Please Help with Bases

I would also appreciate help with modular arithemetic inverses. Thank you!

I learned this in class recently. I will try to help you.

Let's say I want to convert \((1011)_2\) into base 10. By the way, the subscript denotes the base of that number. In this case, I will convert from binary to decimal.

Since this number is in base 2, the nth digit represents (n-1) power of 2 of the number.

In the example I created, \((1011)_2 = 1* 2^3 + 0 * 2^2 + 1 * 2^1 + 1 * 2^0\\ (1011)_2 = 8 + 2 + 1\\ (1011)_2 = 11 \). In base 10, the subscript is usually omitted.

What if I wanted to convert from base 10 to binary? Let's use 11 as our example. Well, here is one method to use. Binary, once again, is base 2. I can use the following iterative formula.

\(11 = bq_0 + r_0\\ 11 = 2q_0 + r_0\\ 11 = 2*5 + 1\) where b is the base, q is the largest multiple without exceeding 11, and r is the whole number remainder.

Now, replace the 11 with q_0 and do this process iteratively until q_i = 0.

\(11 = 2 * 5 + 1\\ 5 = 2 * 2 + 1\\ 2 = 2 * 1 + 0\\ 1 = 2 * 0 + 1\\\)

Now that q_i equals 0, we read the remainders we got backwards, which gives us our answer: 1011.

I hope this made sense to you.