# binary division and remainder

Q=A/B , Q is a real number expressed as a pair of 8 bits:

• most significant 8 bits for the integer part
• least significant 8 bits for the fractional part
• the number is unsigned

for example:

0 0 1 0 1 1 0 1 . 0 1 0 1 0 0 0 0

Can you find the remainder of division if you know B?

example: 44/20

0010 1100 . 0000 0000 / 0001 0100 . 0000 0000

= 0000 0010 . 0011 0011 (2.1999...)

Now, to find the reminder I should multiply(0.19... * 20) 0000 0000 . 0011 0011 * 0001 0100 . 0000 0000

= 0000 0011 . 1111 1100

Now, can I say that the number is 4 because is greater than 3.5? I don't think this works for all the numers

There are exceptions, for example A=2 and B=172, I wonder if increasing the fractional part bits number solves the problem

2/172 :

0000 0010 . 0000 0000 /

1010 1100 . 0000 0000

=0000 0000 . 0001 0010

0000 0000 . 0001 0010 *

1010 1100 . 0000 0000

=0000 0000 . 1100 0001 (should be 2, or at least something greater than 1.5)

## 1 Answer

You should be multiplying the fractional part by the original denominator, not by the original numerator. Look at the same division in base ten: you get $2.2$, and the remainder is $0.2\cdot20=4$, which is correct.

After dividing $a$ by $b$, you have an integer part $n$, say, and a fractional part $\alpha$: $\frac{a}b=n+\alpha$. Multiplying both sides by $b$ gives you $a=nb+\alpha b$, or $\alpha b=a-nb$; clearly $\alpha b$ here is the remainder.