Do binaries in clusters form in the same way as in the field?

We examine the dynamical destruction of binary systems in star clusters of different densities. We find that at high densities (10⁴- 10 ⁵ M_⊙ pc^-3) almost all binaries with separations >10³ au are destroyed after a few crossing times. At low densities [], many binaries with separations >10³ au are destroyed, and no binaries with separations >10⁴ au survive after a few crossing times. Therefore, the binary separations in clusters can be used as a tracer of the dynamical age and past density of a cluster. We argue that the central region of the Orion nebula cluster was ∼100 times denser in the past with a half-mass radius of only 0.1-0.2 pc as (i) it is expanding, (ii) it has very few binaries with separations >10³ au and (iii) it is well mixed and therefore dynamically old. We also examine the origin of the field binary population. Binaries with separations <10² au are not significantly modified in any cluster, therefore at these separations the field reflects the sum of all star formation. Binaries with separations in the range 10²- 10⁴ au are progressively more and more heavily affected by dynamical disruption in increasingly dense clusters. If most star formation is clustered, these binaries must be overproduced relative to the field. Finally, no binary with a separation >10⁴ au can survive in any cluster and so must be produced by isolated star formation, but only if all isolated star formation produces extremely wide binaries.