Math is strange in the U.S. We use Arabic symbols, Latin-root words for counting, and a base ten number system coupled with a base twelve language for the numbers up to 100.
The first twelve numbers have little in the way of logic to describe how the names align with the next value. That would be okay for the ten first, basic numbers, but only if you include zero (a word which didn’t really exist in the Latin, and certainly didn’t have a value). So, 0 means nothing, unless it has another number in front of it.
Put a One in front of a zero, and it becomes One Zero! No, it becomes a different number, ten. There is no separate symbol for ten, though. Okay, so, put a 1 next to a 1, you get OneTen? No.
You get OneOne? No.
You get eleven, which is a special word. Same thing for 12. After 12, a new logic enters, up until One Hundred. This new logic is a special form of the base ten logic, where words meant to represent two digit numbers end in “ty.” So, Twenty, Thirty, Forty, Fifty, Sixty, Seventy, Eighty, Ninety. After One Hundred, you get new logic, where you use the names for the nine number in front of a word mean to represent the type of number. So, One Hundred, Two Thousand. On the next power up, you finally get something truly logical, which is Three Hundred Thousand. But all of the digits AFTER this space follow the logic of the preceding system.
BUT THEN, after Nine Hundred Ninety Nine Thousand Nine Hundred Ninety Nine, we again get a new idea, the million. From here, we apply a similar logic to the previous powers. So, you get One Million, Twenty Million, Three Hundred Million, then we invent a new word, Billion, so you get Four Billion. And the pattern essentially repeats itself after that. BUT, the pattern for the numbers AFTER the major number repeat the earlier patterns.
So, 4,833,511,112, is …
Four Billion, Eight Hundred Thirty-Three Million Five Hundred Eleven Thousand, One Hundred Twelve. Which makes perfect sense AFTER you understand how to leap from Arabic base-10 numerals, to a base 12 language system, and then leap from a base 12 language system to a base 10 language system, and then leap to essentially a base 3 language system that incorporates the previous base 10 language pattern AND base 12 language pattern. HIdden in all of that are all of the cultures, patterns, symbolic logic and language systems that were appropriated and eventually entered modern English.
When you write the number 4833511112, it’s hard for your average person to see at a glance that it’s a number in the billions. Those commas end up being important; they help you see each of the increments in the base three system we’ve moved into …
4 Billion, 833 Million, 511 Thousand, 112.
4,833,511,112 is clearer in our strange language than 483351112. Arguably that’s at least partly because of the way we’re thinking about it, due to our language and it’s constraint of Base 12 to Base 10 to Base 3 but also use Base 10 and Base 12 sometimes, depending on the context.
YEESH!
We use clocks to do complex coordination of time. Our clocks are base 12, and use two rotations. Our Imperial measurements are also base 12. Our symbols for mathematical language are base 10, as the numbers after 100.
Our geometry is arguably also base 12, since a circle is essentially 30 increments of 12, which is close to our concept of yearly time. But, within each daily increment of time, where we think about when we have sunlight and no sunlight, we break it up into multiple base 12 units. We have 60 seconds, which is 5 sets of 12. We have sixty minutes, which is 5 sets of 12 seconds. We then have 12 hours of morning and then 12 hours of night, using 12.00pm to approximate when the sun should theoretically be at the highest point in the sky. It doesn’t always work out that way, but that’s the basic idea.
As we noted, 60 minutes is 60 units, which is another way of saying 5 units of 12. Even though a circle is 30 units of 12 in geometry, when we think of time, we think of 60 distinct units, and each of those units is also broken into 60 distinct units. Maybe each circle of time we use for our clocks should be 360 units, just like we use for geometry? Try counting to six within the space of a single second, and you’ll see a reason why we may have decided so long ago to make that unit into 60, instead of 360. Although, it’s hard to say why we made that unit of time into 60 instead of 30 …
But, did you notice? 60 is a crossover point between a base 12 system (5 units of 12) and a base 10 number system (6 units of 10). It’s the simplest, smallest crossover point between the two language systems.
We have a lot of ways to count, on top of having multiple ways to account for the type of counting we’re doing. I don’t think that math is hard. I think the English language and our cultural conventions make thinking about math extremely difficult, unless you can (perhaps) think only in those Arabic numerals and ignore the verbal and written language behind them.
People don’t think about it much, because by the time they would, all of that complication has already been learned, and internalized. I think all of that confusion may be why math seems hard in the U.S, because learning those weird abstract complications IS hard when you first start, because the logic of our language is in flux within all of the basic concepts you have to first learn.
I think it’s why older generations of teachers came up with the idea of memorizing addition, subtraction and multiplication tables; it’s less about memorizing the numbers, more about learning the logic of our weird number system.