Talk:augend

term, summand, addend

synonyms according to https://en.wikipedia.org/wiki/Summation --Backinstadiums (talk) 10:20, 4 February 2022 (UTC)

If you look at the "See also" section on this entry, we already include those terms. To quote:

(augend) + (addend) = (total)

(summand) + (summand) + (summand)... = (sum)

So you start with the augend, and then add the addend to it. These terms are used when adding two numbers. Summand appears to be used when treating all added terms similarly, or when there are more than two numbers added; it could perhaps be considered a hypernym of the other two words.

From a mathematical standpoint, there is normally not much difference between an augend and an addend, because addition is commutative over the real numbers. That is to say, $a+b=b+a$ for any real numbers a and b; the order in which you add terms is arbitrary. (This applies not just to the real numbers, but to any field such as the complex numbers. In fact, it even goes for a lot of sets of mathematical groups that aren't fields, like semirings such as the natural numbers, abelian groups under addition, like vector spaces, etc.)

However, there are contexts where it makes sense to distinguish between the "first number" and the "second number". One context in which it might make sense to distinguish the two is when you're referring to a procedure/algorithm to compute addition. This can be either a mental math algorithm used by humans:

pg. 69 "The fourth strategy of "counting on first” occurs when a child states the value of the augend and then counts out the number of the addend. For example, to solve a 2 + 3 problem, a child would start at "2” and count up 3 numbers — "2, 3, 4, 5” and say that the answer is 5."

Or an algorithm implemented in an electrical circuit or programming language, where they could refer to separate memory registers or distinct arguments to a function:

[1] "Figure 9.6 depicts a program segment illustrating a numerical example of adding two operands: a 32-bit augend and a 16-bit addend […] "

Furthermore, there are mathematical objects for which addition is not commutative, such as the ordinal numbers, and here it is important to distinguish the two:

[2] "The sum of the two ordinal numbers is always greater than the augend, but greater than or equal to the addend. If we have $\alpha +\beta =\alpha +\gamma$, we always have $\beta =\gamma$. In general $\alpha +\beta$ and $\beta +\alpha$ are not equal."

I hope this helps. 70.172.194.25 02:31, 20 February 2022 (UTC)