Ah, beta, we put that zero as a placeholder, because you are not multiplying by just '2' but by '20' in your example of 46 × 23. It keeps the values of each digit in their proper place. When I first helped my grandson with this, he was puzzled too, until I showed him with my ration book—counting for the whole month is not the same as counting for a single day. My old eyes may need those new spectacles, but this much I see clearly: that little zero holds the place of a whole tens column, just as my notebook holds the place for my mother's fading lullabies.
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