1.5.3 - Arithmetic Catalog

1.5.3

Arithmetic catalog

IMPETUS

You may remember tedious arithmetic drills in grade school, or entire homework pages filled with such problems. Most people look back on such excersizes with a sense of boredom. A few individuals however find that an obessive streak in them makes them predisposed to numbers. I liked the way numbers would pop up in different contexts, because it was proof of a consistent reality (mathematics) which had no tangible existence. Somehow this paradoxical facet of math always fascinated me. Numbers likewise have sometimes taken a vaguely "mystical" quality, and some numbers stand out more than others.

Numbers produced through arithmetic operations such as products and exponents are perhaps the more interesting. A good number catalog should include some of these, as the ideas presented here are the basis for many extensions later leading to some truly staggering numbers !

CATALOG CONTENT

There are 300 mathematical expressions contained in this catalog. Each is of one of the following three forms ...

a+b , axb , a^b

If you don't recognize the last operation ( ^ ), the "carrot" symbol stands for exponentation, which is usually notated by writing the 2nd number ( b ) smaller than a, and writing it in the upper right corner near a.

a and b are both counting numbers less than or equal to 10. Dispite the fact that both addition and multiplication are communitive ( ie. a+b = b+a and axb = bxa ) , both arrangements are listed for completeness sake. Thus there are 10x10x3 possible expressions ( 300 ).

Addition is the combining of two groups. Multiplication is having b groups of a.

Some people forget what exponents are, because they aren't used as much in elementary calculations. Exponents are best described as repeated multiplication. Specifically we can define ...

a^b = a x a x a x ... x a ( with b a's )

where a and b are both counting numbers. Note that a^b is not always equal to b^a. For example ...

2^3 = 2 x 2 x 2 = 2 x 4 = 8

3^2 = 3 x 3 = 9

However in certain rare instances they are equal to each other (ie. 4^2 = 2^4 ). All this means is that exponentation is not communitive, and we can not make any generalization about flipping the order of the inputs.

HOW TO USE CATALOG

Rather than construct a conventional arithematic table, like those you would see in a grade school math textbook, I have ordered the expressions by numerical size.

The first column is a counting number which is the result of one or more arithmetic operations. To the right, the second column lists ( in a special order) all the relevent mathematical expressions that are equivalent to it. This is therefore a catalog of the numbers that result, not the expressions themselves.

This list contains 103 cataloged numbers, so it is quite managable. I might choose to expand on this catalog, or create other arithmetic catalogs later.

ARITHMETIC CATALOG

Number Arithmetic Expressions

1 1x1 , 1^1 , 1^2 , 1^3 , 1^4 , 1^5 , 1^6 , 1^7 , 1^8 , 1^9 , 1^10

2 1+1 , 1x2 , 2x1 , 2^1

3 1+2 , 2+1 , 1x3 , 3x1 , 3^1

4 1+3 , 2+2 , 3+1 , 1x4 , 2x2 , 4x1 , 2^2 , 4^1

5 1+4 , 2+3 , 3+2 , 4+1 , 1x5 , 5x1 , 5^1

6 1+5 , 2+4 , 3+3 , 4+2 , 5+1 , 1x6 , 2x3 , 3x2 , 6x1 , 6^1

7 1+6 , 2+5 , 3+4 , 4+3 , 5+2 , 6+1 , 1x7 , 7x1 , 7^1

8 1+7 , 2+6 , 3+5 , 4+4 , 5+3 , 6+2 , 7+1 , 1x8 , 2x4 , 4x2 , 8x1 , 2^3 , 8^1

9 1+8 , 2+7 , 3+6 , 4+5 , 5+4 , 6+3 , 7+2 , 8+1 , 1x9 , 3x3 , 9x1 , 3^2 , 9^1

10 1+9 , 2+8 , 3+7 , 4+6 , 5+5 , 6+4 , 7+3 , 8+2 , 9+1 , 1x10 , 2x5 , 5x2 , 10x1 , 10^1

11 1+10 , 2+9 , 3+8 , 4+7 , 5+6 , 6+5 , 7+4 , 8+3 , 9+2 , 10+1

12 2+10 , 3+9 , 4+8 , 5+7 , 6+6 , 7+5 , 8+4 , 9+3 , 10+2 , 2x6 , 3x4 , 4x3 , 6x2

13 3+10 , 4+9 , 5+8 , 6+7 , 7+6 , 8+5 , 9+4 , 10+3

14 4+10 , 5+9 , 6+8 , 7+7 , 8+6 , 9+5 , 10+4 , 2x7 , 7x2

15 5+10 , 6+9 , 7+8 , 8+7 , 9+6 , 10+5 , 3x5 , 5x3

16 6+10 , 7+9 , 8+8 , 9+7 , 10+6 , 2x8 , 4x4 , 8x2 , 2^4 , 4^2

17 7+10 , 8+9 , 9+8 , 10+7

18 8+10 , 9+9 , 10+8 , 2x9 , 3x6 , 6x3 , 9x2

19 9+10 , 10+9

20 10+10 , 2x10 , 4x5 , 5x4 , 10x2

21 3x7 , 7x3

24 3x8 , 4x6 , 6x4 , 8x3

25 5x5 , 5^2

27 3x9 , 9x3 , 3^3

28 4x7 , 7x4

30 3x10 , 5x6 , 6x5 , 10x3

32 4x8 , 8x4 , 2^5

35 5x7 , 7x5

36 4x9 , 6x6 , 9x4 , 6^2

40 4x10 , 5x8 , 8x5 , 10x4

42 6x7 , 7x6

45 5x9 , 9x5

48 6x8 , 8x6

49 7x7 , 7^2

50 5x10 , 10x5

54 6x9 , 9x6

56 7x8 , 8x7

60 6x10 , 10x6

63 7x9 , 9x7

64 8x8 , 2^6 , 4^3 , 8^2

70 7x10 , 10x7

72 8x9 , 9x8

80 8x10 , 10x8

81 9x9 , 3^4 , 9^2

90 9x10 , 10x9

100 10x10 , 10^2

125 5^3

128 2^7

216 6^3

243 3^5

256 2^8 , 4^4

343 7^3

512 2^9 , 8^3

625 5^4

729 3^6 , 9^3

1000 10^3

1024 2^10 , 4^5

1296 6^4

2187 3^7

2401 7^4

3125 5^5

4096 4^6 , 8^4

6561 3^8 , 9^4

7776 6^5

10,000 10^4

15,625 5^6

16,384 4^7

16,807 7^5

19,683 3^9

32,768 8^5

46,656 6^6

59,049 3^10 , 9^5

65,536 4^8

78,125 5^7

100,000 10^5

117,649 7^6

262,144 4^9 , 8^6

279,936 6^7

390,625 5^8

531,441 9^6

823,543 7^7

1,000,000 10^6

1,048,576 4^10

1,679,616 6^8

1,953,125 5^9

2,097,152 8^7

4,782,969 9^7

5,764,801 7^8

9,765,625 5^10

10,000,000 10^7

10,077,696 6^9

16,777,216 8^8

40,353,607 7^9

43,046,721 9^8

60,466,176 6^10

100,000,000 10^8

134,217,728 8^9

282,475,249 7^10

387,420,489 9^9

1,000,000,000 10^9

1,073,741,824 8^10

3,486,784,401 9^10

10,000,000,000 10^10

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The arithmetic catalog can be broken up into 3 parts. The first part , 1-20 , is where all the additive expressions are. There are no additive expressions past 20, because the largest possible additive expression is 10+10 which is equal to 20. The 2nd part , 21-100 , contains only multiplicative and exponential expressions. The bulk of the multiplicative expressions are here, although a few are within the first part as well. After 100, there are no more multicative expressions, because 10x10 = 100. Lastly the third part , 125 - 10,000,000,000 , covers the remaining exponential expressions. The list ends at ten billion because 10^10 is the largest expression of all.

One interesting thing to note is that in terms of range, the 3rd part is by far the largest. Addition and multiplication only produces very small values compared to exponents. In fact if you take note of the largest expressions of addition, multiplication, and exponentation ( 20 100 10,000,000,000 ) you can clearly see that the size is accelerating. 10x10 is 5 times larger than 10+10 , but 10^10 is a hundred million times larger than 10x10 ! This is not a coincedence, but rather a consequence of the fact that the operations are built as extensions of each other. Multiplication is built out of repeated addition, and exponents are built from repeated multiplication. So naturally they will transcend binary expression using the previous operation.

Another interesting thing to observe, is how many entries each part actually includes.

the 1st part contains 20 entries, the 2nd part contains 26 entries, and the 3rd contains 57 entries. The reason for this is that at the lower number ranges, more expressions will overlap, and thus there will be less entries covering that area.

Also notice that the number of equivalent expressions seems to peak at 10, and directly around 10 one can make out a wave like pattern. After this the number of expressions drops very low, to either 1 or 2 expressions per entry. Again one can make the observation that this is do to the fact that as expressions become more unweildly ( rapid growth ) they are less likely to land on the same numbers. You can literally observe that most of the exponential expressions are solitary, especially towards the end of the list.

The next catalog tracks usage of various counting numbers ...

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