Can bash do floating-point arithmetic without using an external command?

No.

Bash cannot perform floating point arithmetic natively.

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This is not what you're looking for but may help someone else:

Alternatives

1. bc

bc allows floating point arithmetic, and can even convert whole numbers to floating point by setting the scale value. (Note the scale value only affects division within bc but a workaround for this is ending any formula with division by 1)

$ echo '10.1 / 1.1' | bc -l
9.18181818181818181818
$ echo '55 * 0.111111' | bc -l
6.111105
$ echo 'scale=4; 1 + 1' | bc -l
2
$ echo 'scale=4; 1 + 1 / 1' | bc -l
2.0000

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2. awk

awk is a programming language in itself, but is easily leveraged to perform floating point arithmetic in your bash scripts, but that's not all it can do!

echo | awk '{print 10.1 / 1.1}'
9.18182
$ awk 'BEGIN{print 55 * 0.111111}'
6.11111
$ echo | awk '{print log(100)}'
4.60517
$ awk 'BEGIN{print sqrt(100)}'
10

I used both echo piped to awk and a BEGIN to show two ways of doing this. Anything within an awk BEGIN statement will be executed before input is read, however without input or a BEGIN statement awk wouldn't execute so you need to feed it input.

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3. Perl

Another programming language that can be leveraged within a bash script.

$ perl -l -e 'print 10.1 / 1.1'
9.18181818181818
$ somevar="$(perl -e 'print 55 * 0.111111')"; echo "$somevar"
6.111105

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4. Python

Another programming language that can be leveraged within a bash script.

$ python -c 'print 10.1 / 1.1'
9.18181818182
$ somevar="$(python -c 'print 55 * 0.111111')"; echo "$somevar"
6.111105

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5. Ruby

Another programming language that can be leveraged within a bash script.

$ ruby -l -e 'print 10.1 / 1.1'
9.18181818181818
$ somevar="$(ruby -e 'print 55 * 0.111111')"; echo "$somevar"
6.111105

"Can bash also do floating-point arithmetic without using an external command?"

Nope.

robert@pip2:/tmp$ echo $((2.5 * 3))
bash: 2.5 * 3: syntax error: invalid arithmetic operator (error token is ".5 * 3")