Understanding the logic behind conversion from decimal to other numeral systems via division with remainder

Question

I want to understand why does the conversion from decimal to other numeral systems works via division with remainder? Like for example I want to convert 13 decimal to binary. I start to divide by 2 and I get 6 and reminder 1. Here the logic is clear. There is 6 times 2 in 13 and 1 that will be the first number in binary which is also 1. Then we need to divide 6 by 2 to check is there а 2 in binary. In this step I’m lost. How does it work? Why dividing 6 by 2 show me if there is 2 in binary number?

P.S. My English is awful. Sorry about that.

Answer 1

Anton:

 

Actually, you presented some of the stronger English that I have seen here, so have no fear! We tutors aren't pointing and giggling behind your back.

As for conversions, you might find it easier to follow this method:

If you want to work in a particular base, start by listing the powers of that base until it exceeds your target number. In your example, You targeted 13. The powers of 2 below that are: 2, 4, 8, 16...ah, too big! We'll stop at 8. Now, we perform what I learned as the "hat trick", in which we write the divisor, the current value, and the quotient. It'll be weird, but once you get it you'll find it a very flexible method:

 

8  into   13   goes 1                        Now, we find the remainder:

            - 8

4  into     5   goes 1

             -4

2  into     1   goes 0

             -0

1  into     1   goes 1

              -1

              0                                    When we reach remainder zero, we're done. Your answer is 1101₂

 

Let's review one more example to make this clear:    40₁₀   =   ________₃ ?

 

We list the powers of 3: 3, 9, 27, 81 -- and now we are too high, so the process starts at 27:

 

27  into  40  goes  1

            -27

9   into   13  goes  1

              -9  

3   into     4  goes  1

              -3

1   into     1  goes 1

              -1

               0                                        So your answer is 1111₃  Remember that none of the digits can reach your new base, which means my answer could have had 0, 1, 2 but not 3 or higher. Also, we call it "hat trick" because we drew a shape like thisL

           ___________

______|                  |___________      with the divisor on the left, current value in the middle, and quotient on the right. You wind up with a stack of them as you find each digit.