A long time ago, a mathematician lived in binary land where an ancient law decreed that all numbers could use only two numerals. The mathematician wanted her people to have the same richness of numbers as the people who lived far away in decimal land. The mathematician didn't think it was fair for her people to have only two numerals when the people in decimal land had ten. Between her binary land and decimal land lay the vast territories of ternary land, then quaternary land, then pentary land, then hexal land, then septal land, then octal land, then nigon land, and finally decimal land. Each land had its own allowed number of numerals, but that's another story.

She didn't want to break her ancient law, so she thought hard to devise a number system with two numerals. The two numerals she chose were 0 and 1. Either numeral could go into something she called a place. We will see how useful this invention is shortly. She called her first place the 1's place and put it to the far right. If 0 were in the place, then the value would be 0 x 1 = 0 and if 1 were in the place, then the value would be 1 x 1 = 1. Well that was a start; she could write 0 as 0 and 1 as 1, but what about 2? She wasn't allowed to put a 2 in the 1's place, because she didn't have a 2. Her idea of a place would pay off here. She would have to go on to the next place after a short rest.

She called the next place to the left, the 2's place. If 0 were in the 2's place, then the value would be 0 x 2 or 0, and if 1 were in the two's place, then the value would be 1 x 2 or 2. Well, now she could write 2 as 10 if 10 meant to add up the values of the two places; 10 would equal 1 x 2 + 0 x 1 = 2. What about 3? That would be 11 = 1 x 2 + 1 x 1 = three. She was doing well but couldn't go any further with just two places because she had filled them with the largest numerals she had-they were both 1! She needed a longer rest this time. See if you can figure out what she did before you read on.

She created a third place to the left of the second, and because the next number she needed was 4, it seemed logical to let it be the 4's place; 100 would be 1 x 4 + 0 x 2 + 0 x 1 = 4. The biggest number she could write now was 111 = 1 x 4 + 1 x 2 + 1 x 1 = 7. Can you figure out how to write 5 and 6?

Next she created a fourth place to the left of the third, and because the next number she needed was 8, it seemed logical to let it be the 8's place; 1000 would be 1 x 8 + 0 x 4 + 0 x 2 + 0 x 1 = 8. The biggest number she could write now was 1111 = 1 x 8 + 1 x 4 + 1 x 2 + 1 x 1 = 15. Can you figure out how to write the numbers between 8 and 15?

Well, she continued this process for quite some time until she got to the eighth place and found that the value of the was 128; 10000000 would be 1 x 128 + 0 x 64 + 0 x 32 + 0 x 16 + 0 x 8 + 0 x 4 + 0 x 2 + 0 x 1 = 128. The biggest number she could write was 11111111 = 1 x 128 + 1 x 64 + 1 x 32 + 1 x 16 + 1 x 8 + 1 x 4 + 1 x 2 + 1 x 1 = 255. Including 0, that would be 256 different numbers!

While the mathematician rested from that exhausting job, a young man in town built a machine out of thousands of switches that could be on or off. He knew of the mathematician's great work and thought that his switch in the off position could represent 0 and in the on position could represent 1. He called each 0 or 1 a bit of information, and from the mathematician's work, he knew if he put 8 bits together, he could represent 256 different things, like letters and other symbols. He called the 8bits a byte. Pretty soon everyone in binary land was using the young man's machine, based on the mathematician's binary number, system to type and save stories and add numbers because the machine was much better at doing those routine things and remembering them.

The young man and the mathematician became very famous and respected by all. The people in binary land were very happy because they had a machine that helped them be more productive. The invention soon spread to ternary land, quaternary land, pentary land, hexal land, septal land, octal land, nigon land, and finally decimal land. But the people in binary land were the first and they lived happily ever after.