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Chapter 3 · A Peek Beyond the Point

Extending Indian place value to the right of the point

यह वीडियो हिंदी में भी · Watch in Hindi

The decimal place value system9 min

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9 min.

Also recorded in Hindi.Englishहिन्दी

This is the explanation the previous two were saving the point for. Tenths and hundredths were built without it deliberately.

The idea

Place value works by position alone, and that is exactly what breaks the moment fractional places are allowed: written bare, the digits 705 could name three different quantities and nothing in them says which (Part I, §3.4, p.61). The decimal point is not a new number idea, it is the repair — a mark that fixes where the units place sits, after which every other place is determined. And the choice of ten is not forced by nature either: the chapter shows the unit split into four and into sixteen and says plainly that this works too (Part I, §3.4, p.59). Ten is chosen because the system already to the left of the point counts in tens, and a system that changes its rate halfway would not be one system.

What you should be able to do

  • State that a unit may be split into any number of equal parts, and give the chapter's reason for always choosing ten
  • Extend the ten-to-one chain of places rightwards past the units and name the places it produces
  • Explain why the chain has no last place in either direction
  • Say why the system is called decimal, and give the word's ancestry as the chapter gives it
  • Show, with a worked case, that writing digits without a separator leaves the quantity ambiguous
  • Use the decimal point to write quantities that would otherwise be confused, and read the point as the marker of where the units place ends
  • Fill in a place value table for a given decimal number and read off what each digit contributes
  • Read a decimal number aloud digit by digit after the point, and say why the digits after the point are not read as one whole number
  • Convert a count of one size of part into decimal form, as with 234 tenths

Words to know

TermDefinition in one lineFirst introduced
decimal pointthe mark set between the units place and the tenths placeprinted and set in bold in Part I, §3.4, p.62
separatorwhat the chapter calls the point's jobprinted in Part I, §3.4, p.62
periodthe chapter's alternative name for the mark itselfprinted in Part I, §3.4, p.62
decimal systemthe way of writing numbers built on repeated splitting by tenprinted in Part I, §3.4, p.61
decimal notationa quantity written using the pointprinted in Part I, §3.4, p.62
decimal numbera number written in that notationprinted in Part I, §3.4, p.62
place valuewhat a digit is worth because of where it sitsprinted in Part I, §3.4, p.59
Indian place value systemthe whole-number system this chapter extends rightwardsprinted in Part I, §3.4, p.59
one-thousandthone of the ten parts a hundredth is cut intoprinted in the plural, as one-thousandths, in Part I, §3.4, p.63; the singular arrives later, at Part I, §3.5, p.67
thousandthsthose parts, countedprinted in Part I, §3.4, p.61, in the How Big? questions
daśhathe Sanskrit word for ten that the chapter links the name toprinted in italics in Part I, §3.4, p.61
decimal placeone of the positions to the right of the pointprinted in Part I, §3.8, p.78, in the account of a rival notation
base tenthe usual name elsewhere for a system that groups by tenan added term; not printed in this chapter, which says the system is built on the number ten

Where people slip up

  • "You are not allowed to split a unit into four." The chapter says the opposite in as many words and prints the quarter and sixteenth scales to prove it. An explanation that turns the ten-split into a law has inverted the section: the point is that ten is chosen, for a reason, and the reason is stated.
  • "The decimal point is a new operation." It is a mark of position. Nothing is multiplied, divided or joined by it.
  • "The point sits in the middle of the number." It sits immediately right of the units place, and the number is not symmetric about it. There is no place between the units and the tenths, and no place answering to the units on the right-hand side.
  • "7.05 is seven and five." It is seven units and five hundredths, with an empty tenths place. The chapter's three readings of 705 exist for exactly this student.
  • "After the point, read the digits as one whole number." The page rules that out for 0.274 and explains why: those three digits name three different sizes of part, so each is said on its own.
  • "234 tenths cannot be written with a point because 234 is bigger than 10." It becomes 23.4. Counts larger than the base are normalised, exactly as 23 hundreds becomes 2300.
  • "The places stop at thousandths." The chapter's last chain runs to ten-thousandths and says the extension continues at both ends.
  • "Decimal is a European idea because the word is Latin." The chapter's own account is more careful than that: it gives the Latin word for ten, links it to Sanskrit, and notes the related words in eight Indian languages. The history of the notation itself is taken up at Part I, §3.8, p.78.

Worked answers to this chapter’s exercises · this video explains Figure it Out · 3 Q16

Transcript1,304 words

Everything so far has been built by splitting a unit into ten, and then splitting the pieces into ten again. But nobody has ever said why it has to be ten. So try four instead, and see what happens. Here is a unit cut into four equal parts. Quarters. And here is a pencil, lying against that scale. It reaches two whole units, and then two of the quarters. Two and two quarters. That is a perfectly good reading.

Now put the same pencil against a scale marked in tenths. Two units and five tenths. Also a perfectly good reading, of exactly the same pencil. And the quarter scale is not a dead end either. You can keep going. Take one quarter, and cut that into four. Now count how many of those tiny parts fill one whole unit. Four quarters, and four little parts in each of them.

Sixteen. Sixteen of them fill the unit exactly. So there is a second scale here, built entirely out of fours, and it works. It measures. It splits again when you need it to. Nothing about it is broken. Which means splitting into ten was never something you were forced into. It is a choice. And a choice needs a reason. Here is the reason, and it has nothing to do with ten being a better number than four.

Look at what sits to the left of the unit. Ten ones make one ten. Ten tens make one hundred. Ten hundreds make one thousand. The whole of that side already counts in tens. It always has. Now imagine splitting the unit into four, and keeping the left-hand side as it is. You would have a number that groups by ten going one way and by four going the other.

The rate would change halfway along, right at the unit. And a system whose rate changes halfway is not really one system. It is two, stuck together. So the unit is split into ten, because that is what keeps it all one thing. Let us draw that side properly, because we are about to extend it. A row of places. One. Ten. A hundred. A thousand. Going left, each one is ten times the one before it.

One times ten is ten. Ten times ten is a hundred. A hundred times ten is a thousand. And going right, each one is a tenth of the one before it. A thousand divided by ten is a hundred. A hundred divided by ten is ten. Ten divided by ten is one. That is the whole rule of the chain. Multiply by ten going left, divide by ten going right.

It is one rule, and it holds between every neighbouring pair. So here is the obvious question. What happens if you do not stop at one? Keep going right. Divide by ten again. One divided by ten is one tenth. There is the next place, and we already know what lives in it. Divide again. One tenth divided by ten is one hundredth. And again. One hundredth divided by ten is one thousandth.

Nothing new has been invented here. The same rule was simply not stopped. The places to the right of the unit are not a different kind of thing from the places to the left. They are the same chain, carrying on past the point where people usually put the pen down. Ten of each one makes one of the next, all the way along, in both directions. And there is no last place at either end.

Watch what that looks like. Here is a line from zero to one, marked in tenths. Take the stretch between six tenths and seven tenths, and blow it up. It is marked in hundredths now. Sixty hundredths at one end, seventy at the other. Take the stretch between sixty six hundredths and sixty seven, and blow that up. Thousandths. Six hundred and sixty, up to six hundred and seventy. Every one of those lines is the same stretch of the same line, magnified.

And you could keep doing that for ever. Nothing ever runs out of room. This whole way of writing numbers has a name, and the name is just the number ten. The word decimal is built on an ancient word meaning ten. You can watch that word travel. In Latin it is decem. In classical Sanskrit it is dasha. In Greek, deka. Three languages, three sounds, and one very old word underneath all of them.

Its descendants are still the everyday word for ten in dozens of living languages today. So decimal does not mean point, and it does not mean fraction. It means ten, and it is telling you the one thing you most need to know about the system. Everything in it, on both sides, groups by ten. Now a problem, and it is the reason the rest of this video exists. Writing out every part in words is slow. Four units and two tenths.

So let us do what place value already does, and let position carry the meaning. Just write the digits. Four, then two. Forty two. Except that is not what we meant at all. We meant four units and two tenths. And the digits four and two, sitting side by side, could be either of them. Four tens and two units. Or four units and two tenths. Two completely different quantities, and the digits alone do not say which.

It gets worse with three digits. Take seven, zero, five. Seven hundreds and five ones. That is seven hundred and five. Or seven tens and five tenths. Seventy, and a half. Or seven units and five hundredths. Just over seven. Three quantities. The largest is a hundred times the smallest. And all three of them get written with exactly the same three digits. It is not even limited to three readings. Slide the units place further and you get another quantity every time.

In fact every one of those readings is the number seven hundred and five, times some power of ten. The digits fix which number. What they do not fix is where the units place sits. So say where it sits. That is the whole repair, and it is one mark. Put a dot immediately after the units digit. Seven hundred and five, with nothing after the dot, stays seven hundred and five.

Seventy, dot, five. Seven tens, no ones, and five tenths. Seven, dot, zero five. Seven units, no tenths, and five hundredths. Same three digits every time. The dot is what decides between them. It is called the decimal point, and notice what it is not. It is not an operation. Nothing is being multiplied or divided or joined. It is a separator, and it separates the units place from the tenths place.

And it does not sit in the middle of the number. There is no place between the units and the tenths for it to be a place in. One last thing, and it is about saying these out loud. Here is a number. Zero, point, two, seven, four. It is tempting to read the tail as one whole number. Point two hundred and seventy four. Do not. Those three digits are not one number sitting there.

The two is tenths. The seven is hundredths. The four is thousandths. Three different sizes of part, so you say them one at a time. Point two, seven, four. And finally, a count. Two hundred and thirty four tenths. Can that even be written with a point? Of course. Ten tenths make a unit, so two hundred and thirty tenths are twenty three units, and four are left. Twenty three, point four. A count bigger than ten was never a problem. It just needed trading, the way it always has.

Where this fits

Taken from the notes each video was made from, not from the reading order — these are the ideas this one rests on and the ones that later rest on it.

Builds on

Comes up again in

The book

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