We define an extension of λ-calculus with linear combinations of terms, with coefficients taken in a fixed rig **R**. We extend β-reduction on these terms as follows: *at* + *u* reduces to *at′* + *u* as soon as term *t* reduces to *t*′ and a is a non-zero scalar. We prove that reduction is confluent. Under the assumption that **R** is a positive rig (*i.e.* a sum of scalars is zero iff all of them are zero), we show that this algebraic λ-calculus is a conservative extension of ordinary λ-calculus: two ordinary λ-terms equalized by the reduction of algebraic λ-calculus are β-equal. Last, we prove that under some reasonably minimal conditions on **R**, simply typed algebraic λ-terms are strongly normalizing.