additive inverse

Thân thiện

The student writes the additive inverse of -5 on the chalkboard.

Definition

1. Noun:
   • (Mathematics): An additive inverse is one of a pair of numbers that, when added together, result in a sum of zero. For any number a, its additive inverse is -a.

Usage

• The term is used in arithmetic and algebra to describe the relationship between a number and its opposite.
• It is a fundamental concept for understanding subtraction and solving equations.

Examples

• Noun:
  • The additive inverse of 7 is -7, because 7 + (-7) = 0.
  • To solve the equation x + 5 = 0, you find the additive inverse of 5, which is -5.
  • Zero is its own additive inverse.

Advanced Usage

• In Abstract Algebra: The concept extends beyond numbers to elements in other algebraic structures (like groups, rings, and fields), where every element must have an additive inverse for the structure to be valid.
  • In modular arithmetic, the additive inverse of 3 modulo 7 is 4, because (3 + 4) mod 7 = 0.

Variants and Related Words

• Inverse (Noun): A more general term for something that reverses or undoes the effect of another. The additive inverse is a specific type of inverse.
• Opposite (Number) (Noun): A common, less formal term for the additive inverse of a number.
• Negation (Noun): The operation of finding the additive inverse. For a number , its negation is .

Synonyms

• Opposite number
• Negative (when referring to the inverse of a positive number)

Related Concepts (Not Phrasal Verbs or Idioms)

• Additive Identity: The number zero, which when added to any number leaves it unchanged (e.g., a + 0 = a).
• Multiplicative Inverse: A different concept where two numbers multiply to give 1 (e.g., the multiplicative inverse of 5 is 1/5).

Học thuật

Noun

1. (mathematics) one of a pair of numbers whose sum is zero; the additive inverse of -5 is +5