My fun number system :D
Potato was walking home from Mars one day, and he imagined the wonders of a number system that lets M+N=MN (where M and N are digits). Let's explore!
That would mean 1+1=11 and 7+2=72. For now, let's just think about sums of single digit numbers. In this world, every two-digit number has a unique sum that generates it. For example, 24 can be broken into 2+4. And addition is not commutative. 2+3=23 is different from 3+2=32.
We would want to know what something like 23+481 is equal to, so let's find out. Well, consider something like 4+8+1. Of course, the 4+8 just turns into 48, so we have 48+1. And it would make sense to just stick the 1 at the end like we do with one-digit numbers, and it turns out nothing goes wrong! So the original problem's answer is 23481. This also means that addition is associative. That is to say, A+B+C is equal to AB+C and A+BC.
Subtraction is easy, just take the digit out. But the question is, do you take it off the end of the number or can it be anywhere? That is, what if you have 234-3? Can you subtract this? Well, let's think about it this way. If you have 121-1, then which 1 do you take out? This is why we should have a rule for it. If the number you want to take out is the last number, you can do it. If not, sorry. So 121-1=12, then. You can't subtract 1 from 132, though. It works the same for two-digit numbers. 4121-21=41. But you can't subtract 21 from 4213.
Now let's say we want to compute 3 * 4. Well, multiplication is just repeated addition, so basically this is just 3 + 3 + 3 + 3 = 3333. Note that we could have had 4+4+4=444, but let's just define multiplication as M * N = a string of M's that is N long. So A * 3 = AAA and A * 5 = AAAAA. Multiplication is not commutative, but we saw that with 3 * 4. Multiplication is also not associative. For example, 2 * 3 * 2 could be turned into 222 * 3 = 222222222 or 2 * 33 = a really big number. This means that we will have to use order of operations on expressions that have only multiplication, like 4 * 5 * 6 * 7. We would do first come, first serve, so multiply 4 * 5 = 44444, then 44444 * 6 = a big number, and so on. We can do the same for two-digit numbers. For example, 43 * 2 = 4343. It's also interesting to note that multiplying by 1 still has no effect on the number.
I mean, you can probably guess how this works, but it's not really fair that we would have sections for everything else and not this. Let's say we want to find 333333 รท 3. We need to find what you can add three times to get 333333. Well, it's 33! That was kind of easy. I'm not exactly sure how you would divide 4 into 333333, though. Maybe it would be 3 with remainder 33? That's a bit weird.
Click to see Part 2.