I was going off the rough estimate that 0-9 is 10 digits and 10x10 is 100 but that reduces to 50 thanks to commutativity. I think 45 if you don’t count zeroes—I don’t know where you get 36 from.
I also use the term “memorize” pretty loosely—I remember that memorizing times tables was a thing but not so much for plus—but addition is simple enough that most people can kind of intuit what 7+4 heuristically if they’re sat down and forced to do arithmetic as small children for long enough. (Also I’ve never had the patience for memorization; I just rely on my brain to cache things that I use frequently and it ended up working for times tables. Also other things.)
But I do want to acknowledge that Arabic numerals make multiplication nearly as easy as addition, which is a staggering achievement over Roman numerals.
Though I will say, on the other hand, that the Romans weren’t that stupid and neither were their medieval successors prior to the adoption of Arabic numerals. They could add things up and we’ve discussed in a parallel thread how that algorithm works. The Roman numeral system isn’t as optimized informationally—let’s not underestimate the sheer awesomeness of seamlessly expressing numbers as large as 108730026190037365462849562635965—but that would be useless to most cultures that used Roman numerals.
I would even question one more thing. To someone who doesn’t know a numeral system to begin with, do Arabic numerals actually make addition harder? I mean, very small children (and programming languages like JavaScript if you accidentally express one number as a string) sometimes make the mistake of thinking 11+8=118, but in Roman numerals that’s just like saying XI + VIII = XIVIII, which is also wrong, but not as wrong as 118. A Roman child could easily be taught no, that’s XVIIII since V’s go before I’s, and then maybe reduce to XVIV. A child today is like, “wait wtf are places?” Roman numeral users never have to learn the concepts of places, carry, or borrow, which honestly sounds like a good trade off for a civilization that doesn’t have to do multiplication and division that easily.
I think I'd calculated combinations, but dubiously ignored where both were identical - that's where I ended up with 36.
OTOH do we memorise any of the x+1 combinations? I hope we don't, but perhaps we do. I genuinely can't say at this point. I was trying to work out how I processed sums such as 8+7 earlier, and concluded that so far I can tell, I work out the difference of one of those numbers from 10, subtract it from the other, then it's a very simple addition - ie that becomes 10+5. But I'm now unsure if that's what I do as a general rule, and am even less sure what other people may do.
Times table I vaguely recall learning by rote in formative school, but that's an awfully long time ago, and trying to self-analyse my mechanisms for multiplications is highly challenging. It feels like I try to move those back to multiples of 10 or 100, again, too.
I recall reading aeons ago that the only intuitive interface is the nipple - beyond that, everything is learned. So what makes for an intuitive or sensible mathematical representation of things is probably so arbitrary as to be pointless arguing about. It feels that the kinds of things we do, day to day, with numbers, that base-10 arabic number system is optimum, but that may simply be the lack of exposure to a better system.
As a parent of young children and former math teacher, yes, we do absolutely memorize those sums. It usually happens at an early enough age that the process of doing so evaporates early in life and becomes part of your base mental code. I spent a significant part of my teaching life teaching high school math for college students (and occasionally grade school math for college students) and there are very much people who never managed to get that memorization step completed. What's really fascinating is that it's an orthogonal skill to higher mathematics. I've seen students who needed to use a calculator to do 6+5 and yet managed to be able to solve algebraic problems. This is, I must add, uncommon, but it's less because one skill depends on the other but rather because the failure to gain the basic math skill leads to an unwillingness to try to gain the more abstract math skill.
I agree that Arabic numerals are better, but I did want to make the point that Roman numerals aren’t quite as bad as you might think having already learned Arabic numerals.
Your method for summing 8+7 is what I think I do for things like 7+4 (since I can visualize 7 as “three less than 10” and 4 as “one more than three” all in the same thought to reach 11), but for 7+8 my brain noticeably spits out 15 immediately and only a moment later does it actually do the processing you mention.
I also use the term “memorize” pretty loosely—I remember that memorizing times tables was a thing but not so much for plus—but addition is simple enough that most people can kind of intuit what 7+4 heuristically if they’re sat down and forced to do arithmetic as small children for long enough. (Also I’ve never had the patience for memorization; I just rely on my brain to cache things that I use frequently and it ended up working for times tables. Also other things.)
But I do want to acknowledge that Arabic numerals make multiplication nearly as easy as addition, which is a staggering achievement over Roman numerals.
Though I will say, on the other hand, that the Romans weren’t that stupid and neither were their medieval successors prior to the adoption of Arabic numerals. They could add things up and we’ve discussed in a parallel thread how that algorithm works. The Roman numeral system isn’t as optimized informationally—let’s not underestimate the sheer awesomeness of seamlessly expressing numbers as large as 108730026190037365462849562635965—but that would be useless to most cultures that used Roman numerals.
I would even question one more thing. To someone who doesn’t know a numeral system to begin with, do Arabic numerals actually make addition harder? I mean, very small children (and programming languages like JavaScript if you accidentally express one number as a string) sometimes make the mistake of thinking 11+8=118, but in Roman numerals that’s just like saying XI + VIII = XIVIII, which is also wrong, but not as wrong as 118. A Roman child could easily be taught no, that’s XVIIII since V’s go before I’s, and then maybe reduce to XVIV. A child today is like, “wait wtf are places?” Roman numeral users never have to learn the concepts of places, carry, or borrow, which honestly sounds like a good trade off for a civilization that doesn’t have to do multiplication and division that easily.