Question

how are integers and there oppisites related in mathy

Answer 1

Short answer: the "opposite" of an integer n is its additive inverse, written −n, and it satisfies n + (−n) = 0. Every integer has a unique opposite, and taking opposites is just reflecting numbers across 0 on the number line.

Key properties (with examples):
- Definition: if n is an integer, its opposite is −n. Example: opposite of 7 is −7; opposite of −4 is 4.
- Additive inverse: n + (−n) = 0. Example: 5 + (−5) = 0.
- Opposite of the opposite: −(−n) = n. Example: −(−3) = 3.
- Zero: 0 is its own opposite: −0 = 0.
- Absolute value: |n| = |−n| (opposites have the same magnitude). Example: |6| = |−6| = 6.
- Distributes over addition: −(a + b) = (−a) + (−b). Example: −(2 + 3) = −5 = (−2) + (−3).
- Multiplication rules: (−a)·b = −(a·b) and (−a)(−b) = a·b. Example: (−2)·3 = −6; (−2)(−3) = 6.
- Ordering flips sign: if a > 0 then −a < 0 (and vice versa).

Algebraic viewpoint: under addition the integers Z form an infinite cyclic group; the opposite operation n ↦ −n gives each element its inverse. The map n ↦ −n is an involution (doing it twice gives the original) and is a homomorphism of the additive group.

If you want, I can show proofs of any of these properties or give practice problems.