A grafting number^{[1]} is a number whose digits, represented in base b, appear before or directly after the decimal point of its p'th root. The simplest type of grafting numbers, where b=10 and p=2, deal with square roots in base 10 and are referred to as 2nd order base 10 grafting numbers.
Integers with this grafting property are called grafting integers (GIs).^{[2]} For example, 98 is a GI because:
- $ \sqrt{98} = \mathbf{9.8}9949 \, $
The 2nd order base 10 GIs between 0 and 9999 are:
$ n $ | $ \sqrt{n} $ | $ n $ | $ \sqrt{n} $ |
---|---|---|---|
0 | 0 | 764 | 27.6405499... |
1 | 1 | 765 | 27.6586334... |
8 | 2.828427... | 5711 | 75.5711585... |
77 | 8.774964... | 5736 | 75.7363849... |
98 | 9.899495... | 9797 | 98.9797959... |
99 | 9.949874... | 9998 | 99.9899995... |
100 | 10.0 | 9999 | 99.9949999... |
More GIs that illustrate an important pattern, in addition to 8 and 764, are: 76394, 7639321, 763932023, and 76393202251. This sequence of digits corresponds to the digits in the following irrational number:
- $ 3 - \sqrt{5} = 0.76393202250021019... $
This family of GIs can be generated by Equation (1):
- $ (1)\ \ \ \lceil (3 - \sqrt{5}) \cdot 10^{2n-1} \rceil, n \geq 1 $
$ 3 - \sqrt{5} $ is called a grafting number (GN), and is special because every integer generated by (1) is a GI. For other GNs, only a subset of the integers generated by similar equations to (1) produce GIs.
Each GN is a solution for $ x $ in the Grafting Equation (GE):
- $ (GE)\ \ \ (x \cdot b^{a} )^{1/p} = x+c $
$ a, b, c, p $ are integer parameters where $ p\geq 2 $ is the grafting root, $ b\geq 2 $ is the base in which the numbers are represented, $ a\geq 0 $ is the amount the decimal point is shifted, and $ c\geq 0 $ is the constant added to the front of the result.
When $ 0 < x < 1 $, all digits of $ x $ represented in base $ b $ will appear on both sides of the Equation (GE).
For $ x=3 - \sqrt{5} $ the corresponding values are $ p = 2, b = 10, a = 1, c = 2 $.
References
This page uses content that was added to Wikipedia. The article has been deleted from Wikipedia. The original article was written by these Wikipedia users: See → Talk:Grafting number#Contributors. As with Mathematics Wiki, the text of Wikipedia is available under the Creative Commons Attribution-Share Alike License 3.0 (Unported) (CC-BY-SA). |
- ↑ http://www.numberphile.com/videos/98grafting.html
- ↑ http://roberttanniru.weebly.com/grafting-numbers.html