# In a congruence are the numbers remainders or residues?

If we have $$a \equiv b \pmod{n}$$ then $$a$$ and $$b$$ are congruent to each other modulo $$n$$, correct?

What do we "call" $$a$$ and $$b$$? Because sometimes these numbers can be negative. Would they be remainders? Residues? Do we say these remainders (residues?) belong to the same "congruence class mod $$n$$"? Are remainders and residues synonymous?

Trying to get the terminology / lingo right here. When do we use which words? How do we describe what all this notation is representing / saying?

• Yes; see congruence relation : "two numbers $a$ and $b$ are said to be congruent modulo $n$". $a$ and $b$ are numbers and $n$ is called the modulus of the congruence. – Mauro ALLEGRANZA Oct 31 '18 at 13:35
• Are $a$ and $b$ remainders? Residues? – Sean Hill Oct 31 '18 at 13:37
• "Like any congruence relation, congruence modulo $n$ is an equivalence relation, and the equivalence class of the integer $a$, denoted by $a_n$, is the set $\{ \ldots, a − 2n, a − n, a, a + n, a + 2n, \ldots \}$. This set, consisting of the integers congruent to a modulo $n$, is called the congruence class or residue class or simply residue of the integer $a$, modulo $n$." – Mauro ALLEGRANZA Oct 31 '18 at 13:38
• Thus, residue of $a$ modulo $n$ is the set of all integers congruent to $a$ modulo $n$. – Mauro ALLEGRANZA Oct 31 '18 at 13:40
• @Sean "residue" is an overloaded (ancient) term. It can mean either a representative of a congruence class or the entire class. It can also mean "congruent", e.g. older texts often write "$b$ is a residue of $a$ modulo $n$" for $a\equiv b\pmod{n}.\,$ It predates modern (abstract) algebra so one can't expect it to be completely consistent with modern syntax and semantics. – Bill Dubuque Oct 31 '18 at 14:02