As I’ve mentioned before, I am a 6th grade dropout, not only that but I was “homeschooled” for most of my elementary school years, so my math abilities are pretty non-existent. To correct this problem, I have started studying mathematics; **from zero. **Here is where I will put my notes.

## Behold the power of 0

Something that was pointed out by Scott Flansburg: 0 is a number. No really – it’s not nothing – it’s something. Programmers, often, start counting at zero, but you’d be amazed how your mind changes when you get rid of 10, and start counting at 0. Don’t count: 1,2,3,4,5,6,7,8,9 and 10!** Instead, start at 0 and count like this: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.**

Once you do that, you **notice that counting is circular!**

0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
… |

0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
0 |
1 |
2 |
… |

As you continue, the stack can grow, but the **units** can only ever be 0-9, and the **tens** can only ever be 0-9, and on and on.

That leads to an interesting ideas: **Numbers have no size!**

There’s no such thing as a big number! There are only **long** numbers.

The **length** of a number is **only a problem of memory!**

## The cycle of reflexive addition

The cycle of reflexive addition shows you that at a certain point, the **units** column rolls over, and the same units operations begin again. 2 + 2 = 4 and 12 + 2 = 10 + **(2+2)**.

You’ll notice that: 2,4,6 and 8 roll over after 6 operations and 1, 3, 7 and 9 after 9 operations.

0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |

… |
2 |
4 |
6 |
8 |
10 |
12 |
14 |
16 |
18 |

… |
3 |
6 |
9 |
12 |
15 |
18 |
21 |
24 |
27 |

… |
4 |
8 |
12 |
16 |
… |
24 |
28 |
32 |
36 |

… |
5 |
10 |
15 |
20 |
… |
30 |
35 |
40 |
45 |

… |
6 |
12 |
18 |
24 |
… |
36 |
42 |
48 |
54 |

… |
7 |
… |
21 |
… |
… |
… |
49 |
… |
63 |

… |
8 |
… |
24 |
… |
… |
… |
56 |
… |
72 |

… |
9 |
… |
27 |
… |
… |
… |
63 |
… |
81 |

… |
10 |
… |
30 |
… |
… |
… |
70 |
… |
90 |

… |
11 |
… |
33 |
… |
… |
… |
77 |
… |
99 |

This is incredibly useful to know when counting by numbers. As you’re counting, notice how tension in your mind increases right up to the rollover, and releases with the roll over! 4, 8, 12, 16, 20, 24. Try a nice one, like 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77 … 84, 91, 98, 105, 112, 119… and so on.

## Magical, Mystikal 9

Nine is an interesting number. If you multiply any number by 9, and then sum the individual digits of the result, they will collapse into 9. 9 x 9 = 81 = 8+1 = **9. **As a curious aside, 9 x *n *where *n *is a single digit 1-8 will always be one less the number plus whatever it takes to get to 9. 9 x 3 = **2**7. 9 x 7 = **6**3. 2 is one less than 3, and 6 is one less than 7!

Obviously 9 x 10 = 90, and 9 + 0 = 9. The fun thing is 9 x 489,756 = 4,407,804 = 4 + 4 = 8 + 0 + 7 = 15 = 1 + 5 = 6 = 6 + 8 = 1 + 4 = 5 + 0 + 4 = **9**. You just keep drilling down, if you ever get above 9, combine and continue.