7

What Are Multi-Digit Whole Numbers?

A multi-digit number contains more than one digit.

Examples include:

47

326

5,804

72,915

4,608,231

The value of each digit depends on its place in the number.

Understanding place value is essential for accurate addition and subtraction.

Reviewing Place Value

Consider:

4,582

The digits represent:

• 4 thousands = 4,000
• 5 hundreds = 500
• 8 tens = 80
• 2 ones = 2

So:

4,582 = 4,000 + 500 + 80 + 2

 

 

 

Place value tells us which digits must be combined when adding or subtracting.

Why Digits Must Be Aligned

When numbers are written vertically, digits with the same place value must be placed in the same column.

For example:

  4,582
+ 2,316
-------

The:

• ones are under ones
• tens are under tens
• hundreds are under hundreds
• thousands are under thousands

This is why careful alignment is important.

Addition

Addition combines quantities.

The numbers being added are called addends.

The answer is called the sum.

For example:

245 + 132 = 377

245 and 132 are the addends.

377 is the sum.

 

 

 

6

Adding Without Regrouping

Consider:

2,341 + 4,526

Write the numbers vertically:

  2,341
+ 4,526
-------

Start with the ones.

1 + 6 = 7

Tens:

4 + 2 = 6

Hundreds:

3 + 5 = 8

Thousands:

2 + 4 = 6

Therefore:

2,341 + 4,526 = 6,867

Why We Usually Start from the Right

When using the standard addition algorithm, we normally begin with the ones column.

This is because a column may produce a value of 10 or more.

When this happens, part of the value must be regrouped into the next place-value column.

Starting from the right allows this regrouping to happen naturally.

What Is Regrouping?

Suppose we add:

8 + 7

The result is:

15

But 15 means:

1 ten + 5 ones

So we write:

5

in the ones place and regroup:

1 ten

into the tens column.

 

 

 

6

Regrouping does not change the value.

It simply expresses the number using a different combination of place values.

Addition with Regrouping

Calculate:

347 + 286

   347
+  286
------

Ones:

7 + 6 = 13

Write 3 ones and regroup 1 ten.

Tens:

4 + 8 + 1 = 13

Write 3 tens and regroup 1 hundred.

Hundreds:

3 + 2 + 1 = 6

Therefore:

347 + 286 = 633

Understanding the Regrouping

The calculation:

347 + 286

can also be written using expanded form.

347 = 300 + 40 + 7

286 = 200 + 80 + 6

Combine:

Hundreds:

300 + 200 = 500

Tens:

40 + 80 = 120

Ones:

7 + 6 = 13

So:

500 + 120 + 13

Regroup:

500 + 100 + 20 + 10 + 3

= 600 + 30 + 3

= 633

The standard algorithm is a faster way of recording this place-value process.

Worked Example 1

Calculate:

2,758 + 1,674

  2,758
+ 1,674
-------
  4,432

Ones:

8 + 4 = 12

Write 2 and regroup 1 ten.

Tens:

5 + 7 + 1 = 13

Write 3 and regroup 1 hundred.

Hundreds:

7 + 6 + 1 = 14

Write 4 and regroup 1 thousand.

Thousands:

2 + 1 + 1 = 4

Therefore:

2,758 + 1,674 = 4,432

Adding Numbers with Different Numbers of Digits

Consider:

4,826 + 375

Align the place values:

  4,826
+   375
-------

Do not write:

  4,826
+ 3,750

because this changes the value of 375.

 

 

 

Correct calculation:

4,826 + 375 = 5,201

Adding Three or More Numbers

The same place-value rules apply when adding several numbers.

Calculate:

1,245 + 738 + 2,106

Align carefully:

  1,245
    738
+ 2,106
-------
  4,089

Therefore:

1,245 + 738 + 2,106 = 4,089

Using Number Properties

Sometimes rearranging addends makes calculations easier.

Consider:

275 + 438 + 725

Instead of calculating in the original order:

275 + 725 = 1000

Then:

1000 + 438 = 1438

Therefore:

275 + 438 + 725 = 1,438

This uses the commutative and associative properties of addition.

Subtraction

Subtraction finds the difference between quantities.

In:

853 − 421 = 432

853 is the minuend.

421 is the subtrahend.

432 is the difference.

More simply, we can think:

starting amount − amount removed = amount remaining

 

 

 

6

Subtraction Without Regrouping

Calculate:

7,865 − 3,421

  7,865
- 3,421
-------

Ones:

5 − 1 = 4

Tens:

6 − 2 = 4

Hundreds:

8 − 4 = 4

Thousands:

7 − 3 = 4

Therefore:

7,865 − 3,421 = 4,444

Why Regrouping Is Needed in Subtraction

Suppose we need to calculate:

42 − 17

In the ones column we encounter:

2 − 7

We cannot remove 7 ones from 2 ones using whole numbers.

So we regroup one ten.

42 can be represented as:

4 tens + 2 ones

or:

3 tens + 12 ones

 

 

 

5

Now:

12 − 7 = 5

and:

3 − 1 = 2

Therefore:

42 − 17 = 25

Regrouping Does Not Change the Number

This is an important idea.

42

can be represented as:

4 tens + 2 ones

or:

3 tens + 12 ones

Both represent exactly the same quantity.

Similarly:

500

can be represented as:

5 hundreds

or:

4 hundreds + 10 tens

Regrouping changes the representation, not the value.

Subtraction with Regrouping

Calculate:

563 − 278

Start with:

  563
- 278
-----

Ones:

We cannot calculate 3 − 8 using whole numbers.

Regroup one ten:

63 becomes 5 tens and 13 ones

Now:

13 − 8 = 5

Tens:

We now have:

5 − 7

Regroup one hundred.

The 5 hundreds become:

4 hundreds

and the tens become:

15 tens

Then:

15 − 7 = 8

Hundreds:

4 − 2 = 2

Therefore:

563 − 278 = 285

Checking with Addition

Addition and subtraction are inverse operations.

If:

563 − 278 = 285

then we can check:

285 + 278 = 563

Since this is true, our subtraction is correct.

 

 

 

5

Regrouping Across Zeros

Zeros can make subtraction more challenging.

Consider:

4,002 − 1,675

We cannot calculate:

2 − 5

There are also no tens available to regroup directly.

So we must move left until we find a nonzero digit.

Worked Example 2: Across Zeros

Calculate:

4,002 − 1,675

The 4 thousands can be regrouped.

One thousand becomes:

10 hundreds

One of those hundreds becomes:

10 tens

One of those tens becomes:

10 ones

After regrouping, we effectively have:

• 3 thousands
• 9 hundreds
• 9 tens
• 12 ones

Now subtract:

Ones:

12 − 5 = 7

Tens:

9 − 7 = 2

Hundreds:

9 − 6 = 3

Thousands:

3 − 1 = 2

Therefore:

4,002 − 1,675 = 2,327

Be Careful with Zeros

A common mistake is to treat each zero independently.

Instead, remember that place values are connected.

For example:

1 thousand = 10 hundreds

1 hundred = 10 tens

1 ten = 10 ones

 

 

 

5

This relationship explains subtraction across zeros.

Worked Example 3

Calculate:

8,000 − 3,468

After regrouping:

  8,000
- 3,468
-------
  4,532

Check:

4,532 + 3,468 = 8,000

Therefore, the subtraction is correct.

Estimation

Estimation gives an approximate answer.

It is useful for:

• predicting the size of an answer
• checking calculations
• making quick decisions
• detecting calculator or arithmetic errors

One common method is rounding.

Estimating Addition

Suppose:

3,842 + 2,176

Round to the nearest thousand:

3,842 ≈ 4,000

2,176 ≈ 2,000

Estimate:

4,000 + 2,000 = 6,000

Exact answer:

3,842 + 2,176 = 6,018

The exact answer is close to the estimate.

Therefore, it appears reasonable.

 

 

 

5

Estimating Subtraction

Suppose:

7,891 − 3,164

Round to the nearest thousand:

7,891 ≈ 8,000

3,164 ≈ 3,000

Estimate:

8,000 − 3,000 = 5,000

Exact answer:

7,891 − 3,164 = 4,727

The exact result is reasonably close to:

5,000

Choosing a Useful Place to Round

You do not always need to round to the nearest thousand.

For:

486 + 312

rounding to the nearest hundred works well:

500 + 300 = 800

Exact:

486 + 312 = 798

For:

58 + 43

rounding to the nearest ten is more appropriate:

60 + 40 = 100

Exact:

101

Choose a rounding place that gives a useful estimate without unnecessary work.

Compatible Numbers

Another estimation strategy is to use compatible numbers.

These are nearby numbers that are easy to calculate mentally.

For:

397 + 602

we can think:

400 + 600 = 1000

The exact answer is:

999

For:

804 − 297

think:

800 − 300 = 500

Exact answer:

507

Estimation Is Not the Exact Answer

Suppose:

5,126 + 2,891

Estimate:

5,000 + 3,000 = 8,000

This does not mean the exact answer is 8,000.

The exact calculation is:

5,126 + 2,891 = 8,017

Use the symbol:

≈

for an approximate value.

So:

5,126 + 2,891 ≈ 8,000

Reasonableness

An answer is reasonable if it makes sense compared with what we expect.

Suppose someone calculates:

4,216 + 3,705 = 79,210

Estimate:

4,000 + 4,000 = 8,000

The claimed answer:

79,210

is nowhere near 8,000.

Therefore, the answer is clearly unreasonable.

 

 

 

5

Another Reasonableness Check

Suppose:

9,203 − 4,881

Before calculating, we know the answer should be roughly:

9,000 − 5,000 = 4,000

If we obtain:

14,084

we immediately know something is wrong.

Subtraction of a positive number should also produce an answer smaller than:

9,203

Worked Example 4: Addition and Estimation

Calculate:

6,487 + 2,756

Estimate first:

6,500 + 2,800 ≈ 9,300

Now calculate exactly:

  6,487
+ 2,756
-------
  9,243

The exact answer:

9,243

is close to the estimate.

Therefore, the answer is reasonable.

Worked Example 5: Subtraction and Estimation

Calculate:

12,403 − 5,879

Estimate:

12,400 − 5,900 ≈ 6,500

Exact calculation:

12,403 − 5,879 = 6,524

The exact answer is very close to our estimate.

Mental Math vs Standard Algorithms

Different calculations may require different strategies.

For:

4,000 + 3,000

mental math is efficient.

For:

27,583 + 48,769

a standard written algorithm may be more reliable.

For:

5,000 − 4,998

mental counting is efficient:

2

For:

83,104 − 47,968

a written method may be more appropriate.

Strong mathematicians choose methods based on the numbers involved.

Real-World Application: Shopping

Suppose a store sold:

2,847 items

on Saturday and:

3,569 items

on Sunday.

How many items were sold altogether?

We need addition:

2,847 + 3,569

  2,847
+ 3,569
-------
  6,416

Therefore:

6,416 items

were sold.

 

 

 

Real-World Application: Attendance

A stadium has:

25,000 seats

If:

18,746

people attend an event, how many seats remain empty?

We need subtraction:

25,000 − 18,746

= 6,254

Therefore:

6,254 seats

remain empty.

Real-World Application: Population

Suppose a town had:

48,735 people

and its population increased by:

3,842

people.

New population:

48,735 + 3,842

= 52,577

The new population is:

52,577 people

Real-World Application: Distance

A driver plans to travel:

1,250 km

and has already travelled:

786 km

Distance remaining:

1,250 − 786

= 464 km

Therefore:

464 km remain

in the journey.

 

 

 

4

Real-World Application: Budget

Suppose a project has a budget of:

$12,500

and has spent:

$7,846

Amount remaining:

$12,500 − $7,846

= $4,654

Therefore:

$4,654 remains

in the budget.

Real-World Application: Fundraising

A school wants to raise:

$20,000

It has already raised:

$13,675

How much more is needed?

$20,000 − $13,675

= $6,325

Therefore:

$6,325 more

is required.

Identifying the Operation

Word problems do not always say:

add

or:

subtract

You need to interpret the situation.

Addition is often appropriate when quantities are:

• combined
• increased
• collected together
• accumulated
• totaled

Subtraction is often appropriate when finding:

• how many remain
• how much more
• the difference
• how much was removed
• how far remains

Be Careful with Keywords

Keywords can help, but they should not replace understanding.

For example:

"How many more students are in School A than School B?"

This requires subtraction because we are comparing quantities.

But:

"School A gained 250 more students this year."

If we know the previous population and want the new population, we use addition.

Always think about what is happening to the quantities.

Worked Example 6: Two-Step Problem

A warehouse began with:

15,600 boxes

It received:

3,850 more boxes

and then shipped:

7,425 boxes

First add:

15,600 + 3,850 = 19,450

Then subtract:

19,450 − 7,425 = 12,025

Therefore:

12,025 boxes remain

in the warehouse.

Worked Example 7: Comparing Quantities

City A has:

84,215 residents

City B has:

76,849 residents

How many more residents does City A have?

Subtract:

84,215 − 76,849

= 7,366

Therefore:

City A has 7,366 more residents.

Worked Example 8: Total Distance

A delivery truck travels:

184 km

in the morning,

237 km

in the afternoon,

and:

96 km

in the evening.

Total:

184 + 237 + 96

Use a convenient order:

184 + 96 = 280

Then:

280 + 237 = 517

Therefore:

517 km

were travelled.

 

 

 

Worked Example 9: Missing Value

Suppose:

? + 2,765 = 8,400

To find the missing amount, subtract:

8,400 − 2,765

= 5,635

Therefore:

5,635 + 2,765 = 8,400

Worked Example 10: Change Over Time

A library began the year with:

32,450 books

It purchased:

2,875 new books

and removed:

1,420 old books

First:

32,450 + 2,875 = 35,325

Then:

35,325 − 1,420 = 33,905

Therefore, the library now has:

33,905 books

Addition and Subtraction Are Connected

Consider:

425 + 286 = 711

This gives related subtraction facts:

711 − 425 = 286

and:

711 − 286 = 425

 

 

 

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This relationship can be used to check answers.

Checking an Addition Answer

Suppose:

3,745 + 2,186 = 5,931

Check by subtraction:

5,931 − 2,186 = 3,745

The original answer is confirmed.

Checking a Subtraction Answer

Suppose:

9,284 − 3,617 = 5,667

Check:

5,667 + 3,617 = 9,284

Therefore, the subtraction is correct.

Using a Number Line

Addition and subtraction can also be represented on a number line.

Addition means moving toward larger numbers.

Subtraction means moving toward smaller numbers.

 

 

 

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For example:

450 + 275

could be represented as:

450 → 650:

+200

650 → 720:

+70

720 → 725:

+5

Therefore:

450 + 275 = 725

Counting Up for Subtraction

Sometimes subtraction is easier by finding the distance between two numbers.

Consider:

1,000 − 783

Count up:

783 → 800:

17

800 → 1,000:

200

Total:

17 + 200 = 217

Therefore:

1,000 − 783 = 217

This can be more efficient than regrouping across several zeros.

Choosing an Efficient Strategy

Consider:

6,482 + 3,719

A standard algorithm is appropriate.

Consider:

6,000 + 4,000

Mental math is faster.

Consider:

10,000 − 9,998

Count up:

2

Consider:

5,436 − 2,879

A written subtraction algorithm may be more reliable.

The goal is not to use one method for every calculation.

The goal is to choose an efficient and accurate method.

Common Mistakes

Mistake 1: Misaligning digits

Incorrect alignment changes place values.

Always line up:

• ones
• tens
• hundreds
• thousands

Mistake 2: Forgetting a regrouped value

If 1 ten is regrouped into the next column during addition, remember to include it.

Mistake 3: Regrouping without changing the next column

If you regroup one hundred into ten tens, the hundreds digit must decrease by 1.

Mistake 4: Subtracting the smaller digit from the larger digit regardless of position

For:

52 − 38

you cannot simply calculate:

8 − 2

because the operation is:

2 − 8

Regroup first.

Mistake 5: Difficulty with zeros

Remember that you may need to regroup through several place-value positions.

Mistake 6: Treating an estimate as an exact answer

An estimate is approximate.

Use:

≈

when appropriate.

Mistake 7: Ignoring reasonableness

If:

4,000 + 3,000

produces an answer near:

70,000

something is clearly wrong.

Did You Know?

The standard addition and subtraction algorithms work because of our base-ten place-value system.

 

 

 

4

In base ten:

10 ones = 1 ten

10 tens = 1 hundred

10 hundreds = 1 thousand

10 thousands = 1 hundred thousand

This is exactly why regrouping works.

When we "carry" or "borrow," we are really exchanging one place-value unit for ten units of the next smaller place.

Understanding this makes the algorithms much easier to understand than simply memorizing steps.

Key Terms

• Whole number: Zero or a positive counting number.
• Digit: Symbol from 0 to 9 used to write numbers.
• Place value: Value of a digit based on its position.
• Addend: Number being added.
• Sum: Result of addition.
• Difference: Result of subtraction.
• Regrouping: Rewriting a quantity using different place-value units.
• Standard algorithm: Organized written method for performing a calculation.
• Estimate: Approximate answer.
• Rounding: Replacing a number with a nearby value that is easier to use.
• Compatible numbers: Nearby numbers chosen to make calculations easier.
• Reasonableness: Whether an answer makes sense based on the original problem.
• Inverse operations: Operations that undo each other.
• Expanded form: Writing a number as the sum of its place values.

Key Relationships

10 ones = 1 ten

10 tens = 1 hundred

10 hundreds = 1 thousand

10 thousands = 1 hundred thousand

Addition:

addend + addend = sum

Subtraction:

starting quantity − quantity removed = difference

Checking subtraction:

difference + subtrahend = original quantity

Checking addition:

sum − one addend = other addend

Key Takeaways

• Multi-digit addition and subtraction depend on a strong understanding of place value.
• Digits with the same place value must be aligned when using standard written algorithms.
• In the standard addition algorithm, calculations normally begin with the ones column.
• When a column totals 10 or more, regroup into the next place-value column.
• Regrouping changes the representation of a number but not its value.
• Multi-digit subtraction may require regrouping from a larger place value.
• Subtraction across zeros may require regrouping through several columns.
• Addition and subtraction are inverse operations and can be used to check each other.
• Estimation can predict the approximate size of an answer.
• Rounding and compatible numbers are useful estimation strategies.
• Comparing an exact answer with an estimate helps determine whether the answer is reasonable.
• Mental math may be more efficient for simple or specially structured calculations.
• Standard written algorithms are useful for more complicated multi-digit calculations.
• Real-world problems may involve addition, subtraction, or several operations.
• Understanding the situation is more reliable than simply looking for keywords.
• Multi-digit addition and subtraction are used in budgeting, shopping, population data, travel, inventory, measurement, attendance, and many other everyday situations.
• A strong solution should include the correct operation, accurate calculation, appropriate units or context, and a check that the answer is reasonable.