# Sigma Notation

Sigma notation provides a way to compactly and precisely express any sum, that is, a sequence of things that are all to be added together. Although it can appear scary if you’ve never seen it before, it’s actually not very difficult. Here’s what a typical expression using sigma notation looks like: $\displaystyle\sum_{k=a}^b f(k)$

We would read this as “the sum, as k goes from a to b, of f(k).” In plain English, what this means is that we take every integer value between a and b (inclusive) and substitute each one for k into f(k). This results in a bunch of values which we add up.

Let’s go through each part of that and see what they mean in more detail:

• $\Sigma$: this is a capital sigma, the eighteenth letter of the Greek alphabet. It is not an ‘E’! Sigma corresponds to the English letter ‘S’; ‘S’ is for ‘sum’.
• k: The k on the left side of the equals is called the index variable or the index of summation, or sometimes just the index. It will take on all the integer values between a and b (inclusive).
• a, b: a is the starting index and b is the ending index.
• f(k): this is the expression that describes each term in the sum. For each value of k between a and b, f(k) will be some value which gives one term in the sum.

If you’re still confused, don’t worry; an example should make things clear! $\begin{array}{rcl} \sum_{k=2}^5 (k^2 + 1) &=& (2^2 + 1) + (3^2 + 1) + (4^2 + 1) + (5^2 + 1) \\ &=& 5 + 10 + 17 + 26 = 58 \end{array}$

See how that works? We took every value of k between 2 and 5 inclusive, and substituted each into the expression $(k^2 + 1)$; then we added everything up.

As a bonus, once you understand sigma notation, you understand Big Pi notation for free: a Big Pi ( $\Pi$) works exactly the same as a Big Sigma, except it denotes multiplication instead of addition (‘P’ is for ‘product’). For example: $\begin{array}{rcl} \prod_{k=2}^5 (k^2 + 1) &=& (2^2 + 1) \cdot (3^2 + 1) \cdot (4^2 + 1) \cdot (5^2 + 1) \\ &=& 5 \cdot 10 \cdot 17 \cdot 26 = 22,100. \end{array}$

### 97 Responses to Sigma Notation

1. Pingback: minor mathhelp needed

2. Observer says:

Slight error in summation example: first iteration (2 squared minus 1) shows minus sign where it should be plus sign.

3. Brent says:

Hmm? I see a plus sign. The image gets kind of squashed, though, so maybe for some reason you’re not seeing the crossbar on the plus?

4. John says:

Is there a formula to express a certain number in sigma notation? if i gave you a number like 19018537475 could you express it in sigma notation and show me how you do it? please help me on this!!!

5. Brent says:

John: there are lots and lots of ways to express any number using sigma notation. For example, I could write your number as $\displaystyle \sum_{i=1}^{19018537475} 1$,

or as $\displaystyle \sum_{i=1}^{3803707495} 5$.

There are probably many more complicated ways to write it that require more cleverness to come up with. However, this is sort of a strange thing to do with sigma notation! It should be used to make writing complicated things simpler, not to make simple things complicated.

6. John says:

haha thanks

7. Dave Potter says:

Using Sigma notation how could I express 24 using exactly thre 7’s?

8. Dave Potter says:

Sorry previous question should read exactly three 7’s

9. Brent says:

Dave: I try not to answer questions of the form “solve this puzzle for me”. In any event, the question seems somewhat ambiguous — what are you allowed to use *besides* the 7’s?

10. Dude says:

Dave:

Sigma i = 0 to 2 (7 + i)

11. steve the pirate says:

12. Math is #1 says:

This is really helpful! thanks a lot!

13. telamonides says:

Thank you for the explanation!

14. Mia says:

Is there a certain way how to figure out k? I am practicing for a test and my arithmetic series goes like so “8+5+2-1-4-7-10-13” My mind always wants to say that k is equal to a sub n minus three, but I know that’s not correct. Do you have any tips on how to find out k? And would b or the ending index be 8 in this series? Thanks!

15. Maze says:

What if the k is too large for manually adding until the kth term?

16. rea says:

what if sigma y=0? 3
Σ (x-y+3)^3
y=0

hope you understand. thankyou :))

17. Tony says:

it is quite easy especially for youngsters thanks tony SA

18. ahorsealone says:

This is pretty neat. So does two sigma signs sort of work like two for loops in MatLab?
If in each loop Im just performing an additive operation?

• Brent says:

Yes, that’s right!

19. Rachel says:

How would you find the sum of:
∑_(i=1)^n▒〖i^2+3i+4〗

• Brent says:

20. AshutoshPV says:

sigma a^2 * ( b – c )

• AshutoshPV says:

Actually I know the answer….but I don’t know how to arrive at the answer

• Brent says:

I don’t understand the question. Sigma by itself does not mean anything, you have to specify what index you are summing over. Perhaps if you say what the answer is supposed to be that might help.

21. zakia hussain says:

∑ (i= 3 to n) for (i^2-3)
anothr ques
∑(i=0 to n) fr ( i^2 +5)

• Brent says:

Hi, you should be able to solve this using my explanation here.

22. suman mullick says:

hi, i find ur explanations really very interesting. i wud like to know if you have written something on the definition or explanation of definite integrals. If u havnt, can you plz tell me whr i shud look for it?

23. Nancy K says:

Thank you for this, really useful for someone like me whose maths has all fallen out of my head in the 10+ years since I last studied it. I do have one question – I’ve got an example that includes X bar on both sides of the equation, and I’m not certain what it means; does it simply represent the mean value of X for the given index?

• Brent says:

Hmm, I don’t know, there’s not necessarily a single standard meaning for X bar. Check towards the beginning of the book or document where you found it to see if it defines the notation it is using, I guess.

• Nancy K says:

It’s part of a course, and the lecturer just kind of casually dropped it in there! I shall approach him for clarification. Thank you again, this is a really nice resource for folks like me 🙂

24. Savannah says:

This helped me understand the concenpt of sigma notation so much better! Thank you!

25. Angel says:

i am really confused 😦

• Brent says:

• srinivas kota says:

thank u

26. Jorge De La Cerda says:

Very well explained. Thank you!

27. mathMatrix 101 says:

Thankx to this . ^___^

28. bob@hotmail.com says:

I’m still trying to understand the whole thing, as described above. One thing I don’t understand is why you towards the top wrote Sigma ‘E’ with b above, k=a below, f(k) to the right, and then a little bit later in your writing, moved the small numbers to the right of the ‘E’. Sorry if this makes me seem obtuse, but are you just writing it another way with the top and bottom numbers just to the right of the sigma symbol corresponding to those that were above and below it before, or does this new style of formula mean something else?

• Brent says:

Ah, good question! They mean the same thing. Writing the top and bottom numbers just to the right of the sigma symbol is just a different style of writing it which takes less vertical space and is often used in the middle of a paragraph (instead of on a separate line all by itself).

29. Derek Owens says:

Will this be taught in calculus next year? I am only in Alg II Trig right now but needed to know how to read the equation for lissajou curve for a project I’m doing–requiring me to look into dirac delta functions, which led me to need to know what sigma notation is. I’m worried we won’t be taught this in school, the same way we were not taught phi in geometry and one on one tutoring would help.

• Brent says:

I don’t know about your particular school, of course, but typically sigma notation is indeed covered in most calculus courses. If I were you, though, I would stop worrying too much about what you are or aren’t taught in school—the fact is that there are infinitely more things you could know than what there is time to teach in school! Just learn whatever seems interesting or important to you—as it seems you are already doing. No need to wait for it to be officially covered in school.

30. Bart says:

I’ve been studying an equation with a sigma in it.
There is nothing written above the sigma such as the b in your example, and down below instead of k=a (or equivalent) there is just a j.
Any advice on how to understand it?

• Brent says:

Hi Bart, in that case j is the “index variable” (just like the k in my example), but the a and b are left implicit. It means something like “sum over EVERYTHING”, where you are supposed to infer what “everything” means from the context.

31. Bart says:

Thanks Brent, tremendous service. Let’s see if I get it. In the formula following the sigma the writer has a variable H subscript j. So he means: do the summing for every instance of H.
Key point, if I understand correctly: the j is an arbitrary identifier with no special meaning; he had no need to define beforehand what he meant by j; he could have used any other letter instead providing it didn’t appear anywhere else in the equation.

• Brent says:

Yes, that’s exactly right!

32. cgdermot says:

What about numbers to the left of Sigma? Is it – / on the left, with + x to the right ?

• Brent says:

I am not entirely sure I understand your question. Perhaps you are asking about the meaning of something like $s \sum_{k=a}^b t$? In this case the $s$ is not part of the sigma expression at all. It just means the same thing it usually means to put something next to something else—multiplication. That is, it means $s \times \left(\sum_{k=a}^b t \right)$.

• cgdermot says:

That explains things perfectly. Thank you for the help. Brushing up on my 3d Math for HLSL. Finding mathlesstraveled.com to be a great resource. Thanks again.

33. ibrahim says:

34. Madison says:

How would you write this group of numbers: 3+9+27+81+243 in sigma notation?

• Brent says:

Well, first, do you notice any patterns?

35. someone says:

math’s beauty? you must be joking 😀

• Brent says:

Why do you think I am joking?

36. Chase McDonald says:

I’m a student currently in Algebra “D” and this stuff is out of my league but due to my curiosity I decided to research it and now within 5 minutes of learning about it I now understand the basics of what it is and how it works, I love how you’ve explained it so easily and formally to where it is easy to understand. Also, how to verbally say it in an equation. Thank’s a million!

37. Chase McDonald says:

Algebra I “D” to the above comment (in other words i’m a freshman in highschool) Thanks again 😀

38. lohith says:

The indexes number, r in the equations correspond to the
column, row indexes in the specified matrix.
p0(out putg) = sigma r:ITM(0,r)=1 [P(out putg)[0,r]]

39. lohith says:

What does that statement mean?

40. Joe says:

• Brent says:

Sure. I published it on February 21, 2007.

41. Wazza says:

Thanks Brent. Is there a sigma notation or some other for describing counting the members of a set that meet some criteria. For example a set of documents K and we want to describe counting the documents where the read date R is within L days of the publish date P.

• Brent says:

You can use set comprehension notation to describe such a set, for example, $\{ D \mid D \in K, |\mathrm{read\_date}(D) - P| \leq L \}$; if you want the size of a set you can use notation such as e.g. prefixing the set with $\#$. Alternatively, you could use Iverson brackets (https://en.wikipedia.org/wiki/Iverson_bracket) with Sigma notation, e.g. $\sum_{D \in K} [|\mathrm{read\_date}(D) - P| \leq L]$.

42. Angel says:

Hello,

I understand that the Big Pi function is only defined for non-negative integers. However, the gamma function extends it so that all complex values of z are defined as long as z does not equal real negative integer. Is there such an analogous function that extends the Sigma function such that it is defined for lower and upper bounds equal to complex numbers? I read an essay on an extension of the definition of summation such that it is defined for any complex upper bound n or lower bound m, but the essay was too complicated due to its formality and I simply did not understand it. Perhaps I can provide you the link, and you can make an article on this — such that students that are still stuck on calculus 1 can understand it? Thank you.

• Brent says:

Fascinating! I had not seen this before. I don’t know whether I can find an appropriate way to motivate and explain this, but I will ponder it.

43. Katie Millard says:

I have no idea what is going on here but am looking for a mathematical way of writing ‘one in a million’. Can anyone help?? Many thanks

• Brent says:

Hi Katie, I would express “one in a million” mathematically using the fraction 1/1000000, which fairly accurately captures the idea of having only one thing out of a total of one million things.

44. Jas says:

I’m trying to decode my Biometry homework and this was super helpful (unlike the book)… Thanks a bunch! (:

45. Suraj says:

What is summation of (50 choose j) where the starting value of j is 0 and the end value is 50?

• Brent says:
46. Ganesh says:

What does summation without index and upper limit means

• Brent says:

Usually it means that the summation happens over the “obvious” variable and range of summation. What that means is of course highly dependent on context.

47. Preacher Doom says:

How Do Brackets affect sigma notation. Today I saw a question that had the exact same expression but the other differed only with that it was in brackets. I was asked to give the difference, and then I gave the answer Zero on the presumption that they’re totally similar. But the correct answer was 12. Please help.

• Brent says:

Hi there,

Brackets are used for lots of things, so I can’t say offhand what the difference might be. If you posted the exact expressions you saw I might be able to guess.

48. Kevin says:

Hi Brent,

could you explain how to take a constant outside of a summation and bring it inside the summation?

additionally I’m looking to see how bringing a constant outside a summation may effect an unknown end index number like b= N-1

many thanks

Kevin

49. Justin says:

So it is just like a for() in programming? I just understand it like that.

• Brent says:

It is very similar, but specifically for adding (you could also use a for loop in many other ways).

To be concrete, $\sum_{i=a}^b f(i)$ could be implemented by (Java syntax):
 int sum = 0; for (int i = a; i <= b; i++) { sum += f(i); } 

50. Eric Ng'eno says:

Brilliant
Accessible

51. Stephanie Caldwell says:

Now if I can just remember it later.

52. Pingback: python - Codifica sigma formula?

53. Pingback: Formal PIE | The Math Less Traveled

54. Ve'Laura Bradshaw says:

Thank you!

55. aeaskyme says:

Thanks. I clearly understand it. Although it didn’t state the shortcut on how to sum big ending indexes.

• Brent says:

I don’t think I’m familiar with that shortcut. Can you explain what you mean?

56. Jose Marquez says:

Thanks for this! Super helpful for someone getting into statistics such as myself!

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