Book of Number

 Book of Number

Forewarning

beware what you count
you may come to value only what counts

number is clean
life is not

what fits in digits
leaves much behind
what’s left behind
can be what matters

not all things can be named in number
not all truths can be measured
not all measures tell the truth
not all truths can be proven

yet with number
we carve sky from sea
predict the falling fruit
and follow stars home

so reckon well
but reckon with care
number is sharp
and sharp things cut

First Things

we see one
we see another
we see the same

we match what we see
to what we know
and give it a name

we count
to remember
to compare
to tell

but not all things are counted
some must be measured

we count fingers
we measure time

we count stones
we measure distance

we count flocks
we measure water
and cold
and heat

number is not what we have
it is what we know
what we tell
what we show

counting makes steps
measuring makes lines

and from these lines
we make a tool

we call it the number line

it begins with one
then two
then three
and up we go
into the many

but there is a space
between what is
and what is not

a space with a name

zero is not nothing
but the name of nothing

a mark
a breath
a pause in the beat

a place where something could be
but isn’t

zero is the fulcrum
between more and less
between gain and debt
between warm and cold
the point where water freezes

below zero
we find the cold
the absence
the pull

we name these
negatives

not because they are bad
but because they turn the other way
measuring from presence
counting from absence

number runs both directions
outward
from zero
into the more
and into the less

number relates to number
each to all others
counting the distance between

Number

we count with symbol
we count with name
a finger to each

0 zero no finger
1 one a finger raised
2 two a pair held wide
3 three a grasp with space
4 four the claw of paw
5 five the open hand

6 six five and one
7 seven five and two
8 eight five and three
9 nine five and four

a finger remains
its number ten
but may we not open symbol
to count toward infinity?

Place

10
ten
the last finger

1 and 0
ten and none
side by side
they name the next

where each number sits
tells power

leftward
each place is ten times more
rightward
each place is ten times less

between tens and tenths
we set a dot of meaning
marking the place of ones

the fulcrum of number
between the great
and the small

00.0
is zero tens
and zero ones
and zero tenths

11.1
is one ten
and one one
and one tenth

99.9
is nine tens
and nine ones
and nine tenths

what then is 999.99?

how does a shift of the dot
left or right
transform a number?

with only ten numbers
marks from zero to nine
and one small sign
we count the infinite

from infinitely large
to infinitely small
in infinitesimal steps

place
is the magic
of infinity
counted from fingers

Fractions Ratios Degrees

draw a circle
whole and round
without beginning
without end

slice it
whole into halves
halves into quarters
quarters into eighths

1 → 2⁄2
1⁄2 → 2⁄4
1⁄4 → 2⁄8

1⁄2 one of two
2⁄4 two of four
3⁄8 three of eight
we name these fractions

fractions are written a⁄b
a parts of a whole
cut into b pieces

fractions
divide a whole
to name the pieces
to count the pieces
in relation to that whole

a ratio
is a comparison
between two things
like fractions
yet unlike

ratios are written a/b
a things to b things
a things per b things
a things divided by b things

this to that 
5 fingers per hand
1 hand to 5 fingers
10 fingers to two hands
10 fingers / 2 hands

how does order change use?
how does meaning change use?

a⁄b
can mean a part of a divided whole
or a relation between two things

a fraction of the circle
a ratio of the pack

some ratios stay simple
1⁄2 = 0.5
3⁄4 = 0.75
3⁄8 = 0.375

some stretch on without end
1⁄3 = 0.333…
and endless thirds
the closer we look

***

circle back

draw a circle
and slice it into 360 pieces

we name each piece a degree

we mark degrees  
with the ° symbol  
to count slices of turn  
fractions of a full spin
pointing direction
measuring angle

90 degrees = 90° = 90°⁄360° = 1⁄4 of a circle
3/4 of a circle =  3 x 90°/360° = 270°⁄360° = 270 degrees

***

circle back

draw a circle
mark a point along it
measure its run
point back to point
we name this circumference

measure straight
from point to center
we name this the radius
continue to its opposite edge
twice the radius
to find the diameter

what is the ratio of round to span?
of circumference to diameter?
what is this mystery?

not a whole
not a fraction
not a number that ends
not a number that repeats

we name this ratio pi
we give it a symbol π

 if you would know number follow pi

Power and Root

to raise to a power
is to stack
the same number
upon itself

2 × 2 = 2² = 4
2 × 2 × 2 = 2³ = 8
10 × 10 = 10² = 100
10 × 10 × 10 = 10³ = 1000

powers build
slow at first
a quickening
then leaping away

***

a point
has no dimension
any number raised to 0 power
is one alone
n⁰ = 1

a line
is a point given length
any number raised to 1 power
is itself
3¹ = 3
a row of 3

a square
is a line given breadth
3 × 3 = 3² = 9
a grid of 3 rows and 3 columns

a cube
is two dimensions
square plus depth
3 × 3 × 3 = 3³ = 27
a block of 3 by 3 by 3

a tesseract
is 3 dimensions plus one
cube plus a shift
3 x 3 x 3 x 3 = 3⁴ = 81
a hyper cube of 3 by 3 by 3 by 3
a cube over time?

but not beyond number

what may we not
build as a shape?
what may we not
build as a matrix?

***

number flows away from zero
less as well as more
what of negative number
when applied as powers?

10^(-1) = 1/10¹ = 1/10
10^(-2) = 1/10² = 1/100

falling away
to infinitely small

***

to seek a root
is to ask
what number
built this power?

the root
is the side of the square
is the side of the cube
is the side of the tessaract
and beyond

we set root degree a
before root symbol √
and number last b
a√b

the square root of 25
25^(1/2) = 2√25 = 5
because 5 × 5 = 5² = 25

the cube root of 27
27^(1/3) = 3√27 = 3
because 3 × 3 × 3 = 3³ = 27

roots unbuild

***

powers build
roots unbuild

yet they are bound
one to the other

some as whole number
some as fractions
some as ratio

the square root of 10
10^(1/2) = 2√10
the cube root of 8
8^(1/3) = 3√8

a ratio of powers
is a root
is a fraction of powers
is a power

49^(1/2) = 2√49 = 7

power and root
root and power

***

some powers stretch
toward infinity
toward infinitly small

we count
and measure
and compare

farther than we can count
farther than we can grasp

what is the difference
between countable infinity
and that infinity
infinitely divided?

***

draw a triangle
with legs at right angle
call them a and b
opposite a long diagonal
call it c

if a = 3
if b = 4
if c = 5

does a² + b² = c²?
is this true for any value?
how would you know?

***

draw a square
of side one
sides a = b = 1

draw a diagonal
corner to corner
name it c

what is the length
of that diagonal?

c² = a² + b²
c² = 1² + 1²
c² = 2 
c = 2√2

try as you may
you will not find
any ratio of number
equal to 2√2

a⁄b ≠ 2√2

this number runs on
without repeating
without end

irrational
and yet real
between the corners
of a simple square

what other numbers
what other kinds
will you discover?

Operators and Order

when we reckon number
speak with care
speak in order

some reckon first
some wait their turn

we agree on order
to reckon true

***

some symbols hold others
do these together
do these first

( and )
we name parentheses
cupped hands
holding a sequence
to be taken first
to be taken as one

***

some symbols are instructions
do this
do that

we call these operators

some are paired
each a reflection
one of the other
complementary
dancing a unity

 + and −
plus and minus
add and subtract

× and /
times and per
multiply and divide

^ and √
raise and root
exponentiate and extract ... find the root?

***

not all steps
are taken in the order written

this is the order of operations
the steps of the dance

p parentheses ( )
e exponents ^
m multiply ×
d divide /
a add +
s subtract −

pemdas
an order of intention

***

look

2 + 3 × (4 + 1)² / 5

goes not
left to right
but deep to shallow

first
(4 + 1) = 5
then
5² = 25
then
3 × 25 = 75
then
75 / 5 = 15
finally
2 + 15 = 17

***

without rules
a scribble of symbols
means many things
without agreement
meaning is lost

not law but language
not nature but pact

we reckon
in common order
to find common truth

Logarithms

winds rise
but not in straight line
we name them
on a scale
that rises exponentially
in magnitudes of power

a gentle breeze
rustles the leaves
a strong breeze
shakes the branches
a gale
bends the trees
a storm blows them over

each increase
not in equal measure
but in power raised

feel the wind
count its strength
not in measure
but in magnitude

***

some sequences
grow number
step by step

add one
add one more

1, 2, 3, 4, 5, 6...

counting arithmetically
in even steps
each added
to the one before

regular
and even

***

some sequences
grow number
leap by leap

raise one
raise one more

10⁰ = 1
10¹ = 10
10² = 100
10³ = 1000
10⁴ = 10000...

counting geometrically
in growing bounds
each a multiple
of the one before

we name such leaps
orders of magnitude
each leap
a power of ten

***

to ask
how many times
must we raise ten
to reach a number
is to ask for its logarithm

we call it log for short

log₁₀(1) = 0
log₁₀(10) = 1
log₁₀(100) = 2
log₁₀(1000) = 3
log₁₀(10000) = 4...

to ask
how many times
must we raise two
to reach a number
is to ask for its logarithm

log₂(1) = 0
log₂(2) = 1
log₂(4) = 2
log₂(8) = 3
log₂(16) = 4...

logarithms are leaps
expressed as steps

the shadow of exponential growth
cast onto a number line

***

on a ruler
each mark
lies equal distance
from its neighbors

1, 2, 3, 4…
equal spacing

take two rulers
number lines
laid side by side
offset one from the other

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10...
   0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10...

two and zero align
five and three align
with distance we add
with distance we subtract
one scale measured
against its twin

***

but a logarithmic scale
does not step numbers evenly
but in leaps

between 1 and 2
a wide step
between 2 and 3
a shorter one
from 9 to 10
a mere breath

equal distance
here means
equal ratio
between positions
spaced in leaps

1, 2, 3...
are no longer evenly spaced

1 to 2
2 to 4
4 to 8
equal steps
of multiplication

this is the ruler
of logs

***

logarithms turn
multiplication into addition
division into subtraction

they let us measure ratios
as distance

leaps
counted in steps

***

sliding rulers compare
not two points
but four
a aligned with b
c aligned with d
each pair offset
by scaled distance

three are known
and set to match
the fourth is revealed
in answer to the question posed
reckoning value
from distance along a scale

two numbers here
two numbers there
the distance between
one pair
matches the next

on a straight line
distance is difference
(b − a) = (d − c)
adding and subtracting

on a log line
distance is ratio
(b / a) = (d / c)
multiplying and dividing

how we slide
how we group
tells us
what we find

***

logarithms are the inverse of powers
where powers leap
logs count steps

ask now

what are the fractional values
of logarithms?

what is log₁₀(5)?

what might this number describe?

***

what happens
when we add logs?

what is
log₁₀(1) + log₁₀(10)?

what happens
when we subtract logs?

what is
log₁₀(100) − log₁₀(10) = ?

what might these numbers describe?

***

what happens
when we divide logs?

what is
log₁₀(10) / 2?
what number
has a log of 0.5?

what happens
when we multiply logs?

what is
2 × log₁₀(10)?
what power
does it become?

what might these numbers describe?

***

what is the spacing
along a logarithmic ruler?

why does spacing shrink
as values grow?

if you slide two such
side to side
what means distance?

what might be accomplished
sliding one scale
against its twin?

***

if logs are steps
and powers are leaps
how might one measure the other?

what might be raised
what might be rooted
from rulers of ratio?

***

logarithms
untangle powers
flatten growth
reveal the steps
behind the leap

to see clearly
when things grow fast
when things span far
when things change
by orders not steps

we turn to logs
to measure what swells
to track what accelerates
to name
the thunder
in the storm

Form and Function

what goes in
determines
what comes out

one to one
we name a function
set to set
we name a functional relationship

***

a function relates
members of one set
to members of another
in one way only

we call this mapping

each set of inputs
to a set of outputs
and no other

like parents to number of offspring
like distance traveled to time and speed
like hours of waking to hours of sleep

***

within each function
is a rule
a method
a process
an ordering of instructions

we call this
an algorithm

the algorithm is the box
the function is the bridge

some algorithms are short
a single step
some algorithms run long
ranging far and wide

if the steps are clear
and always end
the rule can be followed

what if  an algorithm
can end but does not?
what if an algorithm
cannot end?
how can we tell
if it will end?

***

a function is not
the numbers it acts on
nor the steps inside
it is the
relationship itself

how one thing
determines another

anything
that takes input
and gives output
by a rule

***

to see a function
we plot its path
in an ordered field

one line horizontal
we name the x axis
one line vertical
we name the y axis

crossing at
the zero of each
positives count upward and right
negatives count downward and left
a pact of convention

this flat field
we call a plane
we name this plane
a graph

***

to plot a point
we take two counts
one along each axis
x and y
(x,y)

so far over or back
so far high or low
a relation of two

one by one
point by point
we draw the shape
of relation

a point
a curve
a wave
a climb
a drop
a broken step
a sudden stop

each function
its shape

***

a third axis
brings depth
new possibility
relations of three

a volume forms
where once was plane
a space for every
x and y and z

what shapes are possible
in this deeper space?

***

what if we add another
axis upon axis?
how far might we go?

how many paths
can cross and twist
in space
beyond our seeing?

what if each point
holds many answers?

***

some scales count
in even steps
others grow
by leaps and bounds

what happens
if we scale an axis
logarithmically?

what shapes might we reveal?
what might they mean?

Algebra

we begin with constants
what is known
what is guessed
what is chosen
what is steady in the wind

Number is constant
0, 1, 1.5, 1/3, 3√8

some numbers
are given symbol

π ~ a circles circumference to diameter
c ~ the speed of light
g ~ the pull of Earth

all are constant
their value fixed

***

when number is unknown
we devise a vessel
we name a variable

we use a letter

x, y, z
a, b, c
A, D, C

to hold what is
to hold what may be discovered
to hold what may be assigned

standing in place of what is unknown
standing in place of what might vary

***

we build expressions
with constants
with variables

x + 3
2a
y × y
d / t
πD

expressions are forms
not yet rules
not yet questions

***

we balance expressions
setting one
= equal to the other
we name these equations

1 = 1
a = 2
π = C / D
2ab = 2 × a × b
(a + b)^2 = a^2 + 2ab + b^2

one side balanced
against the other

some equations are true
1 = 1
some equations are false
1 = 2

some equations are true
but only for some value
x/1 = 1
where x = 1

some equations are true
only for certain values
x^2 = 1
where x = (1, -1)

an equation
is an assertion
of balance that one expression equals another in common value

If one side
balances the other
we call this true

if one side does not
balance the other
we call this false

some equations are undefined
x / 0 = undefined

why is x / 0 not infinite? x/x =1 for all numbers with one exception which is excepted?

***

some equations
assert a truth
no matter the values

addition is commutative
a + b = b + a
and associative
(a + b) + c = a + (b + c)

multiplication is the same
a × b = b × a
(a × b) × c = a × (b × c)

but subtraction is not
a − b ≠ b − a
(a − b) − c ≠ a − (b − c)

nor division
a / b ≠ b ÷ a
(a / b) ÷ c ≠ a ÷ (b ÷ c)

when order matters
keep the order

yet

subtraction is
adding a negative
a − b = a + (−b)

division is
multiplying by a fraction
a / b = a × (1 / b)

a root is number
raised to a fractional power
2√a = a^(1/2)

every doing
has its undoing

every action
has a reversal

when subtraction is addition
is it commutative?
is it associative?

when division is multiplication
is it commutative?
is it associative?

***

some equations assert a balance that holds only when variables have certain values we name these conditional to solve an equation
is to find the unknown
is to make the variable
stand alone balanced with value

x + 3 = 7
subtract 3 from both sides
x + (3 − 3) = 7 − 3
x = 4

we subtract 3
to isolate x
we subtract
from both sides
to preserve the balance

when x is buried deep
this may require many steps
on the path to solution

***

2x = 10
divide both sides by 2
2x / 2 = 10 / 2
x = 5

each step
a doing which undoes
each step
preserves balance

this is algebra's power

***

what is x?
what does it count?
what kind of thing
do we mean?

10 of what?

number alone
is a ghost
a shape
with no substance

a unit is one kind of thing

we think in units
we label with units

some are a count
14 days
5 fingers

some are ratios
asserting a relation
true when inverted
28 days / 1 lunar.month
1 lunar.month / 28 days

these expressions
form equations
5 fingers / 1 hand x 2 hands = 10 fingers
(1000 paces / 1 day) x 10 days = 10000 paces

units combine when we multiply
2 persons x 5.hours = 10 person.hours
with a ratio of
2.5 hours / 1 person

how might we use
such a notion?
do 10 hour.persons make sense?

units only mix with meaning
yet some meanings surprise
odd pairings expand the mind

units reduce when we divide
10 person.hours / 4 persons = 2.5 hours

numbers are clear
yet the world is complex

If a task requires 10 person.hours
how long will 1 person require?
how long will 10 persons require?
how long will 100 persons require?

are all numbers practical?
can other numbers
other ratios
help us predict?

***

these are bare bones
in the bone setters art

to find the unknown
from hints
from method
from relations in flux

is a mystery at the heart of number

if you would find number
follow algebra

Algorithms

not all questions
are solved at once
some unfold over time

not all answers
are stated in balance
nor found in balance
some emerge by doing

an algorithm
is a method
is a program of action
is a set of steps to follow

a function
that runs a path
from inputs to outputs
among the myriad possibilities

each step followed
each outcome built

some algorithms
are simple
a straight path
from input to result

some reduce
a complex task
into simpler tasks
bites which may be chewed
each small answer
a piece of the whole

we set an algorithm a task

it answers
if the steps are clear
if the steps are correct
if the task may be completed

are there tasks
which cannot complete?
how do we know them?
can we know beforehand?

***

a variable
in algebra
is a mystery

a variable
in algorithms
is a container

we assign it
we change it
we use it again

algebra asks
what is x?

algorithms command
what x shall be
how x shall change
when x shall change

***

order matters

some steps must come
before others
as one lays a fire
small to large

what must happen first?
what depends on what?

what must be done
what must be known
before decision?

***

some paths repeat

while this is true
keep doing these

until this is true
keep doing these

we name these loops
we name looping iteration

some paths branch

if this is true
then do these things
else do those things

we name these tests of truth
conditionals
if then else
while
until

***

consider a sequence
where each step
builds the next

start with 1
and 1 again
each number after
is the sum of the two before

1, 1, 2, 3, 5, 8, 13...

what sequences result
when starting from
some other two numbers?
the same or different?

where do you see these sequences
occurring in the world around?

this algorithm is iterative
repeating a loop of doing

1. let a and b be two values
   
2. write a and b
3. let c = a + b
   
4. let a = b
   
5. let b = c
6. write c
7. until weary repeat from step 3
   
8. end

this algorithm is recursive
calling on itself for answers

let the rule
to find the number
at position n

be as follows

if n is 1

return the first value

if n is 2

return the second value

else

call the rule again

once with n − 1

once with n − 2

add the answers

and return that sum

each recursion
calls on itself twice

asking a simpler question
until what is known
returns

answers
rise back
like bubbles
from the deep

iteration repeats the steps
recursion repeats the pattern

iteration walks the loop
recursion calls itself 
asking less of itself
with each call

***

an algorithm is not magic
but memory and motion

can a set of rules build a path?
can a set of rules find the shortest way?
what if the river rises?
what if the steps take too long?

what makes an algorithm effective?
what makes an algorithm efficient?
what makes an algorithm good?

if you would
shape questions into method
shape method into action
shape action into answers
follow the algorithm