Base OneBase One
L. Van WarrenL. Van Warren
Start at the beginning…Start at the beginning…
What is a placeholder?What is a placeholder?
• 100 + • 10 + • 1 • 100 + • 10 + • 1 11 22 88
• 102 + • 101 + • 100 • 102 + • 101 + • 10011 22 88
• 100 + • 10 + • 1 • 100 + • 10 + • 1 11 22 88
+ • 10 + • 1+ • 10 + • 1100100 22 88
+ + • 1+ + • 1100100 2020 88
+ ++ +100100 2020 88
Consider PlaceholdersAs Dimensions…
Consider PlaceholdersAs Dimensions…
Three
881122
How many placeholders?
How many placeholders?
22
ones
tens
hundreds
128
ones
tens
hundreds
128
How many dimensionsin a cube?
How many dimensionsin a cube?
How many dimensionsin a square?
How many dimensionsin a square?
How many dimensionsin a line segment?
How many dimensionsin a line segment?
How many dimensionsin a point?
How many dimensionsin a point?
00
How many dimensionsin a point?
How many dimensionsin a point?
How many symbols in base one?
How many symbols in base one?
How many symbols in base one?
How many symbols in base one?
How do we count in base one?
How do we count in base one?
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Base 1 – CountingBase 1 – Counting
How do we add inbase one?
How do we add inbase one?
Additionis equivalent to
Grouping
Additionis equivalent to
Grouping
Groupingis a powerful
idea.
Groupingis a powerful
idea.
Additionis “closed”
in base one.
Additionis “closed”
in base one.
Meaning we canrepresent any sum
in the same counting system.(given sufficient time)
Meaning we canrepresent any sum
in the same counting system.(given sufficient time)
How do we subtractIn base one?
How do we subtractIn base one?
Subtractionis equivalent to
Separating
Subtractionis equivalent to
Separating
Subtractionis not “closed”in base one.
Subtractionis not “closed”in base one.
The antistickrequires another bit
that makes base two!
The antistickrequires another bit
that makes base two!
The antistick isa negative stick with
the annihilation property:
The antistick isa negative stick with
the annihilation property:
How do we multiplyIn base one?
How do we multiplyIn base one?
The second numbertells us the
number of copiesof the first.
The second numbertells us the
number of copiesof the first.
And visa versaAnd visa versa
This demonstrates thatmultiplication in base one
is commutative.
This demonstrates thatmultiplication in base one
is commutative.
Multiplication is repeated addition.
Multiplication is repeated addition.
Multiplication in base oneis “closed”.
Multiplication in base oneis “closed”.
Base One SquaresBase One Squares
Squaring isa special case
of multiplication.
Squaring isa special case
of multiplication.
Choosing a different symbol reminds us of the
meaning of “squaring”.
Choosing a different symbol reminds us of the
meaning of “squaring”.
How do we findthe square rootin base one?
How do we findthe square rootin base one?
“Find the number thatwhen multiplied by itself
gives the original number.”
“Find the number thatwhen multiplied by itself
gives the original number.”
If the number isa perfect square,the square root is
the length of one side.
If the number isa perfect square,the square root is
the length of one side.
How do we divideIn base one?
How do we divideIn base one?
Divisionis repeatedsubtraction.
Divisionis repeatedsubtraction.
Division, like subtractionis not closedin base one.
Division, like subtractionis not closedin base one.
Base One TrigonometryBase One Trigonometry
We must introducesome new symbols.We must introducesome new symbols.
These symbolsare not used for counting.
These symbolsare not used for counting.
These symbols arecontainers forcopies of our
counting symbol.
These symbols arecontainers forcopies of our
counting symbol.
Name relationships.Name relationships.
rr sin(q) =sin(q) =
xx
rr
qq
yy
yy
sin(q) =sin(q) =
rrsin(q) =sin(q) =
xx
rr
qq
yy
yy
xx cos(q) =cos(q) =
xx
rr
qq
rryy
cos(q) =cos(q) =
xx
rr
qq
xxcos(q) =cos(q) = rr
yy
tan(q) =tan(q) = xx
xx
rr
qq
yy
yy
tan(q) =tan(q) =
tan(q) =tan(q) =
xx
rr
qq
yyxxyy
Base One ProbabilityBase One Probability
Ask some friendsto show you a number
by holding up their hands.
Ask some friendsto show you a number
by holding up their hands.
Then plot the result in base one.
Then plot the result in base one.
Number shown:
Number ofappearances
Questions?Questions?