Addition

Addition is the simplest of the four arithmetic operations: it combines two or more numbers into one, called the sum. We read $a + b = c$ as "a plus b equals c". The numbers $a$ and $b$ are the addends, and $c$ is the sum.

The names in addition

Every part of the operation has a name — worth knowing, because they turn up in every word problem:

• addend — each of the numbers being added ($a$ and $b$),
• sum — the result of the addition ($c$).

For example, in $3 + 5 = 8$ the numbers $3$ and $5$ are addends and $8$ is the sum.

The laws of addition

Addition obeys three laws that make counting easier.

Commutativity

The order of the addends does not matter:

$a + b = b + a$

So $28 + 5$ can be computed as $5 + 28$ — and it is easier to add a small number to a large one than the other way round.

Associativity

When adding three or more addends, the way you group them does not change the result:

$(a + b) + c = a + (b + c)$

So you can first add the addends that make a "round" number. In $7 + 3 + 8$ it is convenient to start with $7 + 3 = 10$, then $10 + 8 = 18$.

Identity element

Adding zero does not change the number:

$a + 0 = a$

Zero is called the identity element of addition.

Column addition

Larger numbers are added in columns — digit under digit, right to left. When the digits in a column add up to $10$ or more, you write the ones digit and carry the ten into the next column.

Take $47 + 38$. Add the ones: $7 + 8 = 15$ — write $5$, carry $1$. Add the tens: $4 + 3 + 1 = 8$. The result is $85$.

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 8Score: 0

11 + 12 =ResultCheck

Common mistakes

• Forgetting to carry — the most common column-addition error. If the digits in a column sum to two digits, the left one always moves to the next column.
• Misaligning columns — digits must be right-aligned (ones under ones), especially when the numbers have different lengths.
• Losing the zero — $a + 0 = a$, not $0$.