Positional value and base radix

[bob]

You see how the spot where a digit sits changes everything in a number system once you start counting positions from the right. I caught onto this fast when I first played with different counting methods back in school projects. And the value multiplies by the base each time you shift leftward which makes the whole setup work like a power game. But you notice the radix decides how many symbols fit before you carry over to the next spot. Or perhaps you tried converting a simple decimal like twelve into binary yourself and saw the positions flip the meaning completely.
I recall you asking about why machines stick to base two instead of ten and it boils down to hardware using on off switches that match that radix perfectly. You get the positional value kicking in strong because the leftmost bit holds the biggest weight like eight or sixteen depending on the length. And radix stays fixed for a given system so binary sticks to powers of two while decimal uses powers of ten without mixing them up. Maybe you wonder how hex comes into play with its base sixteen letting each spot pack more info than binary alone. Then the positional twist shows up again since A through F represent ten to fifteen and each place multiplies accordingly.
You might test this by writing out one hundred in base ten as one times ten squared plus zero times ten plus zero but switch the radix to eight and watch the digits rearrange fast. I always found it handy when debugging memory addresses because the base changes how you read the same bits. But you avoid confusion by keeping the radix clear from the start otherwise the positions lose their meaning right away. Also the architecture relies on this setup for registers and memory where each location follows the same power rule. Perhaps you see now why adding in different bases needs carrying based on that radix limit.
Or think about floating point numbers where the positional idea extends to fractions after the point with negative powers of the base. I tried explaining this to another junior once and they got stuck until they listed out the powers manually. You benefit from grasping it early since processor instructions often deal with bit positions tied to the radix. And the whole system avoids ambiguity by defining the base upfront in every calculation. Then you realize errors pop up if someone mixes radices without converting positions first.
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