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June 7

Binary counting

I understand that there are two types of binary counting:

Without power

Decimal	Binary
0	0
1	1
2	10
3	11
4	100
5	101
6	110
7	111
8	1000
9	1001
10	1010
11	1011
12	1100

With Power

Decimal	Binary
0	0⁰ = 1
1	2⁰ = 1
10	2¹ = 2
100	2² = 4
1000	2³ = 8
10000	2⁴ = 16
100000	2⁵ = 32
1000000	2⁶ = 64
10000000	2⁷ = 128
100000000	2⁸ = 256
1000000000	2⁹ = 312
10000000000	2¹⁰ = 1024

I further understand that without power is used a lot in mathematics and with power is used a lot in computing.

But are there really only two types of binary counting and anyway how did mathematicians "discovered" these two counting-ways? In other words, why did mathematicians reached the conclusion that 10 is 2 and 11 is 3? I as a layman with only elementary mathematics knowledge don't know what problem in the real life it originally solved to say that 2 is 10 and 11 is 3.

2A10:8012:17:CDC6:BD35:377C:2239:E724 (talk) 17:35, 7 June 2023 (UTC)[reply]

I'm afraid you have misunderstood something. The system you call "with power" is not used at all. The binary representation of numbers in computers is as described under "without power". The only difference with the usual way binary numbers are represented in writing is that the words in computer have a fixed word length (the number of bits in the word), which includes the leading zeros that are conventionally not written, So the decimal number 13, which is 1101 in binary representation, is represented in a 16-bit word as 0000000000001101. And, of course, the bits have no shape, like the 0 is not round. The two bits correspond to different voltage levels, or magnetization levels, or any other physical quantity that can be digitally controlled in logic gates.

The origin of the decimal system is that the human species has ten fingers. Not all cultures used the decimal system for representing numbers. The ancient Sumerians came up with a sexagesimal system (base 60), which was passed on to the Babylonians and is why an hour has 60 minutes and why a minute, both as a quantity of time and as an arc, has 60 seconds. Other cultures use a duodecimal system (base 12). To mathematicians describing arithmetic in the decimal system it is obvious that the choice for base 10 is somewhat arbitrary, and that this can be generalized to other bases, including base 2.

The earliest electronic computers were decimal computers; they worked in base 10. The designers soon realized that using base 2 resulted in simpler, cheaper and faster arithmetic logic units. Almost all commercial computers released after 1960 were binary computers. --Lambiam 00:21, 8 June 2023 (UTC)[reply]

If you delete the line 0 ｜ 0⁰ = 1 from the table under "With Power" and swap the two columns, but not the headings, you get this:

Decimal	Binary
2⁰ = 1	1
2¹ = 2	10
2² = 4	100
2³ = 8	1000
...	...

Note that this is compatible with the entries of the "Without power" table. Numbers that are a power of ten, like one hundred, or a million, are written in the decimal system as a 1 followed by only 0s. In the binary system, numbers that are written as a 1 followed by only 0s correspond to the powers of 2. --Lambiam 00:37, 8 June 2023 (UTC)[reply]

In math, you also consider unary representations of numbers, with the empty sequence (or the placeholder 0) standing for zero, and e.g. 111 standing for 3, 11111 for 5, 1111111111 for 10, and so on. This allows you to use Peano arithmetic to recursively define the most common operations. --Stephan Schulz (talk) 00:08, 9 June 2023 (UTC)[reply]

Thanks. 2A10:8012:17:CDC6:E969:E01A:6E7B:49 (talk) 01:44, 8 June 2023 (UTC)[reply]