An example can be $$ 0010000.010 * 0001000.010 $$ which will give $$1000110.000100$$

But how was the operation done? More precisely, how was the decimal point placed there? Is the rule for placing the decimal point in the previous example exclusive for that example because the two numbers have equal number decimal places (3) ? or there is a general rule for placing the decimal point whether or not the number of decimal places are the same? (like will the rule you'll provide me still apply when multiplying 1011 * 0.010 ? Kindly place all zeros even the useless ones (after the last non-zero digit in the decimal part).

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  • $\begingroup$ Decimal binary numbers??? $\endgroup$ – bof Mar 2 '17 at 3:07
  • $\begingroup$ There's literally no difference from the multiplication algorithm you were taught in third grade, except for the digits you're allowed to use. $\endgroup$ – Matt Samuel Mar 2 '17 at 3:07
  • $\begingroup$ This course is digital systems, i know that i can convert it to decimal then multiply it then find the answer, but the professor might want you to show him the operation without converting to decimal then multiplying then reconverting the result to binary. $\endgroup$ – Joe Mar 2 '17 at 3:09
  • $\begingroup$ bof, yea sir they exit in life $\endgroup$ – Joe Mar 2 '17 at 3:09
  • $\begingroup$ LITRALLY NO BODY ANSWERED MY QUESTION RIGHT! :(((( $\endgroup$ – Joe Mar 2 '17 at 3:58

From DelightfulyLinux blog post:

Multiplying Binary Fractions

Align both rows by the least significant bit and multiply the same way as in decimal multiplication. The final position of the radix point is the sum of the number of radix point places from both factors.

That is why in your example final result has 6 bits in fractional part - because the sum of the fractional bits from both operands is $$ 3+3=6 $$


Let's do decimal first. Say $1.5\times 0.24$. But I'm going to write it like this:

$$(1\times 10^0 + 5\times 10^{-1})\times (2\times 10^{-1}+4\times 10^{-2})$$ Those negative powers of $10$ are very annoying. So let's write it as $$10^{-1}(1\times 10^1+5\times 10^0)\times 10^{-2}(2\times 10^{1}+4\times 10^0)$$ But this is just $$10^{-3}(15\times 24)$$ If you know how to multiply integers, you're home free.

Nothing here depended on the base being $10$. For your problem, just replace the $10$ with a $2$.

  • $\begingroup$ Man i know how to do what you've done, all the thing is to know the method without converting to decimal, its a digital systems university course midterm which is not multiple choice and they might tell u operate the multiplication without converting to decimal. $\endgroup$ – Joe Mar 2 '17 at 3:23
  • $\begingroup$ @Joe Could you clarify what you mean by "converting to decimal?" To me, writing the binary number $1.1$ as $1\times 2^0+1\times 2^{-1}$ is not converting to decimal. Converting to decimal would be to write $1.5$. $\endgroup$ – Matt Samuel Mar 2 '17 at 3:25

Here's a simple explanation using a concrete example. If there are 3 decimal places in the first number, you are dealing with thousandths. If there are 7 decimal places in the second number, you are dealing with ten millionths. The product of thousandths and ten millionths gives ten billionths, which has 10 decimal places.

5.021 = $\frac {5021} {1,000}$, 3.3942037 = $\frac {33,942,037} {10,000,000}$

$5.021*3.3942037=\frac {5021} {1,000} * \frac {33,942,037} {10,000,000}$=$\frac {5,021*33,942,037} {10,000,000,000}$


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