Place Value: Concepts

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Now, let's imagine looking at the security footage being stored in the security office; in particular, we'll look at the 24-hour timestamps 00:00:15, 00:15:00, and 15:00:00. (A 24-hour clock means, for instance, that "1 PM" is "13 hours".) The first timestamp, 00:00:15, means "zero hours, zero minutes, and fifteen seconds"; in other words, fifteen seconds after midnight. The second timestamp, 00:15:00, means "zero hours, fifteen minutes, and zero seconds"; in other words, exactly fifteen minutes past midnight. The third timestamp, 15:00:00, means "fifteen hours, zero minutes, and zero seconds"; in other words, exactly three o'clock in the afternoon.

In each of these timestamps, the "15" told the time. But the "15" meant a different thing in each case – 15 seconds, 15 minutes, or 15 hours, depending on the context. The "place" where the "15" was — in the "hours" slot, the "minutes" slot, or the "seconds" slot — told us what the "value" of the "15" was for that particular timestamp.

Note also how important the zeroes were. Without the zeroes, the timestamp "00:00:15" would have collapsed into just "15", as would have the other two timestamps. Those zeroes held all the slots (hours, minutes, and seconds) in their proper places, preserving the meanings of the timestamps. Since the value of "15" varies with the place, it is vitally important that those other places not disappear. The zeroes are crucial to the meaning.

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Now let's think about counting. When we were little kids, we started counting on our fingers. If we were counting up ("tallying") a larger number of things, we might write down tally marks to keep track:

But once we got into bigger numbers, the thicket of tally marks started being unhelpful.

We could count up to ten on our fingers, but what then? We were out of fingers.

But let's say we had some marbles. Every time we used up all of our fingers, we could go back to fists (zero fingers) and we could put a marble down, standing for "one time of all-my-fingers" (that is, one time of ten).

So three marbles and two fingers would be "three times of ten, plus another two fingers", or "thirty-two", as shown below:

When we get ten marbles in the pile, we can't count any more of "all-my-fingers" because we're out of fingers to use for counting the marbles. So let's start a second pile, further away, to stand for "ten 'all-my-fingers' marbles". We'll call these new "ten 'all-my-fingers' marbles" (that is, the new marbles which stand for ten times of ten fingers) "one hundred" marbles. Then this:

...means "two times of ten-times-of-ten, plus six times of all-my-fingers, plus another four fingers", or "two hundred sixty-four".

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Of course, if somebody trips over our piles of marbles, we'll lose count. Once the marbles are no longer in their proper places, we can't tell what values they were meant to represent. It's be just like with the security-tape's timestamps, where we wouldn't have known what each "15" had stood for if somebody had stripped out all the zeroes. This concept that "the value of the marble (or the 15) depends on the pile (or slot) in which the marble (or 15) has been placed" is the point of "place values" in writing number names; namely, the place where the digit has been put tells you the actual value of that digit.