Addition is the operation that puts amounts together to find the total. The numbers you add are called addends, the answer is the sum, and the sign is the plus sign, +. For example, 7 + 5 = 12. On a number line, adding a positive number means moving to the right.
Adding 5 to 7 means starting at 7 and moving 5 steps to the right. The numbers being added are addends; the result is the sum.
Formal definition and notation
For numbersa and b, addition gives the sum a + b. For whole numbers, a + b counts the objects in a group of a joined with a group of b. Addition is commutative (a + b = b + a) and associative ((a + b) + c = a + (b + c)), and 0 is the additive identity: a + 0 = a. Its inverse operation is subtraction.
Subtraction undoes addition: 12 − 5 = 7 because 7 + 5 = 12. This is how you check an addition.
Adding a negative number moves left on the number line: 6 + (−9) = −3.
Parts of an addition
Parts of an addition
addend + addend = sum
addend
each number being added
sum
the total, the answer to an addition
To add multi-digit numbers, line up the place values, add the ones first, and regroup (carry) 10 ones as 1 ten when a column adds to 10 or more.
Addition examples, step by step
Example 1: Add with regrouping
Warm-up
Find 486 + 357.
Ones: 6 + 7 = 13. Write 3, carry 1 ten.
Tens: 8 + 5 + 1 = 14. Write 4, carry 1 hundred.
Hundreds: 4 + 3 + 1 = 8.
Check by subtracting: 843 − 357 = 486.
Answer:486 + 357 = 843.
Example 2: Use the properties to add mentally
Core
Find 38 + 47 + 62 without a calculator.
Swap the order (commutative property): 38 + 62 + 47.
Group the pair that makes a round number: (38 + 62) + 47 = 100 + 47.
Add: 100 + 47 = 147.
Answer: The sum is 147.
Example 3: Addition with a negative number in context
Test-style
At 6 a.m. the temperature is −8degrees Fahrenheit. By noon it has risen 15 degrees. What is the temperature at noon?
“Risen 15 degrees” means add 15: −8 + 15.
Start at −8 on the number line and move 15 to the right.
8 steps take you to 0, and the other 7 take you to 7: −8 + 15 = 7.
Answer: The temperature at noon is 7 degrees Fahrenheit.
Common mistakes
✗ MistakeForgetting to carry: 57 + 38 = 85.
✓ Instead7 + 8 = 15: write 5 and carry 1, so 57 + 38 = 95.
When a column adds to 10 or more, the extra ten belongs in the next column. Estimating first (about 60 + 40 = 100) shows that 85 is too small.
✗ MistakeLining up the left digits: adding 125 and 34 as 125 + 340 = 465.
✓ InsteadLine up the ones: 125 + 34 = 159.
Only digits with the same place value can be added together: ones with ones, tens with tens. Right-align whole numbers (or line up the decimal points).
✗ Mistake−6 + 4 = −10.
✓ Instead−6 + 4 = −2.
Adding a positive number moves right on the number line. From −6, four steps right lands on −2. The answer −10 comes from moving left.
Addition is the operation; an additive inverse is a number that adds with another to make 0, such as 9 and −9.
Quick check: 3 questions
Question 1 of 3
In 15 + 9 = 24, what is 24 called?
Show answer and explanation
Answer: B. The answer to an addition is the sum. 15 and 9 are the addends. A product is the answer to a multiplication and a difference is the answer to a subtraction.
Question 2 of 3
What is 268 + 145?
Show answer and explanation
Answer: D. Ones: 8 + 5 = 13, carry 1. Tens: 6 + 4 + 1 = 11, carry 1. Hundreds: 2 + 1 + 1 = 4. The sum is 413. 303 forgets both carries, 313 forgets the hundreds carry and 403 forgets the tens carry.
Question 3 of 3
Which equation shows the commutative property of addition?
Show answer and explanation
Answer: A. Commutative means the order can change: 9 + 4 = 4 + 9. The second choice is the identity property, the third is the associative property and the fourth shows additive inverses.
Addition: frequently asked questions
What are the parts of an addition problem called?
The numbers being added are the addends and the answer is the sum. In 23 + 18 = 41, 23 and 18 are addends and 41 is the sum. The plus sign, +, shows the operation. Older books sometimes call the first number the augend and the second the addend, but today both are usually called addends.
What words in a word problem mean addition?
Common clue words are sum, total, plus, more than, increased by, combined, in all and altogether. “5 more than a number” is n + 5. Clue words are hints, not rules: always picture what is happening. If a tank already holds 12 gallons and you need its total after adding 9 gallons, you add.
Can adding make a number smaller?
Yes, if you add a negative number. 10 + (−4) = 6, which is less than 10, because adding −4 moves 4 steps left on the number line. Adding a positive number always makes the result larger, and adding 0 leaves it unchanged.
Number LineA number line is a straight line with numbers placed at equal distances along it, so every point stands for a number. Zero sits in the middle, positive numbers go to the right and negative numbers to the left. Numbers increase from left to right, so a number line shows order, distance and adding as moving.Number System · Grade 2, Grade 3, Grade 6NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Whole NumberA whole number is 0 or a positive counting number: 0, 1, 2, 3, and so on. Whole numbers do not include fractions, decimals, or negative numbers, so 12 is a whole number but 8.5, 34 and −3 are not. Every whole number is also an integer.Number System · Grade 3, Grade 4, Grade 5SubtractionSubtraction is the operation that finds how much is left when you take one amount away from another, or how far apart two numbers are. It is written with a minus sign: 9 − 4 = 5. The answer is called the difference. Subtraction undoes addition, so you can check 9 − 4 = 5 by adding 5 + 4 = 9.Operations · Grade 1, Grade 2, Grade 3AddendAn addend is any number that is added to another number. In 8 + 5 = 13, the addends are 8 and 5, and 13 is the sum. An addition can have two or more addends, and you can add them in any order. When one addend is missing, subtract the known addend from the sum to find it.Operations · Grade 1, Grade 2, Grade 3Commutative PropertyThe commutative property says that changing the order of the numbers does not change the result of adding or multiplying them: a + b = b + a and a × b = b × a. For example, 4 + 9 = 9 + 4 and 3 × 5 = 5 × 3. Subtraction and division are not commutative.Operations · Grade 1, Grade 3, Grade 6Associative PropertyThe associative property says that when you add or multiply three or more numbers, the way you group them does not change the result: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c). It lets you pick the easiest pair first. It does not work for subtraction or division.Operations · Grade 1, Grade 3, Grade 6Identity PropertyThe identity property says a number keeps its value when you add 0 or multiply by 1: a + 0 = a and a × 1 = a. So 0 is the additive identity and 1 is the multiplicative identity. For example, 15 + 0 = 15 and 15 × 1 = 15. It is how equivalent fractions work.Operations · Grade 1, Grade 3, Grade 6Negative NumberA negative number is a number less than zero, written with a minus sign, such as −3, −0.5 or −23. On a number line, negative numbers sit to the left of 0. They describe amounts below a starting point: temperatures below zero, depths below sea level, or money owed. Zero itself is neither negative nor positive.Number System · Grade 6, Grade 7DegreeA degree is the standard unit for measuring an angle, written with the symbol °. One degree is 1360 of a full turn, so a full turn is 360°, a straight angle is 180° and a right angle is 90°. An angle that turns through 25 one-degree angles measures 25°.Geometry · Grade 4, Grade 5, Grade 7Place ValuePlace value is the value of a digit based on its position in a number. In 3,507 the 5 is worth 500, because it sits in the hundreds place. Each place is worth 10 times the place to its right: ones, tens, hundreds, thousands, and after the decimal point, tenths and hundredths.Number System · Grade 3, Grade 4, Grade 5Decimal PointA decimal point is the dot that separates the whole-number part of a number from its fractional part. Digits to its left are ones, tens and hundreds; digits to its right are tenths, hundredths and thousandths. In 46.25 the point sits between 6 ones and 2 tenths, and you read it aloud as “and”.Number System · Grade 4, Grade 5, Grade 6SumA sum is the result you get when you add two or more numbers. In 8 + 5 = 13, the numbers 8 and 5 are the addends and 13 is the sum. The order does not change it: 5 + 8 is also 13. Word problems ask for a sum with words like total, in all and altogether.Operations · Grade 1, Grade 2, Grade 3MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4Repeated AdditionRepeated addition is adding the same number again and again. It is one way to understand multiplication: 3 + 3 + 3 + 3 = 4 × 3 = 12. The number of times you add is one factor, and the number being added is the other. It only works when all the addends are equal.Operations · Grade 2, Grade 3Additive InverseThe additive inverse of a number is the value that adds to it to give zero. The additive inverse of 9 is −9, because 9 + (−9) = 0, so it is the same thing as its opposite. To find it, change the sign: the additive inverse of −14 is 14, and 0 is its own.Number System · Grade 6, Grade 7, Grade 8IntegerAn integer is a whole number, its opposite, or zero: the counting numbers, their negatives, and zero. The integers are …, −3, −2, −1, 0, 1, 2, 3, … and go on forever in both directions. Integers have no fractional or decimal part, so 4.5 and 12 are not integers.Number System · Grade 6, Grade 7, Grade 8EstimateAn estimate is an answer that is close to the exact one, and to estimate is to find an answer that is close enough, usually by rounding or using friendly numbers. For example, 49 × 21 is about 50 × 20 = 1000. Estimation helps you check whether an exact answer is reasonable.Problem Solving · Grade 3, Grade 4, Grade 5ProductThe product is the answer you get when you multiply two or more numbers. In 4 × 6 = 24, the product is 24, and 4 and 6 are the factors. In word problems, the words “product of” always mean multiply, so “the product of 3 and 7” is 21.Operations · Grade 3, Grade 4, Grade 5