On this lecture note on page no 4 Corollary 14 the term number of carries has been used. Can anyone explain what does it mean?
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$\begingroup$Think of an addition problem:$$\begin {array}{r} 965257\\ +29568\\ \hline 994825\end {array}$$
There are carries in the ones, tens, and thousands columns, for a total of three. If I write the above in base $5$ I get $$\begin {array}{r} 221342012_5\\ +1421233_5\\ \hline 223313300_5\end {array}$$ and there are carries in the ones, fives, 625's, and 3125's places for four.
If I add $14 + 25$, we get something that looks like
$$\begin{align} &14\\ +&25\\ \hline &39 \end{align}$$
and we see that the ten's digit is the sum of the two ten's digits, and the one's digit is the sum of the two one's digits.
However, in adding $14 + 27$, we have
$$\begin{align} &{\color{#F01C2C} 1}\\ &14\\ +&27\\ \hline &41 \end{align}$$
where I've indicated that we "carry" a $\color{#F01C2C}1$ after we add $4$ and $7$. That is what we mean by "carry": when the naive addition algorithm causes you to "carry" the effect of a lower digit to a higher digit.
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