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Base One L. Van Warren

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Start at the beginning…

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What is a placeholder?

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8 8 1 1 2 2 2 2

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100 + 10 + 1 1 1 2 2 8 8

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10 2 + 10 1 + 10 0 1 1 2 2 8 8

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100 + 10 + 1 1 1 2 2 8 8

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+ 10 + 1 100 2 2 8 8

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+ + 1 100 20 8 8

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+ + 100 20 8 8

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100 20 8 8

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1 1 2 2 8 8

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Consider Placeholders As Dimensions…

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Three 8 8 1 1 2 2 How many placeholders? 2 2

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Three 128128

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128128

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128128

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128128

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ones tens hundreds 128128

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ones tens hundreds 128128

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How many dimensions in a cube?

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How many dimensions in a square?

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How many dimensions in a line segment?

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How many dimensions in a point?

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0 0

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How many symbols in base one?

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How do we count in base one?

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| || |||| Base 1 – Counting

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How do we add in base one?

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+ = ?

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+ =

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Addition is equivalent to Grouping

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Grouping is a powerful idea.

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Addition is “closed” in base one.

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Meaning we can represent any sum in the same counting system. (given sufficient time)

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How do we subtract In base one?

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- = ?

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- =

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Subtraction is equivalent to Separating

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Subtraction is not “closed” in base one.

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The antistick requires another bit that makes base two!

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- =

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The antistick is a negative stick with the annihilation property:

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+ =

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How do we multiply In base one?

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x =

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The second number tells us the number of copies of the first.

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x =

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And visa versa

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x =

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This demonstrates that multiplication in base one is commutative.

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Multiplication is repeated addition.

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Multiplication in base one is “closed”.

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Base One Squares

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Squaring is a special case of multiplication.

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x =

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Choosing a different symbol reminds us of the meaning of “squaring”.

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x =

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Or:

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x =

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How do we find the square root in base one?

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“Find the number that when multiplied by itself gives the original number.”

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If the number is a perfect square, the square root is the length of one side.

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= ?

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=

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How do we divide In base one?

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Division is repeated subtraction.

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= ?

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=

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Division, like subtraction is not closed in base one.

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= ?

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=

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Base One Trigonometry

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We must introduce some new symbols.

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These symbols are not used for counting.

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These symbols are containers for copies of our counting symbol.

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x

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x

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x

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x

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x

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x

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x

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x

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x x

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x x

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x x

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x x

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x x

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x x

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y y x x

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y y x x

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x x y y

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y y x x

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x x y y

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x x y y

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x x y y

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x x y y

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x x y y

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x x r r y y

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x x r r y y

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Name relationships.

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x x r r y y

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r r sin( ) = x x r r y y y y

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r r x x r r y y y y

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x x cos( ) = x x r r r r y y

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x x r r x x r r y y

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tan( ) = x x x x r r y y y y

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x x r r y y x x y y

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Base One Probability

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Ask some friends to show you a number by holding up their hands.

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Then plot the result in base one.

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Number shown: Number of appearances

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Questions?

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