Any indication that a measuring system gives a false reading when the true value of a measured quantity is zero, e.g. the needle on an ammeter failing to return to zero when no current flows. A zero error may result in a systematic uncertainty.
e.g. Zack and Zoe are measuring the rebound height of balls dropped onto a bench. Zack's ruler starts at zero, but Zoe's ruler has a 0.5 cm systematic error at both ends: the scale doesn't begin or finish at the ruler's ends. So all her heights are about 0.5 cm too small.
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e.g. a top-pan balance shows a reading when there is nothing placed on the pan.
