arithmetic - How can you multiply decimal values without using a multiplication or division operator?

I figured out a problem challenging me to write code to do this with integers, but I was wondering how I would have done it if it had been decimals and I couldn't figure it out.

Answer

Let

with $a_1, b_1 \in \Bbb Z$ and $a_2, b_2 \in \Bbb Z$ and $a_2 \le 0$ and $b_2 \le 0$ and

and

Both the numbers $a_0$ and $b_0$ are either integers or actual (finite length) decimals with this representation.

Now

Of course if $a_2 + b_2 \lt 0$ we have to make sure that the form is 'fixed up' to give a pure representation - you have to check/cancel at least one trailing zero (least significant digit) in the product $a_0 b_0$ when represented as a string of digits.

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Example: $a_0 = 5$ and $b_0 = 0.2$:

The exponent is negative and the integer part has $0$ at the far right end. So you remove a zero and increment the exponent.

Since the exponent isn't negative there is nothing to check - the product is an integer and we are writing it in standard $\text{integer or decimal}$ form using $\text{(2)}$.