The symbol ≡ generally means identical to or equivalent to, often signifying a stronger form of equality than the standard equals sign (=). It signifies that two expressions are equal for all values of their variables.
Usage in Mathematics
In mathematics, ≡ has several important applications:
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Congruence Modulo: Perhaps the most common usage is in modular arithmetic. The statement
a ≡ b (mod m)means that a and b have the same remainder when divided by m. In other words, m divides a - b. For example:17 ≡ 2 (mod 5)because both 17 and 2 have a remainder of 2 when divided by 5.11 ≡ -1 (mod 12)because 12 divides 11 - (-1) = 12.
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Identical Functions/Expressions:
f(x) ≡ g(x)means the functions f(x) and g(x) are identical; that is, they produce the same output for every possible input x. This implies that the functions are equal by definition or through a well-established identity. For example:(x + 1)^2 ≡ x^2 + 2x + 1This is an identity because the expressions are equivalent for all values of x.
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Geometric Congruence: In geometry, ≡ can denote congruence between geometric figures, such as triangles. For instance,
△ABC ≡ △DEFmeans that triangle ABC is congruent to triangle DEF, implying that all corresponding sides and angles are equal.
Differences from the Equals Sign (=)
While both ≡ and = indicate equality, there's a subtle difference:
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Equals (=): Generally states that two quantities have the same value in a specific context or under certain conditions. It may or may not hold true for all possible values of the variables involved.
- Example:
x + 2 = 5This is only true when x = 3.
- Example:
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Identical To (≡): Indicates that two expressions are equal in a more fundamental sense. It implies that the equality holds true under all conditions and for all possible values of the variables. It denotes a definition, an identity, or congruence.
- Example: Trigonometric identities such as
sin^2(x) + cos^2(x) ≡ 1which holds true for all values of x.
- Example: Trigonometric identities such as
Summary
In summary, ≡ signifies a stronger form of equality than =, indicating that two expressions are identical, congruent, or equivalent under all conditions or within a specific mathematical context such as modular arithmetic. It conveys a deeper relationship between the expressions.
