How do you convert $6.75$, in base $10$, to base $2$?

Question

Can I get a detailed explanation on how to convert $6.75$ base $10$ to $2$.

I have really tried all I can but I don't still get it.

Answer 1

$6=110_2$ and $0.75=\dfrac34=\dfrac{11_2}{100_2}=0.11_2$ then $6.75=110.11_2$

Answer 2

$2^2 < 6.75 < 2^3$ so $100_2 < 6.75_{10} < 1000_2$

So $6.75 = 100_2 + \color{blue}{2.75}_{10}$.

$2^1 < 2.75$ so $10_2 < 2.75_{10} < 100_2$

$2.75 = 10_2 + \color{blue}{.75}_{10}$.

$6.75 = 100_2 + 10_2 +\color{blue}{.75}_{10}=110_2+\color{blue}{.75}_{10}$

$2^0 \not < .75$

$.75_{10} = 0_2 + \color{blue}{.75}_{10}$

$6.75 = 110_2 + 0_2 + \color{blue}{.75}_{10}=110_2 +\color{blue}{.75}_{10}$

$\frac 12 = 2^{-1} < .75 < 2^0=1$ so $0.1_2 < .75 <1_2$

$.75 = 0.1_2 + \color{blue}{.25}_{10}$

$6.75 = 110_2 + 0.1_2 +\color{blue}{.25}_{10}= 110.1_2 +\color{blue}{.25}_{10}$

$2^{-2} = \frac 14 =.25 < 2^{-1}$ so $0.01_2 = 0.25_{10}< 0.1_2$

$.25 = 0.01_2$

$6.75 = 110.1_2 + 0.01_2 = 110.11_2$

That's it.

Answer 3

$6.75 = 27/4 = 3\cdot 9/4 = (2+1)\cdot(8+1)/4 = 11_2\cdot 1001_2/100_2 = 11011_2/100_2 = 110.11_2$