Math Vale

🔢 Number Kingdom · Number Sense

Place Value & Rounding

Name the value of every digit in a number and round to any place. Place value is the engine that makes all written arithmetic work.

In short

  • Each place is worth ten times the place to its right; position gives a digit its value.
  • The value of a digit is the digit multiplied by its place value, not the digit itself.
  • When rounding, only the digit immediately to the right of the rounding place decides the direction.
  • A zero is a placeholder that keeps every other digit where it belongs.

Ten of these makes one of those

Our number system uses only ten digits, 0 to 9. Everything else comes from position. Each place is worth ten times the place to its right:

  • ones: 1
  • tens: 10
  • hundreds: 100
  • thousands: 1,000
  • ten thousands: 10,000

So in 4,738 the 7 is not worth 7 — it is worth 700, because it sits in the hundreds place. This is the whole reason written addition, subtraction, multiplication and division work the way they do.

Place versus value

Two different questions get asked about the same digit, and they have different answers.

  • "Which place is the 7 in?" — the hundreds place.
  • "What is the value of the 7?" — 700.

The value is always digit x place value. In 4,738: 4 x 1,000 + 7 x 100 + 3 x 10 + 8 x 1.

Written out like that, the number is in expanded form: 4,000 + 700 + 30 + 8. Expanded form is a good check because the parts must add back to the original number.

Zero holds the place

A 0 is not "nothing". In 4,038 the 0 says there are no hundreds, and it keeps the 4 in the thousands place. Remove it and you get 438, a completely different number.

That is why zeros must be written even when they seem to contribute nothing, and why borrowing across a zero needs extra care.

Rounding

Rounding replaces a number with the nearest friendly number, so that a calculation can be done quickly or a result reported sensibly.

The method: 1. Find the place you are rounding to, and locate the two neighbours the number sits between. 2. Look at the digit immediately to the right of that place — the deciding digit. 3. If it is 5 or more, round up. If it is 4 or less, round down. 4. Every digit to the right of the rounding place becomes 0.

Round 4,738 to the nearest hundred: it lies between 4,700 and 4,800; the deciding digit is 3, so it rounds down to 4,700.

Only the deciding digit matters. 4,749 still rounds down to 4,700 to the nearest hundred, even though the 9 looks big — it is too far right to have a say.

Estimating with rounded numbers

Rounding earns its keep as a checking tool. Before computing 4,738 + 2,961, round both: 4,700 + 3,000 = 7,700. Any exact answer far from 7,700 is wrong.

Estimate before you calculate, so the estimate is honest. Estimating afterwards tends to shape itself around the answer you already wrote.

Worked examples

Example 1

In 58,206, what is the value of the digit 8?

  1. Count the places from the right: 6 ones, 0 tens, 2 hundreds, 8 thousands, 5 ten thousands.
  2. The 8 is in the thousands place.
  3. The thousands place is worth 1,000.
  4. Value = digit x place value = 8 x 1,000 = 8,000.

Example 2

Round 4,972 to the nearest hundred.

  1. 4,972 lies between 4,900 and 5,000.
  2. The deciding digit is the tens digit, which is 7.
  3. 7 is 5 or more, so round up to 5,000.
  4. Note the rollover: rounding up from 4,9xx pushes the thousands digit from 4 to 5.

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

In the number 626, what is the value of the digit 2 in the tens place?

Answer: 20

  1. The digit 2 sits in the tens place.
  2. The tens place is worth 10.
  3. 2 x 10 = 20

Problem 2

Difficulty 3 of 5

Round 19,457 to the nearest 1,000.

Answer: 19000

  1. The two neighbours are 19,000 and 20,000.
  2. The deciding digit (just right of the thousands place) is 4, so round down.
  3. 19,457 rounds to 19,000.

Problem 3

Difficulty 4 of 5

Which of these is 79,501 written in expanded form?

  1. 7 + 9 + 5 + 1
  2. 1 + 500 + 9,000 + 70,000
  3. 70,000 + 9,000 + 500 + 1
  4. 7,000 + 900 + 50 + 1

Answer: C. 70,000 + 9,000 + 500 + 1

  1. 79,501 digit by digit: 7, 9, 5, 0, 1
  2. Each digit is multiplied by its place value.
  3. 79,501 = 70,000 + 9,000 + 500 + 1

Common mistakes

  • Answering with the digit (7) instead of its value (700).
  • Looking at the last digit of the number instead of the digit next to the rounding place.
  • Truncating — chopping the digits off — instead of rounding.
  • Forgetting the rollover when rounding up a number full of 9s.

What you should be able to do

  • State the place and the value of a digit in a multi-digit number.
  • Write a number in expanded form and read it back.
  • Round a whole number to the nearest 10, 100 or 1000.
  • Use rounding to estimate a sum or product before computing it.

Where this fits in the curriculum

Common Core

  • 4.NBT.A.1

    Grade 4 — Recognise that a digit in one place is ten times what it represents in the place to its right.

  • 4.NBT.A.2

    Grade 4 — Read and write multi-digit numbers in base-ten numerals, number names and expanded form.

  • 4.NBT.A.3

    Grade 4 — Round multi-digit whole numbers to any place.

Ontario

  • G3.B1.5

    Grade 3 — Use place value when describing and representing multi-digit numbers.

  • G3.B1.3

    Grade 3 — Round whole numbers to the nearest ten or hundred.

  • G4.B1.3

    Grade 4 — Round whole numbers to the nearest ten, hundred or thousand.

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