Number Sentence
A number sentence is a mathematical statement that uses numbers and symbols (such as =, <, or >) to show a relationship, similar to a sentence in language.
Definition
A number sentence is a complete mathematical statement that shows two things are equal or one is greater or less than another, using numbers and symbols. Formally, it contains numbers, operation symbols ($+,-,\times,/$), and a relational symbol ($=,<,>,\le,\ge,\neq$), and is either true or false; an equation (using $=$) is one type, inequalities use $<,>$, and open sentences contain variables. In formal logic, a number sentence corresponds to a closed formula (one with no free variables) in the language of arithmetic, its truth value determined by the standard model $(\mathbb{N},0,S,+,*)$ or by any model of a theory like Peano Arithmetic.
Example
$3 + 5 = 8$, $10 > 7$, and $6 - 2 = 4$ are all number sentences, math "facts" written in symbols. $4 \times 6 = 24$ is a true equation, $15 - 3 > 10$ is a true inequality, $2+2=5$ is false, and $x + 7 = 12$ is an open sentence, true only when $x=5$. The Goldbach conjecture, "every even integer greater than $2$ is a sum of two primes," is a closed arithmetic sentence, likely true (verified up to $4\times10^{18}$) but unproven.
Key Insight
Just as a sentence in English has a subject and verb, a number sentence has numbers and a relationship symbol. Open number sentences (with variables) are proto-algebraic: solving them means finding the value that makes the sentence true, the core goal of algebra. The distinction between a true number sentence and a provable one is the heart of Godel's incompleteness theorems: there exist true arithmetic statements that no formal system of sufficient strength can prove, as long as that system is consistent.