Metric

This is the distance between two objects. You can define a metric for Number Sets that are not Fields like Integers!

Definition

1. $M$ as a set
2. Let $d:M\times M\to R$ be a function
3. The function is metric if:
   1. $d(x,y)\ge 0,\forall x,y\in M$ (Non negative)
   2. $d(x,y)=d(y,x)$ (Commutative/Symmetric)
   3. $d(x,y)=0⟺x=y$ (Zero distance is equivalent to Equality)
   4. $d(x,z)\le d(x,y)+d(y,z)$ (Triangle Inequality)

Metric from Norm

$d(x,y)=|x-y|$ From Absolute Value Functions Induce a Metric Theorem

Types

• Euclidean Metric
• Euclidean Metric for Complex Numbers
• Euclidean Metric for Real Numbers
• Metric for Integer modulo n
  • Hamming Distance

Theorems

• Absolute Value Functions Induce a Metric Theorem