## FANDOM

1,099 Pages

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).