Small isotopic differences between the silicate minerals in planets may have developed as a result of processes associated with core formation, or from evaporative losses during accretion as the planets were built up. Basalts from the Earth and the Moon do indeed appear to have iron isotopic compositions that are slightly heavy relative to those from Mars, Vesta and primitive undifferentiated meteorites (chondrites). Explanations for these differences have included evaporation during the 'giant impact' that created the Moon (when a Mars-sized body collided with the young Earth). However, lithium and magnesium, lighter elements with comparable volatility, reveal no such differences, rendering evaporation unlikely as an explanation. Here we show that the silicon isotopic compositions of basaltic rocks from the Earth and the Moon are also distinctly heavy. A likely cause is that silicon is one of the light elements in the Earth's core. We show that both the direction and magnitude of the silicon isotopic effect are in accord with current theory based on the stiffness of bonding in metal and silicate. The similar isotopic composition of the bulk silicate Earth and the Moon is consistent with the recent proposal that there was large-scale isotopic equilibration during the giant impact. We conclude that Si was already incorporated as a light element in the Earth's core before the Moon formed.