Teach Me Maths

Place Value and Ordering Numbers

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How to find the value of a digit, write numbers in figures, multiply and divide by 10, 100 and 1000, put whole numbers, decimals and negative numbers in order, use the less than and greater than signs, and read a number line.

What place value means

Place value means that the value of a digit depends on its column. The 5 in 50 is worth fifty, but the 5 in 500 is worth five hundred.

Going left, each column is worth 10 times the one before: units, tens, hundreds, thousands, ten thousands, and so on. After the decimal point the columns are tenths, hundredths and thousandths, each one a tenth of the column before it.

In 3528.46 the 5 is in the hundreds column, so it is worth 500. The 4 is in the tenths column, so it is worth 0.4

In the exam you are asked to write down the value of a digit, write a number in figures, put numbers in order of size, or work out a number multiplied or divided by 10, 100 or 1000. These are usually worth 1 mark, near the start of a paper. Place value matters in longer questions too: a decimal point in the wrong place loses marks.

Worked example 1: The value of a digit

Write down the value of the 7 in (a) 172\,540 (b) 9.374

  1. (a) In 172\,540 the 1 is in the hundred thousands column, and the 7 is in the ten thousands column.
  2. So the 7 is worth 7 ten thousands, which is 70\,000
  3. (b) In 9.374 the 3 is in the tenths column, the 7 is in the hundredths column and the 4 is in the thousandths column.

  4. Your turn. Write 7 hundredths as a decimal.

    7 hundredths is 0.07: no units, no tenths, then 7 in the hundredths column

  5. Answer: (a) 70\,000 (b) 0.07

Writing numbers in figures

To write a number from words into figures, write the thousands first, then the rest. After the thousands there must always be exactly three digits: hundreds, tens and units.

Two hundred and six thousand and fifty: the thousands are 206, and fifty fills the last three places as 050. So the number is 206\,050

A zero holds the place of an empty column. Seven thousand and four is 7004, with no hundreds and no tens. Millions work in the same way: three million, forty thousand and six is 3\,040\,006

A number can be split into the values of its digits: 5308.27 = 5000 + 300 + 8 + 0.2 + 0.07. To go the other way, put each value in its own column and write a zero in any empty column.

Multiplying and dividing by 10, 100 and 1000

When you multiply by 10, every digit moves one column to the left, so it is worth 10 times as much. The decimal point stays where it is.

  • Multiply by 10, 100 or 1000: the digits move 1, 2 or 3 places to the left.
  • Divide by 10, 100 or 1000: the digits move 1, 2 or 3 places to the right.

3.45 \times 100 = 345: the 3 units become 3 hundreds, the 4 tenths become 4 tens and the 5 hundredths become 5 units.

Write a zero in any empty column between the digits and the decimal point: 2.6 \times 1000 = 2600 and 7 \div 100 = 0.07

Don't just add a zero to the end. That works for a whole number, 52 \times 10 = 520, but not for a decimal: 3.45 \times 10 is 34.5, not 3.450

Worked example 2: Multiplying and dividing by 100 and 1000

Work out (a) 4.62 \times 100 (b) 85 \div 1000

  1. (a) Multiplying by 100 moves every digit two places to the left.
  2. The 4 units become 4 hundreds, the 6 tenths become 6 tens and the 2 hundredths become 2 units: 462
  3. (b) Dividing by 1000 moves every digit three places to the right. The 8 tens become 8 hundredths and the 5 units become 5 thousandths.

  4. Your turn. Write 8 hundredths and 5 thousandths as one decimal.

    There are no units and no tenths, so write zeros there: 0.085

  5. Check: 0.085 \times 1000 = 85
  6. Answer: (a) 462 (b) 0.085

Putting decimals in order

To compare decimals, line them up by their decimal points. Compare the digits column by column, starting from the left. The first column where the digits are different decides which number is bigger.

It helps to give the numbers the same number of decimal places by writing zeros at the end. A zero at the end of a decimal does not change its value: 0.6 = 0.60 = 0.600

So 0.6 is greater than 0.45. Written as 0.60, it is 60 hundredths, and 0.45 is only 45 hundredths. A decimal with more digits is not always bigger.

If the numbers are measurements, change them to the same units first. For example, 0.3 kg is 300 g, so it is less than 350 g.

Worked example 3: Ordering five decimals

Write these numbers in order of size. Start with the smallest number. 3.2, 3.025, 3.22, 3.205, 3.02

  1. Every number has 3 units, so compare the digits after the decimal point.
  2. Write each number with 3 decimal places: 3.200, 3.025, 3.220, 3.205, 3.020
  3. Look at the tenths. 3.025 and 3.020 have 0 tenths, so they are the two smallest. The other three have 2 tenths.

  4. Your turn. Write down the smallest number in the list.

    3.020 has 0 thousandths and 3.025 has 5, so the smallest is 3.02

  5. For the three with 2 tenths, compare the hundredths, then the thousandths: 3.200, then 3.205, then 3.220
  6. Answer: 3.02, 3.025, 3.2, 3.205, 3.22

Negative numbers and the signs < and >

On a number line, numbers get bigger as you go right and smaller as you go left. Every negative number is less than zero, and less than every positive number.

For negative numbers, the one further from zero is the smaller one. -5 is less than -2, just as -5^\circC is colder than -2^\circC.

Decimals work in the same way. 2.6 is more than 2.06, so -2.6 is further below zero than -2.06. That makes -2.6 the smaller number.

  • < means is less than: 3 < 8
  • > means is greater than: 8 > 3

The open side of the sign always faces the bigger number, and the point faces the smaller one. So -2.6 < -2.06

Worked example 4: Ordering negative decimals

(a) Write these numbers in order of size. Start with the smallest number. -0.8, 1.2, -1.35, 0, -1.5 (b) Write < or > between -1.35 and -1.5

  1. (a) 1.2 is the only positive number, so it is the largest. 0 comes just before it.
  2. That leaves -0.8, -1.35 and -1.5. The one furthest from zero is the smallest.

  3. Your turn. Write down the smallest number in the list.

    1.5 = 1.50 is more than 1.35, so -1.5 is furthest below zero: the smallest is -1.5

  4. Next comes -1.35, then -0.8, which is the negative number closest to zero.
  5. (b) -1.35 is the bigger of the two, so the open side of the sign faces it: -1.35 > -1.5
  6. Answer: (a) -1.5, -1.35, -0.8, 0, 1.2 (b) -1.35 > -1.5

Steps on a number line

On a number line or a scale, only some marks have numbers. To read an unlabelled mark, first work out what each small step is worth.

Find the difference between two labelled numbers, then divide it by the number of gaps between them. Count the gaps, not the marks. From 0 to 1 with 5 gaps, each step is 1 \div 5 = 0.2

There are always more numbers between two decimals. Between 2.3 and 2.4 come 2.31, 2.32 and so on, up to 2.39. To find them, use the next column along.

Worked example 5: Reading a number line

The marks on the number line are equally spaced. Write down the number at A.

  1. From 1.6 to 2 there are 8 equal gaps.
  2. The difference between the labels is 2 - 1.6 = 0.4

  3. Your turn. Work out 0.4 \div 8 to find what one step is worth.

    0.4 \div 8 = 0.05, so each step is worth 0.05

  4. A is 5 steps after 1.6: 5 \times 0.05 = 0.25, and 1.6 + 0.25 = 1.85
  5. Check: halfway between 1.6 and 2 is 1.8, at the fourth mark, and A is one mark past halfway.
  6. Answer: 1.85

Worked example 6: Making numbers from digits

Use each of the digits 0, 2, 5 and 7 once to make a number with two digits before the decimal point and two digits after it. (a) Write down the largest number you can make. (b) Write down the smallest number greater than 50 that you can make.

  1. (a) Put the largest digit in the column worth the most, and so on: 7 tens, 5 units, 2 tenths and 0 hundredths.
  2. That makes 75.20, which is the same as 75.2
  3. (b) To be greater than 50, the tens digit must be 5 or 7. The smaller choice is 5
  4. The digits left are 0, 2 and 7. To keep the number small, put the smallest in the units, the next in the tenths and the largest in the hundredths.

  5. Your turn. Write down the number.

    5 tens, 0 units, 2 tenths and 7 hundredths make 50.27, which is greater than 50

  6. Answer: (a) 75.20 (b) 50.27

Common mistakes

  • Giving the name of the column instead of the value. The value of the 7 in 4.75 is 0.7; writing just the word tenths gets no mark.
  • Ignoring the decimal point when ordering. 0.16 is less than 0.2, even though 16 is more than 2: write both with 2 decimal places, 0.16 and 0.20
  • Just adding a zero to multiply a decimal. Move the digits instead: 1.8 \times 100 = 180, not 1.800 or 100.8
  • Thinking -5 is bigger than -2. For negative numbers, the one further from zero is the smaller.
  • Counting the marks instead of the gaps on a number line. Divide the difference between two labels by the number of gaps between them.
  • Missing a condition in a digit question. Check that your number really is odd, even, or greater than the value the question gives.

Practice questions

More on this topic: Place Value and Ordering Numbers worksheet with full solutions