitive, so if a=b and b=c, then it is also true that a=c.

A symbol with three horizontal line segments (=) resembling the equals sign is used to denote both equality by definition (e.g., A=B means A is defined to be equal to B) and congruence (e.g., 13=1 (mod 12) means 13 divided by 12 leaves a remainder of 1--a fact known to all readers of analog clocks).

Given an expression involving known constants, integration in finite terms, computation of limits, etc., the constant problem asks to determine if the expression is equal to zero (or, equivalently, if the equality a=b holds, since this is equivalent to a-b=0). In general, this is a very difficult (and unsolved) problem.

Equalities that the Wolfram Language cannot establish "out of the box" include

  (1+Sqrt[5])/2 == GoldenRatio
  (1+E)^2 == 1+2E+E^2

See also

Congruence, Constant Problem, Defined, Different, Equality, Equivalence Relation, Equivalent, Hidden Zero, Isomorphism, Transitive, Unequal Explore this topic in the MathWorld classroom

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Cite this as:

Weisstein, Eric W. "Equal." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Equal.html

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