Think about how you count. We start with 0, then go to 1, 2, 3, and so on up to 9. But what happens when we reach 9 and want to count one more? Well, we run out of digits. So, we put a 1 in the next place to the left and reset the rightmost digit back to zero, giving us '10'.
In essence, what you're doing is saying, "I've run out of digits in this 'column,' so I'll add another 'column' to the left." Just like you'd carry over in addition, you "carry over" a digit to the next position.
Now, What About Binary System?
In the binary system, you only have two digits: 0 and 1, so instead of running out of digits after 9, you run out after 1. So, what do you do next? You carry over a 1 to the left, resetting the digit you just counted to 0. In binary, the counting proceeds as: 0, 1, 10, 11, 100, 101, and so on.
Unpacking the Formula in Base-10
Before diving into the formula for binary, let's understand the base-10 system, which is what we use daily. Take the number 5432, for example:
Each digit is multiplied by 10 raised to the power of its position, starting from the rightmost digit at position 0.
Unpacking the Formula in Binary
For binary numbers, the formula simplifies to:
Each digit is multiplied by 2 raised to the power of its position in the number. This simple formula can be used to convert binary numbers to base-10.