Chained Arrow Notation

Chained Arrow Notation is a powerful computable notation created by John Conway. It is based off of the original Up-Arrow Notation. The differences are that they are separated by rightward arrows instead of up arrows, and this notation is much more powerful. The definition of Up Arrow Notation is below:

$$a \uparrow b = a^{b}=\underbrace{a \times (a \times (a \times ....(a\times a)))....)))}_{b\text{ copies}\text{ of} \text{ }a}$$

$$a\uparrow \uparrow b = \underbrace{a\uparrow(a\uparrow(a\uparrow (a\uparrow(.....(a\uparrow a)))))....)))}_{b\text{ copies}\text{ of} \text{ }a}$$

$$a\uparrow\uparrow\uparrow b= \underbrace{a\uparrow\uparrow(a\uparrow\uparrow(a\uparrow\uparrow(........(a\uparrow\uparrow a)))))))....))))}_{b\text{ copies}\text{ of} \text{ }a}$$

$$\text{Define }a\uparrow^{c}b \text{ as }a\underbrace{\uparrow\uparrow\uparrow\uparrow....\uparrow\uparrow\uparrow}_{c\text{ arrows}}b$$

$$\text{In }\text{general, }a\uparrow^{c}b=a\uparrow^{c}(a\uparrow^{c-1}b)$$

Now for the Chained Arrows:

$$a\rightarrow b \rightarrow c=a\uparrow^{c}b$$

$$a\rightarrow b \rightarrow c \rightarrow 1= a\rightarrow b \rightarrow c$$

Rule: 1's and anything that is after it (to the right of it) can be ignored and removed.

$$a\rightarrow b \rightarrow c \rightarrow d=a\rightarrow b \rightarrow(a \rightarrow b \rightarrow (c-1)\rightarrow d)\rightarrow (d-1)$$

$$a\rightarrow b \rightarrow 1 \rightarrow d = a \rightarrow b$$

$$a \rightarrow 1 \rightarrow c \rightarrow d = a$$

$$a\rightarrow b \rightarrow c \rightarrow d \rightarrow 1 = a \rightarrow b \rightarrow c \rightarrow d$$

$$a\rightarrow b \rightarrow c \rightarrow d \rightarrow e = a \rightarrow b \rightarrow c \rightarrow (a\rightarrow b \rightarrow c \rightarrow (d-1) \rightarrow e)\rightarrow (e-1)$$

$$a \rightarrow b \rightarrow c \rightarrow d \rightarrow e \rightarrow f = a \rightarrow b \rightarrow c \rightarrow d \rightarrow (a \rightarrow b \rightarrow c \rightarrow d \rightarrow (e-1)\rightarrow f)\rightarrow (f-1)$$

$$a \rightarrow b \rightarrow c \rightarrow d \rightarrow e \rightarrow f \rightarrow g = a \rightarrow b \rightarrow c \rightarrow d \rightarrow e \rightarrow (a \rightarrow b \rightarrow c \rightarrow d \rightarrow e \rightarrow (f-1) \rightarrow g) \rightarrow (g-1)$$

$$a \rightarrow \rightarrow b = a \rightarrow_{2} b = \underbrace{a \rightarrow a \rightarrow a \rightarrow a \cdots\cdots a \rightarrow a \rightarrow a}_{b \text{ copies }\text{of }\rightarrow}$$

$$a\rightarrow\rightarrow\rightarrow b = a\rightarrow_{3}b = \underbrace {a\rightarrow\rightarrow (a\rightarrow\rightarrow (a\rightarrow\rightarrow \cdots\cdots (a\rightarrow\rightarrow a))))))))\cdots\cdots)))))}_{b \text{ }\rightarrow\rightarrow's}$$

This continues like the Up Arrow Notation: This is the limit of Chained Arrow Notation. \[\Huge{\text{Adding 1 to a } dB \text{ rating will make the sound pressure/intensity multiply by } 10^{0.1}. \\ \text{Adding 0.1 to a } dB \text{ rating will make the sound pressure/intensity multiply by } 10^{0.01}. \\ \text{Adding 0.01 to a } dB \text{ rating will make the sound pressure/intensity multiply by } 10^{0.001}. \\ \text{etc.}}\] 