Add Whole Numbers

Use addition notation

A college student works a part-time job. Last week she worked $3$ hours on Monday and $4$ hours on Friday. To find the total number of hours she worked, add $3$ and $4$:

We read this as three plus four, and the result is the sum of three and four. The numbers $3$ and $4$ are called the addends. A math statement that combines numbers and operations, like $3 + 4$, is called an expression.

Model addition of whole numbers

Addition is really just counting. Base-10 blocks make this visible: a small block represents $1$, and a rod of ten blocks represents $10$.

Example. Model the addition $3 + 4$.

Both addends are less than $10$, so ones blocks work: $3$ blocks plus $4$ blocks, counted together, give $7$ blocks in all. A math sentence showing two equal expressions is called an equation, so this models the equation $3 + 4 = 7$.

Example. Model the addition $5 + 8$.

$5$ blocks plus $8$ blocks gives $13$ blocks — more than $10$. Exchange $10$ of the ones blocks for $1$ tens rod, leaving $1$ ten and $3$ ones: $1$ ten and $3$ ones is $13$, so $5 + 8 = 13$.

Now model a sum where both addends already have two digits.

Example. Model the addition $17 + 26$.

$17$ is $1$ ten and $7$ ones; $26$ is $2$ tens and $6$ ones. Combined, that’s $3$ tens and $13$ ones. Exchange $10$ of those ones for another ten, leaving $4$ tens and $3$ ones — that is, $40 + 3 = 43$. So $17 + 26 = 43$.

Add whole numbers without models

Before adding larger numbers without blocks, know the one-digit addition facts cold — every sum of two digits from $0$ through $9$. Two patterns make them easier to learn.

The sum of any number and zero is the number itself:

This is the Identity Property of Addition; zero is called the additive identity.

Reversing the order of the addends never changes the sum — $2 + 3 = 5$ and $3 + 2 = 5$; $8 + 9 = 17$ and $9 + 8 = 17$. This is the Commutative Property of Addition:

Add whole numbers with the column algorithm

To add numbers with more than one digit, write them vertically so each place value lines up in its own column, then add column by column starting from the ones place.

Example. Add $28 + 61$.

Add the ones ($8 + 1 = 9$), then the tens ($2 + 6 = 8$). Neither column sums to $10$ or more, so no carrying is needed.

But what happens when a column’s sum is $10$ or more? Going back to the base-10 blocks, adding $17 + 26$: the ones columns give $7 + 6 = 13$ ones — more than $10$, so exchange $10$ of them for $1$ ten. Without the blocks, that exchange is written as a small carried $1$ above the tens column.

How to add whole numbers:

1. Write the numbers so each place value lines up vertically.
2. Add the digits in each place value, working from right to left starting with the ones place. If a column’s sum is $10$ or more, write the ones digit of that sum and carry the rest to the next place value.
3. Continue adding each place value from right to left, carrying whenever a column sums to $10$ or more.

Example. Add $43 + 69$.

Add the ones: $3 + 9 = 12$. Write $2$ in the ones place and carry $1$ ten. Add the tens: $1 + 4 + 6 = 11$.

$43 + 69 = 112$.

Example. Add $324 + 586$.

Add the ones: $4 + 6 = 10$ — write $0$, carry $1$. Add the tens: $1 + 2 + 8 = 11$ — write $1$, carry $1$. Add the hundreds: $1 + 3 + 5 = 9$.

$324 + 586 = 910$.

Example. Add $1{,}683 + 479$.

Add the ones: $3 + 9 = 12$ — write $2$, carry $1$. Add the tens: $1 + 8 + 7 = 16$ — write $6$, carry $1$. Add the hundreds: $1 + 6 + 4 = 11$ — write $1$, carry $1$. Add the thousands: $1 + 1 = 2$.

$1{,}683 + 479 = 2{,}162$.

When addends have different numbers of digits, line up the ones places first and let the shorter number’s blank columns act as zeros — never line up the leftmost digits.

More than two numbers can be added the same way — add straight down each column, carrying whenever a column’s total reaches $10$ or more.

Translate word phrases to math notation

Earlier, addition notation was translated into words. Now reverse the process: translate word phrases into math notation.

Example. Translate and simplify: the sum of $15$ and $32$.

The word sum tells us to add; of $15$ and $32$ tells us the addends.

The sum of $15$ and $32$ is $47$.

Example. Translate and simplify: $37$ increased by $26$.

The words increased by tell us to add, and the numbers given are the addends.

So $37$ increased by $26$ is $63$.

Add whole numbers in applications

Word problems follow a simple plan: read the problem to see what’s being asked, write a word phrase for the information needed, translate that phrase into math notation, simplify, and then write a sentence answering the question — with units.

Example. Hao earned test grades of $87, 95, 80, 92,$ and $78$ on the five tests of the semester. What is the total number of points he earned?

We want the total number of points on the tests — the sum of the grades.

Hao earned a total of $432$ points.

Some application problems involve shapes. The perimeter of a figure is the distance around it — the sum of the lengths of its sides.

Example. A rectangular patio has sides of length $14$ feet, $9$ feet, $14$ feet, and $9$ feet. Find the perimeter.

We want the perimeter — the sum of the sides.

The perimeter of the patio is $46$ feet.

Key terms

sum — the result of addition. addend — one of the numbers being added. expression — a math statement that combines numbers and operations. equation — a math sentence stating that two expressions are equal. Identity Property of Addition — the sum of any number and $0$ is that number ($a + 0 = a$). Commutative Property of Addition — changing the order of the addends does not change the sum ($a + b = b + a$). carrying — regrouping $10$ units in one place value as $1$ unit in the next place value to the left, written as a small digit above the column. perimeter — the distance around a geometric figure; the sum of the lengths of its sides.

This section is adapted from Prealgebra 2e, Section 1.2: Add Whole Numbers by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: condensed prose, omitted the addition facts table (0–9) as redundant with earlier material, redrew the base-10 block figure as an inline SVG and the column-addition figures as aligned math blocks, used new example numbers for the word-phrase and applications examples, and converted “Try It” practice problems into interactive exercises with instant feedback.