What Is a Sum in Math?

A sum is the result of adding numbers together. In 12 + 9 = 21 the sum is 21, and the numbers being added are called addends or terms.

When a question says “find the sum of 12 and 9”, it is asking you to add them. That is the whole of the definition, and the reason it is worth stating is that the word is almost always met as an instruction rather than as a noun you had to look up.

One wrinkle: in British schools sum is often used loosely to mean any arithmetic problem at all, as in “do your sums”. In mathematical writing it means addition specifically, and that is the meaning every exam question uses.

Each operation has its own word for its result. Answering with the wrong one loses the mark even when the arithmetic was right, which is the single most common reason this vocabulary is taught at all.

Sum and product are the pair that get swapped most often, because both are symmetric: the order of the numbers does not change either answer, so nothing about the phrasing warns you when you have picked the wrong one. Difference and quotient at least protest, since 9 − 12 and 12 − 9 differ.

The companion guide to products covers multiplication’s side of that pairing, including why a product can come out smaller than the numbers you multiplied.

For a short list, add them. For a long or structured one, there is usually a shortcut worth knowing.

1. Line up place values if you are adding by hand. Units under units, tens under tens.
2. Group to make tens. In 7 + 8 + 3 + 2, pairing 7 with 3 and 8 with 2 gives 10 + 10 = 20 with no carrying.
3. For fractions, find a common denominator first. 2/3 + 3/4 becomes 8/12 + 9/12 = 17/12.
4. For a run of consecutive whole numbers, use the formula rather than adding them one at a time.

That last shortcut is worth its own paragraph. The sum of the whole numbers from 1 to n is n(n + 1)/2. For 1 to 100 that is 100 × 101 / 2 = 5050, which is a good deal faster than ninety-nine additions. The story that Gauss worked this out as a schoolboy may or may not be true, but the reasoning behind it is: pair the first with the last, the second with the second-last, and every pair sums to 101. There are fifty such pairs.

Once a sum has more than a handful of terms, writing them all out stops being practical. The capital Greek sigma, Σ, is the shorthand.

The rule can be anything. Σ over i from 1 to 4 of i² means 1 + 4 + 9 + 16, which is 30. Changing the rule from i to i² is the only difference between the two.

Sigma notation is not a different kind of arithmetic. It is a compact way of writing down which additions to perform, and the summation calculator performs exactly the ones it describes.

Three sums, each using a different one of the methods above.

The arithmetic sequence formula guide covers where that evenly spaced run comes from in the first place.

The last two are the same formula. n(n + 1)/2 is just n/2 × (1 + n) with the first term set to 1.

Outside a maths lesson, most sums are computed by a machine, and two conventions are worth recognising.

• =SUM(A1:A10) adds a range in Excel, Google Sheets and every spreadsheet that copies them. It ignores text and empty cells rather than erroring, which is convenient and occasionally hides a problem: a column of numbers stored as text sums to zero without complaint.
• Σx in statistics means the sum of all the values in a data set. It is the numerator of the mean, so mean = Σx / n. The guide to the mean covers what that division is doing.
• Running or cumulative sums add each value to the total so far, producing a column of subtotals rather than one figure. Bank statements are the everyday example.

The statistical use is the one to watch, because Σx and Σx² look similar and mean different things. The first adds the values; the second squares each value and then adds. They appear side by side in the standard deviation formula and swapping them gives a number that is wrong without looking wrong.

• Multiplying when asked for a sum. The sum of 12 and 9 is 21, not 108.
• Adding fractions across the top and bottom. 2/3 + 3/4 is not 5/7. Denominators have to match before the numerators can be added.
• Using n(n+1)/2 on a run that does not start at 1. For a general arithmetic run use n/2 × (first + last).
• Confusing Σx² with (Σx)². The first squares then adds; the second adds then squares, and they are not equal.
• Trusting a spreadsheet SUM over text. Numbers stored as text are skipped silently.

• A sum is the result of adding; the numbers added are addends or terms.
• Sum, product, difference and quotient name the results of the four operations.
• The sum of 1 to n is n(n + 1)/2; from 1 to 100 that is 5050.
• Any evenly spaced run sums to n/2 × (first + last).
• Σ is shorthand for a sum, not a different operation.

The summation calculator evaluates a sigma expression term by term, and the product guide covers the operation sum is most often confused with.