Counting: bases 2, 10, and 16

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Here are the base 2 numbers from 0d to 15d.

0000 0100 1000 1100 0001 0101 1001 1101 0010 0110 1010 1110 0011 0111 1011 1111

You need to know these 16 numbers in base 2.

The hard way to learn the first 16 numbers in base 2 is to memorize them.

The easier way, and what I do, is to write down the list starting at 0000 and ending at 1111.

Start with 0000.
Add 1 to get 0001.
Add 1, to get 0 with a carry of 1, to get 0010.
Add 1 to get 0011.
Add 1 to get 0100 (i.e., two carries of 1, as in going from 99 to 100 in base 10).
... and so on ...

To help you see the pattern, here is the sequence where 1 is added to each binary number starting at 0000.

0000 0100 1000 1100 + 1 + 1 + 1 + 1 ---- ---- ---- ---- 0001 0101 1001 1101 0001 0101 1001 1101 + 1 + 1 + 1 + 1 ---- ---- ---- ---- 0010 0110 1010 1110 0010 0110 1010 1110 + 1 + 1 + 1 + 1 ---- ---- ---- ---- 0011 0111 1011 1111 0011 0111 1011 1111 + 1 + 1 + 1 + 1 ---- ---- ---- ----- 0100 1000 1100 10000

Here is the list from 0000b to 1111b.

10 16 2 10 16 2 0d 0h 0000b 8d 8h 1000b 1d 1h 0001b 9d 9h 1001b 2d 2h 0010b 10d Ah 1010b 3d 3h 0011b 11d Bh 1011b 4d 4h 0100b 12d Ch 1100b 5d 5h 0101b 13d Dh 1101b 6d 6h 0110b 14d Eh 1110b 7d 7h 0111b 15d Fh 1111b

The bits of a byte can be numbered from 0 to 7, just as post office mailboxes can have numbers. Each box can contain a 0 or a 1.

7 6 5 4 3 2 1 0 --------------------------------- | ? | ? | ? | ? | ? | ? | ? | ? | ---------------------------------

Bit 0, for 2⁰, or 1, is the first bit.
Bit 1, for 2¹, or 2, is the second bit.
Bit 2, for 2², or 4, is the third bit.
Bit 3, for 2³, or 8, is the fourth bit.
Bit 4, for 2⁴, or 16, is the fifth bit.
Bit 5, for 2⁵, or 32, is the sixth bit.
Bit 6, for 2⁶, or 64, is the seventh bit.
Bit 7, for 2⁷, or 128, is the eighth bit.

So, Ch is 1100b is 1*8 + 1*4 + 0*2 + 0*1 is 12d.

dec bin hex base 2 exponential notation 0d 0000b 0h 0*2³ + 0*2² + 0*2¹ + 0*2⁰ 1d 0001b 1h 0*2³ + 0*2² + 0*2¹ + 1*2⁰ 2d 0010b 2h 0*2³ + 0*2² + 1*2¹ + 0*2⁰ 3d 0011b 3h 0*2³ + 0*2² + 1*2¹ + 1*2⁰ 4d 0100b 4h 0*2³ + 1*2² + 0*2¹ + 0*2⁰ 5d 0101b 5h 0*2³ + 1*2² + 0*2¹ + 1*2⁰ 6d 0110b 6h 0*2³ + 1*2² + 1*2¹ + 0*2⁰ 7d 0111b 7h 0*2³ + 1*2² + 1*2¹ + 1*2⁰ 8d 1000b 8h 1*2³ + 0*2² + 0*2¹ + 0*2⁰ 9d 1001b 9h 1*2³ + 0*2² + 0*2¹ + 1*2⁰ 10d 1010b Ah 1*2³ + 0*2² + 1*2¹ + 0*2⁰ 11d 1011b Bh 1*2³ + 0*2² + 1*2¹ + 1*2⁰ 12d 1100b Ch 1*2³ + 1*2² + 0*2¹ + 0*2⁰ 13d 1101b Dh 1*2³ + 1*2² + 0*2¹ + 1*2⁰ 14d 1110b Eh 1*2³ + 1*2² + 1*2¹ + 0*2⁰ 15d 1111b Fh 1*2³ + 1*2² + 1*2¹ + 1*2⁰

A magic number is a number that appears in many places as if it has some special type of meaning.

Many magic numbers in computers are actually base 2 numbers of the form 2ⁿ or 2ⁿ-1.

n 2ⁿ 2ⁿ 0d 1d 0000000000000001b 1d 2d 0000000000000010b 2d 4d 0000000000000100b 3d 8d 0000000000001000b 4d 16d 0000000000010000b 5d 32d 0000000000100000b 6d 64d 0000000001000000b 7d 128d 0000000010000000b 8d 256d 0000000100000000b 9d 512d 0000001000000000b 10d 1024d 0000010000000000b 11d 2048d 0000100000000000b 12d 4096d 0001000000000000b 13d 8192d 0010000000000000b 14d 16384d 0100000000000000b 15d 32768d 1000000000000000b 16d 65536d 10000000000000000b

Notice that the 1 moves to the left one place when the number is multiplied by 2 (in base 2). This is similar to the number 12 multiplied by 10 being 120 in base 10.

n 2ⁿ-1 2ⁿ-1 1d 1d 0000000000000001b 2d 3d 0000000000000011b 3d 7d 0000000000000111b 4d 15d 0000000000001111b 5d 31d 0000000000011111b 6d 63d 0000000000111111b 7d 127d 0000000001111111b 8d 255d 0000000011111111b 9d 511d 0000000111111111b 10d 1023d 0000001111111111b 11d 2047d 0000011111111111b 12d 4095d 0000111111111111b 13d 8191d 0001111111111111b 14d 16383d 0011111111111111b 15d 32767d 0111111111111111b 16d 65535d 1111111111111111b

n 10ⁿ 10ⁿ 0 0d 00001d 1 10d 00010d 2 100d 00100d 3 1000d 01000d 4 10000d 10000d

n 10ⁿ-1 10ⁿ-1 1 9d 00009d 2 99d 00099d 3 999d 00999d 4 9999d 09999d

An earlier Excel version had the following limitations, much later increased.

An Excel spreadsheet can have from 1 to 65526 rows (i.e., 16 bits are used).
An Excel spreadsheet can have from 1 to 256 columns (i.e., 8 bits are used).

The column after Z is AA, then AB, etc. This is a base 26 representation where A is column 1, I is column 9, V is column 22, Z is column 26, AA is column 27, etc. Thus, column IV is

9*26 + 22 = 234 + 22 = 256.

Often, software limits picked by software developers correspond to the above magic numbers, either 1 to 2ⁿ or 0 to 2ⁿ-1.

Here is the idea behind the NCAA basketball tournament.

Initially, there are 64 teams, or 2⁶ teams.
After the first round, there are 32 teams, or 2⁵ teams.
After the second round, there are 16 teams, or 2⁴ teams (sweet sixteen).
After the third round, there are 8 teams, or 2³ teams (elite eight).
After the fourth round, there are 4 teams, or 2² teams (final four).
After the fifth round, there are 2 teams, or 2¹ teams.
After the sixth round, there is 1 team, or 2⁰ teams, and this team is declared the national champion for that year.

So, for a n-round tournament, you start with 2ⁿ teams so that after n rounds, you have 2^n-n = 2⁰ = 1 team left.

Note: To have more than 64 teams in the tournament, one must add another level for some of the teams.