mathemetika+

You're very right. This feeling always comes to me and it always holds me back from learning. I always want to delve into details, I can't be unreasonable. Well, we as humans repeat things most of the time without understanding (without conciousness). I remeber a study on apes (bonobos) on the way they learn compared with the way we learn. I turned out (bonobos) don't just repeat what the others do ! Somehow they do it for a reason, but when the researcher compared their result with a child, it turned out the opposite, repeating and no logical thinking of the behavior. You can simply think of that and prove it to yourself, imagine a child grown in a society let's say in that society each individual unrinate on the street as urinating is not ritual inside homes, just assuming that, you would think that child be different ? absolutely not, at least for the period of his childhood. So repeating is our way of learning, but we suffer if we think of logic. You may agree with me.

I tho When we divide by [latex]2[/latex], what happens is simply shifting bits to the right side. For example: [latex]19_{10} = 10011_{2}[/latex] To convert the decimal 19 t0 binary we divide by two, but watch what happens at each time: *r = remainder 19 / 2 = 9 r 1 | 1001.1 < r 9 / 2 = 4 r 1 | 100.1 < r 4/ 2 = 2 r 0 | 10.0 < r 2/2 = 1 r 0 | 1.0 < r 1/2 = 0 r 1 | .1 < r Each time we divide by [latex]2 [/latex] we move [latex]1[/latex] bit to the right-hand side successively until we run out of bits, just like in the previous example. In terms of converting fractions to binary (e.g. [latex]0.5[/latex], [latex]1.039[/latex]) we use the inverse operation (rater than division, we use multiplication). In this process each time we multiply by [latex]2[/latex], we shift the bits to the left-hand side. I hope the way I explained it, shows that I understood the concept myself. If you think there's an error please correct me.

What I don't understand is simply, why when we divide by two it shifts bits to the right (remainders) and when we multiply by 2 it shift bits to the left (the whole part)

Trying to understand the fundamentals of binary rather than just following steps, I wanted to know why do we multiply by 2 to convert a decimal ([latex]0.5[/latex], [latex]0.25[/latex]) to a binary and why do we divide by [latex]2[/latex] when we want to convert a whole number [latex](200)[/latex] by [latex]2[/latex]? Obviously, it works but how ? Take the following example: Convert [latex]200_{10}[/latex] to binary: Solution: D > B | Remainder ------------------------------------- 200 / 2 = 100 | 0 100 / 2 = 50 | 0 50/2 = 25 | 0 25/2 = 12 | 1 12/2 = 6 | 0 6/2 = 3 | 0 3/2 = 1 | 1 1/2 = 0 | 1 By taking the remainder from **bottom-to-top** [latex]200_{10} = 11001000_2[/latex] > Why this method works ? In other words, what's the secret behind the division by [latex]2[/latex] ? Now converting decimals (e.g. [latex]0.5[/latex], [latex]0.25[/latex]) to binary: Now Suppose we have a decimal like [latex]0.25[/latex] and we want to convert it to binary, one of the method which I know goes like this: Multiplying the decimal by [latex]2[/latex] repeatedly: 0.25 * 2 = {0}.50 | {0} 0.50 * 2 = {1}.00 | {1} 0.00 -------------------------- .01 0.01 For more details about the above method: Decimal to binary conversion with fraction You can see the the two operations are **reversed**, to convert a whole number to a binary we divide by[latex] 2[/latex] and to convert a fraction (decimal) we used multiplication. Add to that the order in which we take the result from bottom-to-top and from top-to-bottom. How that works? >(Why division used to convert whole numbers to binary and why multiplication used to convert decimals (e.g. 0.25) to binary?) How These procedures work ?