# How one knows it is $(n-1)$ versus $(n+1)$

I have had a problem with this concept all my life so I thought I would reach out to the experts for help!

Here is the problem statement:

Quote: "Consider how much work is required to multiply two -digit numbers using the usual grade-school method. There are two phases to working out the product: multiplication and addition.

First, multiply the first number by each of the digits of the second number. For each digit in the second number this requires $n$ basic operations (multiplication of single digits) plus perhaps some "carries", so say a total of $2n$ operations for each digit in the second number. This means that the multiplication phase requires basic $n(2n)$ operations.

The addition phase requires repeatedly adding n digit numbers together a total of $(n-1)$ times. If each addition requires at most $2n$ operations (including the carries), and there are $(n-1)$ additions that must be made, it comes to a total of $(2n)(n-1)$ operations in the addition phase.

Adding these totals up gives about $4n^2$ total operations. Thus, the total number of basic operations that must be performed in multiplying two n digit numbers is in $O(n^2)$ (since the constant coefficient does not matter)."

End

Why is it $(n-1)$ and $2n?$

When you add two $n$ digit numbers, you have to do $n$ additions, one for each digit, but there might be a carry at each digit, so that's another $n$ operations, and there's your $2n$.
Now you have to add $n$ of these $n$-digit numbers, which means you have to do $n-1$ of these additions of pairs of numbers – right? To add two numbers, that's one addition; to add three numbers, that's two additions, and so on.
But in the end, you're going for an estimate $O(n^2)$, so it makes no difference whether you use $n-1$ or $n+1$, and no difference whether you use $2n$ or $17n$ or $n/42$.