Number Sentence
ArithmeticA 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.
Example
$3 + 5 = 8$ is a number sentence. $10 > 7$ is a number sentence. $6 - 2 = 4$ is a number sentence.
Key Insight
A number sentence is like a math "fact" written in symbols. Just as a sentence in English has a subject and verb, a number sentence has numbers and a relationship symbol.
Definition
A number sentence is a mathematical statement containing numbers, operation symbols ($+,-,\times,/$), and a relational symbol ($=, <, >, \le, \ge, \neq$). It is either true or false. An equation (using $=$) is one type; inequalities use $<, >$, etc. Number sentences can contain variables (open sentences).
Example
$4 \times 6 = 24$ (true equation). $15 - 3 > 10$ (true inequality). $2 + 2 = 5$ (false equation). $x + 7 = 12$ (open sentence: true only when $x = 5$).
Key Insight
Open number sentences (with variables) are proto-algebraic: solving them means finding the value that makes the sentence true, the core goal of algebra.
Definition
In formal logic, a number sentence corresponds to a closed formula in the language of arithmetic. A sentence is a formula with no free variables. Its truth value is determined by the standard model $(\mathbb{N}, 0, S, +, *)$ or by any model of a theory like Peano Arithmetic. Godel's first incompleteness theorem produces a sentence that is true in the standard model but not provable in PA.
Example
The Goldbach conjecture ("every even integer $> 2$ is a sum of two primes") is a closed arithmetic sentence. It is likely true (verified up to $4\times10^{18}$) but unproven.
Key Insight
The distinction between a true number sentence and a provable number sentence 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.