Place value and decimal notation

Someone covers you for a takeaway, and you owe them twenty-five pounds. You open the banking app and tap it in. Two, five, point, nought, nought. Now imagine your thumb hits the point one key late. Two, five, nought, point, nought. Same taps, same digits. But two hundred and fifty pounds leaves your account. Not one digit changed. What changed is where the digits were sitting, and that alone made the number ten times bigger. That is this whole topic, in one slip of a thumb.

Let's build the thing that decides all of that. A table of columns.

Every number sits in columns. Start at the right-hand end and walk left. Ones. Tens. Hundreds. Thousands. Ten thousands. Hundred thousands. Millions. Each column you step into is worth ten times the one you just left. Why ten times, every step? Because there are only ten digits to write with, nought up to nine. Once a column is full, the only place left to go is a new column on its left. The columns exist because the digits run out. A digit on its own only tells you how many. The column tells you how many of what. Put the two together and you have that digit's value.

Gentle one to start. Take the number two hundred and fifty. What is the five worth? A: five. B: fifty. C: five hundred. Have a think. I'll wait. It is B. Fifty. The five is sitting in the tens column, so it is worth five tens, and five tens is fifty. Move that same five into the ones column and it is worth five. Same digit. Different column.

Hold on to that, because it runs through this whole chapter. Where it sits is what it means. For a single digit: where it sits in the columns is what it is worth.

That was a small number. Point the same table at a big one.

Three million, four hundred and seventy-two thousand, nine hundred and five. Digit by digit: three, four, seven, two, nine, nought, five. What is the value of the digit seven? Pause the video here and have a go at this one. I'll wait. The answer is seventy thousand. Label the columns from the right-hand end. Five is ones. Nought is tens. Nine is hundreds. Two is thousands. Seven is ten thousands. So the seven is worth seven lots of ten thousand. Seventy thousand. How you write it matters. It asked for a value, and a value is a number: seventy thousand, or the numeral seven nought nought nought nought. Seven ten thousands names the column. It does not give the value.

The columns do not stop at the decimal point. Let's cross it.

Going left, each column is ten times the last. So going right, each column is a tenth of the last. That carries straight on across the point. Tenths. Then hundredths. Then thousandths. Those names do real work. Cut one whole thing into ten equal pieces and one piece is a tenth. Cut that tenth into ten again and each piece is a hundredth. So nought point three five reads as three tenths, and five hundredths. Not thirty-five of anything.

That reading is where people come unstuck. Nought point three five, next to nought point four. The first has more digits, and thirty-five feels far bigger than four. So it gets treated as the bigger number. Read the columns instead. Nought point three five starts with three tenths, then five hundredths on top. Nought point four is four tenths, and nothing after it. Read that way, the count of digits stops mattering. Deciding which of two numbers is larger belongs to the video on ordering positive and negative numbers using inequality symbols. Reading the columns is this one.

Now the noughts, because they work harder than they look. Back to that seven-digit number. There is a nought in the tens column, and it is not there to say nothing. It is holding the tens column open. Take it out and everything right of it slides left: three hundred and forty-seven thousand, two hundred and ninety-five. That is a placeholder. A nought that keeps every other digit where it belongs. Same job after the point. Nought point nought five is not nought point five. That first nought holds the tenths column, pushing the five out into hundredths. And the nought in front of the point says there are no whole ones, so a lonely point is never missed.

Here is what all that column-building buys you.

Multiplying by ten makes every digit worth ten times more. And you know where a digit goes to be worth ten times more. One column left. So multiplying by ten slides the whole row one column left. It falls straight out of the table. Six point three, times ten. The six is in ones, the three is in tenths. Slide both one column left. The six lands in tens, the three lands in ones. Sixty-three. Dividing by ten walks the other way, one column right. Sixty-three divided by ten: the six drops into ones, the three drops into tenths. Six point three, back where it started. One warning about how people say this out loud. You will hear move the decimal point. The point never moves. It lives between the ones column and the tenths column, permanently. The digits are what move. Getting that backwards is how digits end up in the wrong column.

A real one now, out of a bakery. There is a cue to spot first.

When a question asks you to multiply by twenty, or fifty, or two hundred, there is a ten hiding inside. Split it into an easy bit times ten, and most of the work becomes a slide. A bakery uses six point three kilograms of flour per batch, and makes twenty batches. Work out the total flour needed, using the fact that twenty is two times ten. Pause it now and work this one out for yourself. I'll wait. The answer is one hundred and twenty-six kilograms. Do the two first. Double six point three: six doubles to twelve, three tenths doubles to six tenths, so twelve point six. Now the times ten, and that is a slide. The one goes from tens to hundreds, the two from ones to tens, the six from tenths to ones. One hundred and twenty-six. Writing it down: the units travel with the answer, so one hundred and twenty-six kilograms, not a bare number. And nothing is left after the point, so write nothing there. Not point nought. Just one hundred and twenty-six.

The people who mark these papers write a report on every one. Two lines land straight on this. The first. Part (a) was quite well done, although some students became confused with the place value of their answer. Read the end of that again. Confused with the place value of their answer. Not with the method. Not with the sum. The digits landed in the wrong columns. The second comes from a paper question built on those same two numbers, six point three and twenty, where what was wanted at the end was half the total. That is why halving turns up in it. Working with decimals often led to arithmetic mistakes or place value mistakes, and very few students spotted that the twenty could be halved immediately, making the six point three easy to multiply by the ten. Two things in one sentence. Place-value slips rather than arithmetic slips. And almost nobody hunted for the ten hiding inside the twenty. So two habits. When a decimal meets a multiple of ten, hunt for the ten before reaching for long multiplication. And when an answer appears, check the columns before you check the arithmetic.

One more number. This time I set it up, then get out of your way.

The number is nought point nought five nought seven. Two questions on it. What is the digit five worth? And if you multiply the whole number by ten, what happens to every digit? Pause it, and write both of your answers down before I say them. I'll wait. The five is worth five hundredths. Written out, that is nought point nought five. Count the columns out from the point: first is tenths, and there is a nought sitting there. Second is hundredths, and there is your five. Now times ten, and every digit slides one column left. The five moves from hundredths into tenths. The nought after it moves into hundredths. The seven moves into thousandths. You get nought point five nought seven.

Here is the whole video in four lines. One. The columns run ones, tens, hundreds, thousands going left, and tenths, hundredths, thousandths going right, each step left worth ten times more. Two. A digit's value is the digit times its column, which is why that seven was worth seventy thousand. Three. A nought is a placeholder with a real job: it holds a column open so every other digit stays put. Four. Times ten slides every digit one column left, divide by ten slides them one column right, and the point never moves. Underneath all four sits the line for this chapter. Where it sits is what it means. For a digit: where it sits in the columns is what it is worth. That is twenty-five pounds becoming two hundred and fifty.

Next in the chapter: Adding, subtracting, multiplying and dividing positive and negative integers. You can now say what any digit is worth. That video puts whole numbers together and takes them apart, on both sides of zero.

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