Powers of 10, and place value written out for whole numbers and decimals

0:00Here is a number: forty-seven thousand, five hundred and sixty-one. You have taken a number apart like this before. Four times ten thousand. Plus seven times one thousand. Plus five times one hundred. Plus six times ten. Plus one. Add those five terms up — forty thousand, seven thousand, five hundred, sixty, one — and you get the number back. But look at the last term. It is written bare. Just a one, with nothing multiplying it.

0:36That is the only odd thing on the line, and it turns out to be the point. Now cover the digits and look only at the multipliers. Ten thousand. One thousand. One hundred. Ten. Not one of those is an arbitrary number. Ten thousand is four tens multiplied together. One thousand is three of them. A hundred is two. Ten is one. Every multiplier in that line is a power of ten, and nobody ever said so out loud.

1:11Expanded form was a statement about powers of ten long before anybody wrote a single exponent into it. Writing the exponents in does not add anything. It only makes the line say what it was already saying. So here is the same line again, with the exponents written in. Four times ten to the fourth. Plus seven times ten to the third. Plus five times ten squared. Plus six times ten to the first.

1:43And the units term: plus one times ten to the zero. Put the two lines one above the other. Same digits, same terms, same total. But the second line has something the first one did not. Every place now carries a number of its own. And the last term has stopped being the odd one out. Look at what just happened to that units term. It used to be a bare one. Now it is one times ten to the zero.

2:14Which only works if ten to the zero is one. And that is not a convenience anybody arranged. Watch. Leave every other place exactly as it is, and try handing the units place some different multiplier. Try zero. A tenth. A half. Two. Five. Ten. A hundred. And one. Eight candidates. Exactly one of them leaves the sum equal to the number it came from. It is one. So the notation does not choose that ten to the zero is one. It demands it.

2:52One caution. That test only works because the last digit here is not zero. Change the number to forty-seven thousand, five hundred and sixty. Now the units digit is zero. Run the same eight candidates through it. Every single one of them works. Zero times a hundred is nothing. So is zero times a half. With a zero in the units place, nothing is pinned down at all. It is the digit that pins the multiplier, not the place.

3:26Worth remembering, the next time a rule seems to hold because you tested it on a friendly number. Three to try. One hundred and seventy-two. Five thousand, six hundred and forty-two. Six thousand, three hundred and seventy-four. Pause here if you want them. One hundred and seventy-two is one times ten squared, plus seven times ten, plus two times ten to the zero. Five thousand, six hundred and forty-two starts a place higher: five times ten cubed.

4:04So does six thousand, three hundred and seventy-four. That is why there are three of them and not one. The highest place is not a fixed feature of expanded form. It is read off the number in front of you. Now a number with a point in it. Five hundred and sixty-one, point nine zero three. The part above the point you can already do. Five times ten squared, plus six times ten, plus one times ten to the zero.

4:37Then the point, and three more digits. Nine tenths. Zero hundredths. Three thousandths. Which is nine times one tenth, plus zero times one hundredth, plus three times one thousandth. And every one of those multipliers is a power of ten as well. A tenth is ten to the minus one. A hundredth is ten to the minus two. A thousandth is ten to the minus three. So write the whole number out with exponents.

5:10Five times ten squared. Six times ten to the first. One times ten to the zero. Nine times ten to the minus one. Zero times ten to the minus two. Three times ten to the minus three. Now ignore the digits and read the exponents on their own. Two, one, zero, minus one, minus two, minus three. Every step down is a step of exactly one, the whole way along.

5:40Nothing happens at the point. No jump, no gap, no change of rule. The exponents were already counting down before the point arrived, and they carry on counting down afterwards. The point is not a wall between two kinds of digit. It is simply the spot where the exponents cross from zero to minus one. Draw one strip of places and hang both numbers on it. Ten to the fourth, ten cubed, ten squared, ten to the first, ten to the zero, then minus one, minus two, minus three.

6:18Eight places. Forty-seven thousand, five hundred and sixty-one uses the top five and stops at the units. Five hundred and sixty-one, point nine zero three starts three places lower and runs three places past the units. Same strip. A different stretch of it. And now you never have to count from the left-hand end again. A digit's exponent is how many places it sits from the units place — upwards on the left, downwards on the right.

6:53Give me a digit's position and I can tell you its exponent. Give me the exponent and I can tell you where the digit sits. Back to that zero in the hundredths place. Zero times one hundredth is nothing. Drop the term out of the sum and the total does not move. So why write it at all? Because dropping it out of the sum is not what a reader does. A reader drops it out of the notation.

7:23Take five hundred and sixty-one point nine zero three, throw the zero term away, and the terms left are five, six, one, nine, three. Now take five hundred and sixty-one point nine three, which has no zero term to throw away. Its terms are five, six, one, nine, three. The same five, from two different numbers — and they differ by twenty-seven thousandths. The zero contributes nothing to the value and everything to the notation, which is exactly why it gets written.

8:01Here is what that strip actually buys you. Which is bigger: nought point five nought three, or nought point five one? The first one is longer. That is not the question. Line them both up on the strip and read down from the top. Tenths: five and five. The same. Hundredths: nought and one. Different — and the second number wins, right there. You never reached the thousandths, and you never subtracted anything.

8:31The highest place where two numbers differ settles them. Nine point nine nine nine against ten: they differ first at the units place, nine against nought, and ten is the bigger one. More digits is not more number. One more thing, nearly free. Twelve squared is a hundred and forty-four. So what is one point two squared? One point four four. And nought point one two squared? Nought point nought one four four.

9:05Nought point nought one two squared is nought point nought nought nought one four four. And a hundred and twenty squared is fourteen thousand, four hundred. Every one of those is a hundred and forty-four, slid along the strip. And notice how far it slides: move the number one place, and its square moves two. Last thing. Why ten? Not because ten is special. Because we happen to have ten symbols to write digits with, and a positional system needs exactly as many symbols as its base.

9:41Two in base two. Eight in base eight. Sixteen in base sixteen. Allow yourself one symbol too many and the whole thing collapses. Give yourself a single symbol worth ten, and write it after a one. That names twenty. But a two followed by a nought also names twenty. Two writings, one number, and a notation you can no longer read backwards. So the base and the count of symbols are the same number, and that is the only thing ten was ever doing here.

10:16The places, the exponents, the point that turned out not to be a wall — none of it was about ten at all.