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Chapter 4 · Another Peek Beyond the Point
Why multiplying by a decimal below 1 shrinks a number
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“Multiplying makes things bigger” is the first rule anybody learns about multiplication, and it is false.
The idea
"Multiplying makes things bigger" is a rule about counting numbers, and students carry it across the decimal point where it stops being true. What decides whether a product grows or shrinks is one comparison only — which side of 1 each factor sits on — because multiplying by something below 1 is taking a part of the other number. That single idea generates all three cases the chapter tabulates. It also generates the case the chapter does not tabulate, and the trap sitting under its exercises: two factors below 1 do force the product below 1, but one factor below 1 forces nothing at all.
What you should be able to do
- State, before multiplying, whether a product will exceed both factors, fall below both, or land between them
- Justify each of the three outcomes by the position of the factors relative to 1
- Explain why the count of decimal places is irrelevant to this question
- Recognise that a factor exactly equal to 1 leaves the other factor untouched, and that this case sits outside the chapter's table
- Show by a counterexample that a single factor below 1 does not force a product below 1
- Decide which of a list of products fall below 1 without carrying out any multiplication
- Give a pair of decimals whose product is a whole number
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| product | the result of a multiplication — the quantity being compared with its factors here | printed in §4.2, Part II, pp.69–73 |
| counting number | the book's name for the whole numbers used to count, whose behaviour under multiplication sets the expectation | printed in §4.2, Part II, p.72 |
| natural number | the wording used in the two questions about a product landing exactly on a whole | printed in §4.2, Part II, p.70 |
| decimal | a number written with a point, here always positive in this chapter | Class 6; used throughout, Part II, pp.67–96 |
| fraction | the form the chapter falls back on to explain the shrinking | printed in §4.2, Part II, pp.69–72 |
| multiplier | the number you multiply by — the one whose size relative to 1 decides the direction | printed in §4.2, Part II, p.70 |
| multiplicand | the number being multiplied | printed in §4.2, Part II, p.70 |
| greater than 1 | the condition heading the first row of the chapter's situation table | printed in §4.2, Part II, p.72 |
| between 0 and 1 | the condition heading the second and third rows | set inside the situation table, which wraps in extraction — checked p.72 of Part II |
| tenths | the place whose digit alone cannot tell you which side of 1 a number is on | printed in §4.1, Part II, p.67 |
| the pivot at 1 | the explanation's name for the single comparison that decides the outcome | an added phrasing; the book tabulates the cases without naming the idea |
| scaling factor | the explanation's word for a multiplier read as a stretch or a shrink | an added term, not printed in this chapter |
Where people slip up
- "Multiplication always makes a number bigger." The habit the whole topic exists to break. It is true only when you multiply by a number greater than 1, and the chapter's second example — the digits 25 and 8 again, with the point moved — breaks it in one line.
- "If any factor is less than 1, the product is less than 1." False, and it is the dangerous one, because the chapter never states it and never knocks it down — it only prints the numbers and moves on. 18 × 0.12 has a factor below 1 and a product above 2. The true statement is narrower: if both factors are below 1, the product is below both of them and therefore below 1.
- "More decimal places means a smaller number." 12.5 has one decimal place and stands above 1; 0.9999 has four and stands below it. The count of places is the wrong thing to look at; the comparison with 1 is the right one.
- "Multiplying by 0.75 is smaller because 0.75 is a small number." It shrinks because 0.75 is below 1, not because it looks small. 0.99 also shrinks, by almost nothing.
- "There is a fourth case for a factor equal to 1." There is a case, but it is not in the book's table, and it is not a shrink or a stretch — the other factor comes through unchanged. Name it as the boundary rather than presenting it as a printed row.
- "A product with a point in it can never be a whole number." 2.25 × 8 = 18 is on Part II p.72 and refutes it, and Part II p.70 asks the question directly.
- "This is only about decimals." It is the same argument made for fractions in Why multiplying can make a number smaller. The chapter says outright that decimals are only a way of writing fractions, so the behaviour has to carry across.
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Worked answers to this chapter’s exercises · this video explains Figure it Out · 1 Q9, Figure it Out · 4 Q10
Transcript1,395 words
Very early on, everybody learns that multiplying makes things bigger. And for years nothing contradicts it. Three times four is twelve. Seven times nine is sixty-three. Every answer comes out bigger than both numbers you started with. So it stops feeling like a rule and starts feeling like what multiplying is. Then the decimal point turns up, and it is simply false. Zero point two five times eight is two. That is smaller than eight.
Nothing has broken. The old rule was never about all numbers — it was about counting numbers, and we walked it across a border it does not cross. So what is the real rule? Here are three multiplications that look almost the same. Two point two five times eight. Zero point two five times eight. And zero point two five times zero point eight. The first one. Two point two five times eight is eighteen.
Eighteen is bigger than two point two five, and bigger than eight. It cleared both of them. The second. Zero point two five times eight is two. Two is bigger than zero point two five, but smaller than eight. It cleared one of them. The third. Zero point two five times zero point eight is zero point two. Zero point two is smaller than both. It cleared neither. Above both. Between them. Below both. Those are the only three things that ever happen.
So what decides which? The obvious guess is that small-looking numbers shrink things. That is close, but it is not it. Look at what actually changed between the three. Two point two five is above one. Zero point two five is below one. Eight is above one. Zero point eight is below one. That is the entire decision. Which side of one each number is standing on. And it has nothing to do with how many digits come after the point. Twelve point five has one digit after the point and sits above one. Zero point nine nine nine nine has four and sits below.
Length tells you nothing. Position tells you everything. So take the three cases one at a time. Both numbers above one. Three point four, times six point five. Both of them are past the gate. Three point four times six point five is twenty-two point one. Twenty-two point one is past three point four, and past six point five. It has overshot both. And this is the familiar case. It is the one the old habit was built on.
Every whole number bigger than one lives here, and so does every decimal bigger than one. Multiplying really does make things bigger — here. Now both numbers below one. Zero point seven five, times zero point four. Neither of them reaches the gate. Zero point seven five times zero point four is zero point three. Zero point three is under zero point four, and under zero point seven five. It has dropped below both.
And notice one more thing. It is under one as well, and it has to be. If the answer is smaller than the smaller of two numbers, and both of those are already under one, then the answer is under one too. Keep that, because it is the only version of this idea that is actually true. Both factors below one forces the product below one. Both. And now one of each. Zero point seven five, times five.
One below the gate and one above it. Zero point seven five times five is three point seven five. Bigger than zero point seven five. Smaller than five. It has come to rest between them, which is the only place left for it. And you can see why from either side. Multiplying five by something below one drags it down from five. Multiplying zero point seven five by something above one pushes it up from zero point seven five. Two pulls, and the answer stops somewhere in the middle.
But why does a number below one drag things down at all? Because of what multiplying by it actually means. Zero point seven five is three quarters. So zero point seven five of five is three quarters of five. Draw five as a bar. Now take three quarters of that bar. You are taking a part of it. Not the whole thing. A part. And a part of something is smaller than the something. That is not arithmetic, that is just what the word part means.
Every number below one names a part. That is what being below one is. Which leaves exactly one number unaccounted for. One itself. One is not above one and it is not below one. One is the gate. One times zero point four is zero point four. One times six point five is six point five. Nothing moves. No stretch and no shrink — the number comes out exactly as it went in.
And that is what makes one the right place to measure from, rather than some landmark somebody picked. Everything above it stretches. Everything below it shrinks. And it does neither. Which is why the only question ever worth asking here is which side of one. So here is a test. Four products, and one question. Which of them come out below one? Two of them settle instantly. Zero point seven times zero point six. Both below one, so the answer is below both, so it is below one. Done.
Zero point zero seven times zero point zero six. Both below one. Done, the same way. The other two have one factor on each side, so the answer lands somewhere between the two factors. And between zero point six and seven there is plenty of room on both sides of one. So the rule cannot settle those, and it is honest about it. For those two you have to look. Seven times zero point six is four point two. Zero point seven times six is also four point two.
Two below one, two above. And half the work was free. Now the trap, and it is a good one. After all that, it is very tempting to shorten it to this. If either number is below one, the answer is below one. It sounds like the same rule. It is not, and it is false. Here is the pair that finishes it. Eighteen times zero point one two. And zero point one eight times zero point one two.
The same factor, zero point one two, safely below one, sits in both of them. Eighteen times zero point one two is two point one six, which is above one. Zero point one eight times zero point one two is zero point zero two one six, which is below one. One factor below one settles nothing. Both of them settles everything. One more question, because it comes up. Can a product with decimals in it land exactly on a whole number?
Yes, and we have already seen it happen. Two point two five times eight is eighteen. Exactly eighteen. No point, nothing after it. But that was a decimal times a whole number. Can two decimals do it? Two point five times zero point four is one. Two point five is not a whole number. Zero point four is not a whole number. Their product is. The point is not something a number carries around for ever. It is a record of what the denominators were, and denominators can cancel out.
Two point five is twenty-five over ten, zero point four is four over ten, and twenty-five times four is exactly a hundred. So, all of it in one line. Compare each number with one. That is the only comparison there is. Both above one, and the answer passes both. Both below one, and it drops under both. One of each, and it settles between them. Exactly one, and nothing happens at all.
How many digits sit after the point never came into any of that. And the old habit — multiplying makes things bigger — turns out to be a true statement about one stretch of the number line that everybody kept using outside it. That happens constantly in mathematics. A rule learned where it works, carried somewhere it does not. The repair is not another rule to remember. It is knowing which side of one you are standing on.
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
- Multiply as whole numbers, then count the decimal digitsClass 7 · Ch 4, Another Peek Beyond the Point
- Why multiplying can make a number smallerClass 7 · Ch 8, Working with Fractions
Comes up again in
- Estimating first, so a wrong answer is visibleClass 7 · Ch 4, Another Peek Beyond the Point
Either side of this one
- Long division continued past the ones placeClass 7 · Ch 4, Another Peek Beyond the Point