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 nonexistent. 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 09, and the tens can only ever be 09, 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 18 will always be one less the number plus whatever it takes to get to 9. 9 x 3 = 27. 9 x 7 = 63. 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.
Multiplying double digit numbers by bases
What is 32 x 32? or 34 x 28? An easy way to work out these problems is to multiply by bases. So in the case of 32 x 32, we think of the problem like so:
32 
x 
32 
+2 

+2 
32 + 2 
x 
10 
340 
x 
3 
1020 
+ 
+2 x +2 


1024 
That might look complicated, but it’s actually very easy. We don’t think of 32 as 32, we think of it as 30 + 2. So our base is 30. In this example the other side of the multiplication is also, 30 + 2. So we take the +2 from the other side and and add it to 32 to get 34. We multiply by 10 (base 10) and then 3 (to get base 30). Multiply by 10 and then 3 is easy. Next we multiply the +2’s together and get 4, so we add that and get 1024.
Let’s try 33 x 28? Well, base 30 seems the best to use, so that’s 30 + 3 and 30 – 2. Notice the 2. If we add 2 to 33, we get 31, times 10 is 310, times 3 is 930, plus 3 * 2 is 924. Remember, 3 * 2 is 6, and 930 + 6 is 924.
Notice that 28 disappeared? That’s because we thought of it as a base, it’s not 28, it’s 30 – 2.