Advanced Bash-Scripting Guide:
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Chapter 15. Arithmetic Expansion

Arithmetic expansion provides a powerful tool for performing arithmetic operations in scripts. Translating a string into a numerical expression is relatively straightforward using backticks, double parentheses, or let.

Variations

Arithmetic expansion with backticks (often used in conjunction with expr)

z=`expr $z + 3`          # 'expr' does the expansion.

Arithmetic expansion with double parentheses, and using let

The use of backticks in arithmetic expansion has been superseded by double parentheses $((...)) or the very convenient let construction. 
z=$(($z+3))
# $((EXPRESSION)) is arithmetic expansion.  # Not to be confused with
                                            # command substitution.

let z=z+3
let "z += 3"  #If quotes, then spaces and special operators allowed.
# 'let' is actually arithmetic evaluation, rather than expansion.
All the above are equivalent. You may use whichever one "rings your chimes".

Examples of arithmetic expansion in scripts:

  1. Example 12-6

  2. Example 10-14

  3. Example 26-1

  4. Example 26-5

  5. Example A-18

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