Introduction

Before this page: be able to read and write whole numbers and say which digit is which. The KS2 page on number and place value covers that ground.

Ten symbols, 0 to 9, are enough to write every number there is. That is only possible because a digit's position carries meaning of its own: the symbol says how many, and the place it sits in says how many of what. Without that convention, a fresh symbol would be needed for every new quantity, and arithmetic would be an act of memory rather than a method.

Place value is the name of that convention, and almost everything that follows is built on it. Comparing and ordering, adding in columns, multiplying by ten, rounding and standard form are all the same idea applied: work out what each place is worth, and the rest follows.

What is place value?

Place value is the value a digit takes from its position in a number. The same digit written in a different place means a different quantity.

For example, the digit 5 has a different value in each of the following numbers:

• The digit 5 in '5' is worth five ones

• The digit 5 in '50' is worth five tens

• The digit 5 in '500' is worth five hundreds

• The digit 5 in '0.5' is worth five tenths

• The digit 5 in '0.05' is worth five hundredths

• The digit 5 in '0.005' is worth five thousandths

The symbol is the same in every one of them. Everything that separates five thousandths from five hundred is the place it was written in.

The values of the places

Each position in a number always has the exact same meaning in terms of place value:

The digits to the left of the decimal point (.) always make up the whole number part.

The digits to the right of the decimal point always make up the decimal part.

The parts in the diagram above are:

• Hundred Millions — 100,000,000

• Ten Millions — 10,000,000

• Millions — 1,000,000

• Hundred Thousands — 100,000

• Ten Thousands — 10,000

• Thousands — 1,000

• Hundreds — 100

• Tens — 10

• Ones/Units — 1

(Decimal point)

• Tenths — 0.1

• Hundredths — 0.01

• Thousandths — 0.001

• Ten-thousandths — 0.000 1

• Hundred-thousandths — 0.000 01

• Millionths — 0.000 001

• Ten-millionths — 0.000 000 1

• Hundred-millionths — 0.000 000 01

• Billionths — 0.000 000 001

If you write a number with more digits than the places listed above on either side of the decimal point, the value of their place will continue growing or shrinking following a predictable pattern:

• A number further to the left will have a larger place value

• A number further to the right will have a smaller place value

Furthermore;

• Moving one place to the left will make the place value 10 times bigger

• Moving one place to the right will make the place value 10 times smaller

Constructing words for numbers

Take this number apart, digit by digit, and read the meaning of each one:

• The 4 in the thousands place represents 4 lots of 1000, which equals four thousand

• The 5 in the hundreds place represents 5 lots of 100, which equals five hundred

• The 8 in the tens place represents 8 lots of 10, which equals eighty

• The 2 in the ones place represents 2 lots of 1, which equals two

• The 3 in the tenths place represents 3 lots of 0.1, which equals three tenths

• The 0 in the hundredths place represents 0 lots of 0.01, which equals zero hundredths

• The 7 in the thousandths place represents 7 lots of 0.001, which equals seven thousandths

Put these parts together in order to get the full number:

four thousand five hundred and eighty–two, three tenths, and seven thousandths

The decimal part can be restated:

four thousand five hundred and eighty–two, and three hundred and seven thousandths

Both forms describe the same quantity. The second one collects the decimal parts into a single count, and to do that they must first be counted in the same place. Thousandths are the smallest place used here, so every part is rewritten in thousandths: one tenth is one hundred thousandths, and one hundredth is ten thousandths.

The digits were already saying this. Reading 307 as one number and naming it by the place of its last digit gives 307 thousandths directly, which is why the shortcut works: take the place of the final decimal digit, then read all the decimal digits together as a single count of that place.

Read any number

The same reading works on any number.

Constructing numbers from words

The reverse process builds a number from its words. Each group of words names a place, and the places fix where its digits belong.

Suppose we are given:

six million, three hundred and twelve thousand, forty-seven and eighty-six hundredths

Start by separating the whole-number part into groups of three digits:

• six million gives the millions group 6

• three hundred and twelve thousand gives the thousands group 312

• forty-seven gives the final group 047. There are no hundreds in “forty-seven”, so a zero holds the hundreds place. Without it, the thousands group would slide up into the hundreds place and change the value of the number

Putting these groups together gives 6,312,047.

Next, eighty-six hundredths means 0.86. The last digit is in the hundredths place, so the decimal part has two digits. In the diagram, this is shown as 860 thousandths so that the decimal digits still fill a group of three. This does not change the value, because 0.860 = 0.86. The zero at the end is only there to keep the grouping clear, and it can be left out when writing the final number.

Combining the whole-number and decimal parts gives:

6,312,047.86