Locating a decimal on the number line, and comparing two decimals

0:00Here is a number line, and a question that sounds too easy. Where exactly does one point four go? Somewhere between one and two. Everybody knows that much. So take the stretch between one and two and cut it into ten equal pieces. Now count along four of them and stop. That mark is one point four. And notice what the four just did. It did not measure anything. It counted.

0:28Four pieces along, out of the ten this stretch was cut into. That is all a decimal digit ever is. An answer to the question, which piece? Hold on to that. Everything else in this video is that one move, done again. So do it again, one level further down. Look only at the first piece, the stretch from one to one point one. Blow it up until it fills the picture, and cut that into ten too.

1:00Count along four of those, and you land on one point zero four. Same move, same count of four, completely different number. Because this time the four is choosing a piece of a piece. And that is what the zero in the middle is doing. Holding the tenths place open. One point four is four pieces of the first cut. One point zero four is four pieces of the second.

1:27The zero is not decoration. It says which cut the four belongs to. Which sets up an argument worth having. Someone writes zero point two, then writes it again as zero point two zero. Then zero point two zero zero. The same number, they say. Someone else objects. Surely putting zeros on the end makes it bigger? There is a way to settle this that does not involve anybody raising their voice.

1:59Put all three into a place value table and look at each column. Two in the tenths, and nothing anywhere else. Two in the tenths, and nothing anywhere else. And again. Row by row they are identical, so they are one number. The zeros on the right had nothing to push, because there was no digit to their right. But do not conclude from that that zeros never matter. Put one somewhere else and watch what happens.

2:30Zero point zero two puts it between the point and the two, and shoves the two out of the tenths into the hundredths. One more, and zero point zero zero two shoves it into the thousandths, ten times smaller again. So a zero after the last digit changes nothing, and a zero in front of one changes everything. The difference is whether the zero has a digit to its right to push.

2:59Here are seven numbers written with fours, fives and zeros in different arrangements. Sort them into families of equal numbers and you get exactly four families, not seven. Three are four point five, including one wearing a spare zero at each end, and neither zero does a thing. Now put it together and locate a number three places deep. Four point one eight five. Start with a line from zero to ten, cut into ten. The four picks four to five.

3:35Magnify that until it fills the picture and cut it into ten. The one picks four point one to four point two. Magnify again, cut again, and the eight picks four point one eight to four point one nine. Magnify once more, and the five counts five marks along. There is the number. Four digits, four cuts, and every digit answered the same question. Which of these ten? Each level is a tenth as wide as the one above it.

4:07So the number is not sitting somewhere vaguely near four point two. It is sealed inside a box one thousandth wide, and that box is what everything after this depends on. Before going on, a habit worth breaking. It catches nearly everybody. Not every division on a number line is a tenth. Here is a line running from five to ten, cut into ten pieces. Ten pieces, yes. But the stretch is five units long, so each piece is half a unit.

4:42Count five along and you are standing on seven point five, not five point five. Here is another, from eight to eight point one, also cut into ten. Each piece is one hundredth. And a third, four point three to four point eight, cut into ten. Five hundredths each. Three lines, three different divisions, and not one of them says so anywhere. So read the ends, count the gaps, work out what one gap is worth, and only then read anything off it.

5:17Now the second half of this, which turns out to be the first half again. Take two numbers that look nearly identical. Six point four five six, and six point four six five. Put them both on a line, and start zooming in. At the first level, both are in the stretch from six to seven. Together. Zoom in. At the next level both are between four tenths and five tenths. Still together.

5:47Zoom again, and they finally part company. One falls into the piece beyond five hundredths, the other into the piece beyond six. From that moment they are in different boxes, and they never share a box again. The third digit is where they separated, and the third digit is where the answer is. You can do all of that without drawing anything, and it is the same procedure. Start at the biggest place and walk rightwards.

6:19Six against six. The same. Move on. Four tenths against four tenths. The same. Move on. Five hundredths against six hundredths. Different, and there is your answer. Six point four six five is the larger, and the thousandths were never looked at. And here is the case that catches anyone comparing by counting digits instead. Three point eight one against thirteen point eight zero zero. The second is bigger, but not because it has more digits. It is bigger because thirteen beats three, settled in the whole part before a single digit after the point was read.

7:05Which leaves the question you ought to be asking. Why is stopping safe? Once two numbers differ at some place, could the digits after it turn the answer around? No. And the number line has already shown you why not. At the place where they differed, the two went into different pieces, and those pieces do not overlap. Everything afterwards happens inside one piece or the other. And here is the fact that closes it. A tail is always narrower than the piece it sits in.

7:39Fill it with nines. Give one of them nine thousandths, then nine ten thousandths, then nine more. That is still less than one whole hundredth, so it cannot climb out of its own piece. The gap opened at the deciding place can never be closed. That is why the first difference is the last thing you need. One more thing this buys you, not quite the same as comparing. Which of these is closest to one? Zero point nine, one point one, one point zero one, and one point one one.

8:16Closest is not about how many digits differ. It is a distance, and distances get measured. Zero point nine sits one tenth below one. Ten hundredths away. One point one sits one tenth above. Also ten hundredths, on the other side. So those two are exactly the same distance from one, which no digit counting would ever tell you. One point one one is eleven hundredths away, the worst of the four.

8:47And one point zero one is one hundredth away. One hundredth against ten hundredths is not a close race. Measure, and the question answers itself. To finish, a puzzle whose answer lands exactly where this video has been pointing. You have five digits. Four, one, eight, two and five, each used once. Put two before the point and three after, and get as close to twenty five as you can.

9:18There is an obvious candidate coming down from above. Twenty five point one four eight. That is one hundred and forty eight thousandths over. And one climbing up from below. Twenty four point eight five one. That is one hundred and forty nine thousandths short. One hundred and forty eight against one hundred and forty nine. Decided at the third place after the point, by a single thousandth. Which is this whole video in one answer. Compare from the left, stop at the first place that differs, and trust it, because by then the two are already in boxes that will never touch again.