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Skill 6 of 11

Sigma summation

We could easily have many more forces to add. While towing your boat, the stream might get stronger and a stormy wind may appear, so you’ll need help from nearby strangers:

$$\vec F_\text{result}= \vec F_\text{you}+ \vec F_\text{friend}+ \vec F_\text{stream}+ \vec F_\text{stranger1}+ \vec F_\text{stranger2}+ \vec F_\text{stranger3}+ \vec F_\text{stranger4}+ \vec F_\text{wind}$$

We start having a lot of terms to add up here. Let’s invent a simple symbol to mean ‘add them all, no-matter how many there are’. Let’s pick the Greek uppercase letter Sigma $\sum$ and write:

$$\vec F_\text{result}=\sum\vec F$$

Some would call this: sigma notation.[1,2]

If we in the future want to know how much force only the four strangers aided with, we can quickly invent a way of writing that we only want those four added up. We could for example quickly number all the terms: $\underbrace{\vec F_\text{you}}_1+ \underbrace{ \vec F_\text{friend}}_2+ \underbrace{ \vec F_\text{stream}}_3+ \underbrace{ \vec F_\text{stranger1}}_4+ \underbrace{ \vec F_\text{stranger2}}_5+ \underbrace{ \vec F_\text{stranger3}}_6+ \underbrace{ \vec F_\text{stranger4}}_7+ \underbrace{ \vec F_\text{wind}}_8$ – let’s call this numbering indexes1 – and then indicate that we only want to sum up those between number 4 to 7 (from index 4 to 7) in for example this way:

$$\vec F_\text{strangers}=\sum_4^7\vec F$$

There is of course no need to write these numbers when we want to sum up all terms.


References:

  1. Summation Notation’ (web page), Columbia University, www.columbia.edu/itc/sipa/math/summation.html (accessed Aug. 28th, 2019)
  2. The Summation Symbol’ (article), HEC Montreal, www.hec.ca/en/cams/help/topics/The_summation_symbol.pdf (accessed Aug. 28th, 2019)
  3. Online Etymology Dictionary’ (dictionary), Douglas Harper, www.etymonline.com