A 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.
The addends are the parts; the sum is the whole. On a number line, start at one addend and count on by the other to land on the sum.
Formal definition and notation
The sum of the numbersa1, a2, …, an is the single number a1 + a2 + ⋯ + an. Each number being added is an addend. Because addition is commutative and associative, the sum does not depend on the order or the grouping of the addends. In algebra, an expression such as x + 7 is also called a sum, even though it cannot be simplified to one number.
Symbol
Read it as
Meaning
+
plus
the addition sign; a + b is the sum of a and b
Σ
sigma
capital Greek S for sum; Σx means add up all the values of x (used in algebra and statistics)
A product is the result of multiplying. For 6 and 3, the sum is 6 + 3 = 9 but the product is 6 × 3 = 18.
Sumvs.Difference
A difference is the result of subtracting. For 10 and 4, the sum is 14 but the difference is 10 − 4 = 6.
Sumvs.Addend
An addend is one of the numbers you add; the sum is the result. In 2 + 9 = 11, 2 and 9 are addends and 11 is the sum.
Quick check: 3 questions
Question 1 of 3
What is the sum of 19 and 6?
Show answer and explanation
Answer: C.9 + 6 = 15: write 5, carry 1. Then 1 + 1 = 2 tens, so the sum is 25. 15 forgets the carried ten, 13 is the difference, and 114 is the product.
Question 2 of 3
Which expression means the sum of a number and 9?
Show answer and explanation
Answer: A. Sum means add, so the sum of x and 9 is x + 9. 9x is a product, x − 9 is a difference, and x9 is a quotient.
Question 3 of 3
The sum of three numbers is 50. Two of them are 18 and 14. What is the third number?
Show answer and explanation
Answer: D. The known addends add to 18 + 14 = 32, so the third is 50 − 32 = 18. 32 stops after adding the two, 82 adds all three numbers, and 4 is 18 − 14.
Sum: frequently asked questions
What does sum mean in math?
Sum means the answer to an addition problem. When you add 12 and 30, the sum is 42. The word also names the whole expression, so 12 + 30 is called a sum even before you work it out. In word problems, total, in all, altogether and combined usually tell you to find a sum.
Can a sum be smaller than the numbers you add?
Yes, when at least one addend is negative. For example, −9 + 4 = −5, and −5 is smaller than 4. With whole numbers only, the sum is always at least as large as each addend, because adding 0 or more never moves you to the left on the number line.
What does the symbol Σ mean?
The capital Greek letter sigma, Σ, means sum. It is a shorthand for adding a list of numbers. If the test scores are 70, 80 and 90, then Σx = 70 + 80 + 90 = 240. You will meet it in statistics formulas, such as the mean, Σxn, and in algebra with series.
NumberA 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 2AddendAn 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 3AdditionAddition 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.Operations · Kindergarten, Grade 1, Grade 2ExpressionAn expression is a number, variable, or combination of numbers and variables joined by operations, without an equal sign. Examples are 7, 4x + 9 and 3(a − 2). Expressions can be simplified, but they are not solved unless they are part of an equation. You evaluate one by replacing each variable with a number.Algebra · Grade 6, Grade 7, Grade 8Commutative 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 6IntegerAn 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 8MeanThe mean is the sum of the values divided by the number of values. It is the arithmetic average: add every value and divide by how many there are. For 3, 7, 5 and 9, the mean is 24 ÷ 4 = 6. The mean is the balance point of the data, so one outlier can pull it far.Statistics · Grade 5, Grade 6, Grade 7DataData are facts or values collected to answer a question, such as survey answers, measurements or counts. Numerical data are numbers you measure or count, like heights; categorical data are labels or groups, like favorite colors. Data are organized in tables and shown in graphs so patterns are easy to see.Statistics · Grade 2, Grade 3, Grade 6Place 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 5SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2DifferenceA difference is the result of subtracting one number from another. In 12 − 7 = 5, the difference is 5. It tells how much bigger one number is than another, or how far apart they are on a number line. “The difference of a and b” means a − b, in that order.Operations · Grade 1, Grade 2, Grade 3QuotientA quotient is the answer you get when you divide one number by another. In 20 ÷ 4 = 5, the quotient is 5. The number being divided is the dividend and the number you divide by is the divisor. When the division is not exact, the quotient comes with a remainder, or as a decimal or fraction.Operations · Grade 3, Grade 4, Grade 5Number 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 6ProductThe 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 5Whole 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 5Order of OperationsThe order of operations is the agreed sequence for evaluating an expression: parentheses first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. Without it the same expression would have several answers. Many students remember it as PEMDAS.Algebra · Grade 5, Grade 6, Grade 7SubtractionSubtraction 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 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 4