The Two-Argument Exponential Function

I will begin by describing each of the more familiar operators, starting with the most basic.

Addition a+b = a^①b

Addition can be thought of as an iterated act of adding one:

a+b = a+1+1+1+1...

where there are b copies of 1 on the right-hand side of the = sign.

Here is a table of the addition function for arguments from 0 to 6. It is the first part of a larger table that most grammar-school students learn:

a+b	0	1	2	3	4	5	6

0	0	1	2	3	4	5	6
1	1	2	3	4	5	6	7
2	2	3	4	5	6	7	8
3	3	4	5	6	7	8	9
4	4	5	6	7	8	9	10
5	5	6	7	8	9	10	11
6	6	7	8	9	10	11	12

By antidiagonals, this table is sequence A3056.

The rows and columns are all A0027 (omitting one or more initial terms).

The main diagonal is A5843.

The superdiagonal and subdiagonal are both A5408.

Multiplication a×b = a^②b

Multiplication is iterated addition. These two definitions are equivalent:

a×b = ((a+a)+a)+...+a

or:

a×b = a+...+(a+(a+a))

where in each case there are b copies of a on the right-hand side of the = sign.

Here is a table of the multiplication function for arguments from 0 to 6. It is the first part of a larger table that most grammar-school students learn:

a×b	0	1	2	3	4	5	6

0	0	0	0	0	0	0	0
1	0	1	2	3	4	5	6
2	0	2	4	6	8	10	12
3	0	3	6	9	12	15	18
4	0	4	8	12	16	20	24
5	0	5	10	15	20	25	30
6	0	6	12	18	24	30	36

By antidiagonals, this table is sequence A4247.

The rows are A0004, A0027, A5843, A8585, A8586, A8587, A8588, etc.

Each column is the same as the corresponding row.

The main diagonal is A0290.

The superdiagonal and subdiagonal are both A2378.

Exponentiation a^b = a↑b = a^②b

Exponentiation is iterated multiplication. It is normally written a^b but on this page I will use the ↑ symbol, like this: a↑b. These two definitions are equivalent:

a↑b = ((a×a)×a)×...×a

or:

a↑b = a×...×(a×(a×a))

where in each case there are b copies of a on the right-hand side of the = sign.

Here is a table¹ of the exponentiation function for arguments from 0 to 6:

a^b	0	1	2	3	4	5	6

0	1	0	0	0	0	0	0
1	1	1	1	1	1	1	1
2	1	2	4	8	16	32	64
3	1	3	9	27	81	243	729
4	1	4	16	64	256	1024	4096
5	1	5	25	125	625	3125	15625
6	1	6	36	216	1296	7776	46656

By antidiagonals, this table is sequence A3992.

The rows are A0007, A0012, A0079, A0244, A0302, A0351, A0400, etc.

The columns are A0012, A0027, A0290, A0578, A0583, A0584, A1014, etc.

The main diagonal is A0312.

The superdiagonal is A7778 and the subdiagonal is A0169.

Non-Commutativity

Lifting Addition

Lifting Multiplication

Lifting Exponentiation (Converting to Right-Associative)

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2024 Dec 11.

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