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# Lesson 10.0: Sigma Notation

## Symbols

$\text{expression directly above}$
⠨⠔

$\text{expression directly below}$
⠨⠢

$\sum \phantom{\rule{.3em}{0ex}}\text{capital greek sigma}$
⠠⠨⠎

## Review

Grade 1 mode is turned on by grade 1 indicators and by the numeric indicator. When grade 1 mode is turned on by the numeric indicator, it is turned off by a space, hyphen, or dash. When complex algebraic equations contain indicators such as superscript, subscript, fractions, radicals, and/or letters standing alone, it is best to enclose them in grade 1 passage indicators to ensure the symbols are well defined without the need for grade 1 symbol indicators. When a complex expression is comprised of a single symbol sequence, a grade 1 word indicator will be enough to ensure that the various indicators are well defined without the need for grade 1 symbol indicators.

## Explanation

Sigma notation also known as summation notation, is a form of mathematical shorthand for a sum of the values of a variable written as a compact expression. The expression is centered around the Greek capital sigma, with the element being summed appearing directly to the right of the sigma. The index of the summation appears directly below the sigma and the ending value of the summation is placed directly above the sigma.

Braille has four level change indicators: level change up (superscript), level change down (subscript), expression directly above, and expression directly below. Some or all of these indicators may be used in sigma notation. The scope of a level change indicator is the next item. Braille grouping indicators must be used to enclose an expression that includes more than one item.

The Greek capital sigma is formed with three cells, dot six in the first cell, dots four six in the second cell, and dots two three four in the third cell. The expression below the sigma is brailled first, followed by the expression above the sigma, and finally the element(s) to the right of the sigma. The expression directly below indicator is two cells, dots four six in the first cell and dots two six in the second. The expression directly above indicator is dots four six, dots three five. The extent of the expression directly below and expression directly above indicators is the next item. Braille grouping indicators are required if the expression extends beyond one item. The element directly to the right of the sigma is brailled as it is shown in print.

### Example 1

$\sum _{1}^{6}3$
⠠⠨⠎⠨⠢⠼⠁⠨⠔⠼⠋⠼⠉

When a sign of comparison is not on the base line, it is preferable to braille it unspaced from the surrounding content.

### Example 2

$\sum _{n=1}n$
⠰⠰⠠⠨⠎⠨⠢⠣⠝⠐⠶⠼⠁⠜⠝

### Example 3

$\sum _{n=1}^{9}n$
⠰⠰⠠⠨⠎⠨⠢⠣⠝⠐⠶⠼⠁⠜⠨⠔⠼⠊⠝

### Example 4

$\sum _{n=1}^{9}5n$
⠰⠰⠠⠨⠎⠨⠢⠣⠝⠐⠶⠼⠁⠜⠨⠔⠼⠊⠼⠑⠝

### Example 5

$\sum _{p=7}^{20}2p+1$
⠰⠰⠠⠨⠎⠨⠢⠣⠏⠐⠶⠼⠛⠜⠨⠔⠼⠃⠚⠼⠃⠏⠐⠖⠼⠁

### Example 6

$\sum _{I=100}^{150}{I}^{2}$
⠰⠰⠠⠨⠎⠨⠢⠣⠠⠊⠐⠶⠼⠁⠚⠚⠜⠨⠔⠼⠁⠑⠚⠠⠊⠔⠼⠃

It is preferable to keep a mathematical expression entirely on one line. When it is necessary to divide the expression, choice of a runover site should follow mathematical structure. It is best to divide before a sign of comparison or operation.

### Example 7

$\sum _{t=1}^{3}2t=2\left(1\right)+2\left(2\right)+2\left(3\right)$
⠰⠰⠰⠠⠨⠎⠨⠢⠣⠞⠐⠶⠼⠁⠜⠨⠔⠼⠉⠼⠃⠞⠀⠐⠶⠀⠼⠃⠐⠣⠼⠁⠐⠜
⠐⠖⠼⠃⠐⠣⠼⠃⠐⠜⠐⠖⠼⠃⠐⠣⠼⠉⠐⠜⠰⠄

### Example 8

$2\sum _{r=1}^{3}3r$
⠼⠃⠠⠨⠎⠨⠢⠣⠗⠐⠶⠼⠁⠜⠨⠔⠼⠉⠼⠉⠗

### Example 9

$\sum _{k=1}^{30}\left(4k+6\right)$
⠰⠰⠠⠨⠎⠨⠢⠣⠅⠐⠶⠼⠁⠜⠨⠔⠼⠉⠚⠐⠣⠼⠙⠅⠐⠖⠼⠋⠐⠜

### Example 10

$\sum _{k=3}^{7}\frac{k+1}{k}$
⠰⠰⠠⠨⠎⠨⠢⠣⠅⠐⠶⠼⠉⠜⠨⠔⠼⠛⠷⠅⠐⠖⠼⠁⠨⠌⠅⠾