What is the difference between operator “=” and “==” in Bash?

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You must use == in numeric comparisons in (( ... )):

$ if (( 3 == 3 )); then echo "yes"; fi
yes
$ if (( 3 = 3 ));  then echo "yes"; fi
bash: ((: 3 = 3 : attempted assignment to non-variable (error token is "= 3 ")

You may use either for string comparisons in [[ ... ]] or [ ... ] or test:

$ if [[ 3 == 3 ]]; then echo "yes"; fi
yes
$ if [[ 3 = 3 ]]; then echo "yes"; fi
yes
$ if [ 3 == 3 ]; then echo "yes"; fi
yes
$ if [ 3 = 3 ]; then echo "yes"; fi
yes
$ if test 3 == 3; then echo "yes"; fi
yes
$ if test 3 = 3; then echo "yes"; fi
yes

“String comparisons?”, you say?

$ if [[ 10 < 2 ]]; then echo "yes"; fi    # string comparison
yes
$ if (( 10 < 2 )); then echo "yes"; else echo "no"; fi    # numeric comparison
no
$ if [[ 10 -lt 2 ]]; then echo "yes"; else echo "no"; fi  # numeric comparison
no

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