Training Whole Numbers Addition and Subtraction
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Addition and Subtraction

20 min Whole Numbers

Addition and Subtraction of Whole Numbers

Addition and subtraction are the two most fundamental operations in mathematics. Every advanced technique you will study later — from solving equations to computing integrals — ultimately reduces to these two operations performed many times over.

In this lesson you will review the standard algorithms for adding and subtracting whole numbers, paying special attention to carrying (regrouping) in addition and borrowing in subtraction. These procedures rely directly on the place-value concepts from the previous lesson.

You will also explore the key properties of addition — the commutative, associative, and identity properties — which not only simplify arithmetic but form the foundation for abstract algebra.

Addition

Addition combines two or more numbers into a sum.

Properties of Addition
  • Commutative Property: $a + b = b + a$
  • Associative Property: $(a + b) + c = a + (b + c)$
  • Identity Property: $a + 0 = a$

Column-Addition Algorithm: Align numbers by place value, add column by column from right to left, and carry (regroup) when a column sum is 10 or more.

Example 1

Add: $4{,}587 + 2{,}946$

  1. Ones: $7 + 6 = 13$. Write 3, carry 1.
  2. Tens: $8 + 4 + 1 = 13$. Write 3, carry 1.
  3. Hundreds: $5 + 9 + 1 = 15$. Write 5, carry 1.
  4. Thousands: $4 + 2 + 1 = 7$.

$$4{,}587 + 2{,}946 = 7{,}533$$

Example 2

Add three numbers: $1{,}254 + 3{,}078 + 695$

Align by place value and add column by column:

$$1{,}254 + 3{,}078 + 695 = 5{,}027$$

Subtraction

Subtraction finds the difference between two numbers. It is the inverse of addition.

Caution

Subtraction is not commutative: $a - b \neq b - a$ in general.

Column-Subtraction Algorithm: Align by place value, subtract column by column from right to left. When the top digit is smaller, borrow (regroup) from the next column to the left.

Example 3

Subtract: $5{,}003 - 1{,}247$

This requires borrowing across zeros:

  1. $3 < 7$: borrow from tens — but tens is 0.
  2. Hundreds is also 0. Borrow from thousands: $5{,}003$ becomes $4$ thousands, $9$ hundreds, $9$ tens, $13$ ones.
  3. Now subtract: $13 - 7 = 6$, $9 - 4 = 5$, $9 - 2 = 7$, $4 - 1 = 3$.

$$5{,}003 - 1{,}247 = 3{,}756$$

Check: $3{,}756 + 1{,}247 = 5{,}003$ ✓

Example 4

Subtract: $10{,}000 - 4{,}673$

Borrow across all the zeros: think of $10{,}000$ as $9{,}999 + 1$.

$9{,}999 - 4{,}673 = 5{,}326$, then $5{,}326 + 1 = 5{,}327$.

$$10{,}000 - 4{,}673 = 5{,}327$$

Practice Problems

1. $3{,}456 + 2{,}789$
2. $8{,}005 - 3{,}467$
3. $12{,}094 + 7{,}968$
4. $30{,}000 - 14{,}258$
5. $456 + 1{,}234 + 5{,}678$
6. $9{,}001 - 3{,}456$
7. $100{,}000 - 1$
8. $47{,}892 + 53{,}108$
9. $7{,}200 - 4{,}836$
10. $25{,}000 - 8{,}999$
11. $11{,}111 + 22{,}222 + 33{,}333$
12. A checking account has $\$4{,}250$. You deposit $\$1{,}875$ and then write a check for $\$2{,}340$. What is the balance?
13. $60{,}400 - 27{,}538$
14. $999 + 1$
15. $50{,}006 - 28{,}479$
Show Answer Key

1. $6{,}245$

2. $4{,}538$

3. $20{,}062$

4. $15{,}742$

5. $7{,}368$

6. $5{,}545$

7. $99{,}999$

8. $101{,}000$

9. $2{,}364$

10. $16{,}001$

11. $66{,}666$

12. $4{,}250 + 1{,}875 - 2{,}340 = \$3{,}785$

13. $32{,}862$

14. $1{,}000$

15. $21{,}527$

Rounding Whole Numbers Multiplication and Division