Arithmetic with numbers | Arithmetic and Geometry Math Foundations 2 | N J Wildberger

[Music] Good day. I'm Norman Wildberger, and this is the  second of a series of videos on the foundations   of mathematics. Please have a look at the first  video if you haven't already done so. Last time,   we introduced the most important objects in  mathematics: the natural numbers. We said that   a natural number was just a string of ones, so  the sequence of natural numbers looks just like   this: the first one is one, the next one is  1 1, the next one is 1 1 1, and then 1 1 1 1,   and so on. We also mentioned the standard names  for the first few of them: 1, 2, 3, 4, 5, but to   carry on naming them requires the Hindu-Arabic  numeric system, and we don't have that yet. So   it's safer for us to think of a general number  like this as just 1 1 1 1 1 1, a string of ones. Alright, so let's go back in time, maybe  100,000 years, to a water hole in Africa. We might   imagine a 10-year-old boy there who's been sent  by his tribe to count animals, and he sees a bunch   of buffalo and antelope. Let's say buffalo are  these squares and the antelope are the triangles,   and he's pretty good at counting. He's figured out  a way of counting things. He has a piece of bark,   and on the bark, he makes little scratches, one  for each of the animals that he sees. So for   example, counting buffaloes, here's how he would  do it: there's one, there's one, there's one, one,   one, one, one, one. Of course, he has to make sure  not to repeat. Counting antelopes is the same:   there's one, one, one, one, one, one, one, one,  one. So he goes back to his village with this   piece of bark, and he shows the elders these  marks, and they can sense how many buffalo   and antelopes there are. They may not have a  name for these numbers like we do, but still, it gives them a very good idea of what's at the  water hole. Now suppose his father asks him,   how many animals are there in total? Why don't  you go back and count them all? Our young boy,   who's rather clever, realizes that he doesn't  have to go back to the water hole to count them   all because he has counted the buffalo and he has  counted the antelope, and he can use the numbers   that he has obtained to find the number of all the  animals. He does that by what we call adding those   two numbers. How do we do that? Well, he just  copies the first one: one, one, one, one, one,   one, one, one, and then he adds the second one:  one, one, one, one, one, one, one, one, one, one.   That then is the total number of animals at the  water hole,