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
# b

a

b

=

a
×
(
a
×
(
a
×
.
.
.
.
(
a
×
a
)
)
)
.
.
.
.
)
)
)

⏟

b
 copies
 of

a

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

a
↑↑
# b

a
(
a
(
a
(
a
(
.
.
.
.
.
(
a
a
)
)
)
)
)
.
.
.
.
)
)
)

⏟

b
 copies
 of

a

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

a
↑↑↑
# b

a
↑↑
(
a
↑↑
(
a
↑↑
(
.
.
.
.
.
.
.
.
(
a
↑↑
a
)
)
)
)
)
)
)
.
.
.
.
)
)
)
)

⏟

b
 copies
 of

a

{\displaystyle 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}}

Define
a

c

b
 as
a

↑↑↑↑
.
.
.
.
↑↑↑

⏟

c
 arrows

b

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

In
general,
a

c

# b
a

c

(
a

c
−
1

b
)

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

Now for the Chained Arrows:

a
→
b
→
# c
a

c

b

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

a
→
b
→
c
→
# 1
a
→
b
→
c

{\displaystyle 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
→
b
→
c
→
# d
a
→
b
→
(
a
→
b
→
(
c
−
1
)
→
d
)
→
(
d
−
1
)

{\displaystyle a\rightarrow b \rightarrow c \rightarrow d=a\rightarrow b \rightarrow(a \rightarrow b \rightarrow (c-1)\rightarrow d)\rightarrow (d-1)}

a
→
b
→
1
→
# d
a
→
b

{\displaystyle a\rightarrow b \rightarrow 1 \rightarrow d = a \rightarrow b}

a
→
1
→
c
→
# d
a

{\displaystyle a \rightarrow 1 \rightarrow c \rightarrow d = a}

a
→
b
→
c
→
d
→
# 1
a
→
b
→
c
→
d

{\displaystyle a\rightarrow b \rightarrow c \rightarrow d \rightarrow 1 = a \rightarrow b \rightarrow c \rightarrow d}

a
→
b
→
c
→
d
→
# e
a
→
b
→
c
→
(
a
→
b
→
c
→
(
d
−
1
)
→
e
)
→
(
e
−
1
)

{\displaystyle 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
→
b
→
c
→
d
→
e
→
# f
a
→
b
→
c
→
d
→
(
a
→
b
→
c
→
d
→
(
e
−
1
)
→
f
)
→
(
f
−
1
)

{\displaystyle 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
→
b
→
c
→
d
→
e
→
f
→
# g
a
→
b
→
c
→
d
→
e
→
(
a
→
b
→
c
→
d
→
e
→
(
f
−
1
)
→
g
)
→
(
g
−
1
)

{\displaystyle 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
→→
# b
a

→

2

# b

a
→
a
→
a
→
a
⋯
⋯
a
→
a
→
a

⏟

b
 copies
of
→

{\displaystyle 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
→→→
# b
a

→

3

# b

a
→→
(
a
→→
(
a
→→
⋯
⋯
(
a
→→
a
)
)
)
)
)
)
)
)
⋯
⋯
)
)
)
)
)

⏟

b

→

→
′

s

{\displaystyle 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.
\[\Huge{\text{Adding 1 to a } dB \text{ rating will make the sound pressure/intensity multiply by } 10^{0.1}. \\ \text{Adding 0.1 to a } dB \text{ rating will make the sound pressure/intensity multiply by } 10^{0.01}. \\ \text{Adding 0.01 to a } dB \text{ rating will make the sound pressure/intensity multiply by } 10^{0.001}. \\ \text{etc.}}\]
<script type="text/javascript" src="https://www.hostmath.com/Math/MathJax.js?config=OK"></script>