1. Base One L. Van Warren

2. Start at the beginning…

3. What is a placeholder?

4. 8 8 1 1 2 2 2 2

5. 100 + 10 + 1 1 1 2 2 8 8

6. 10 2 + 10 1 + 10 0 1 1 2 2 8 8

7. 100 + 10 + 1 1 1 2 2 8 8

8. + 10 + 1 100 2 2 8 8

9. + + 1 100 20 8 8

10. + + 100 20 8 8

11. 100 20 8 8

12. 1 1 2 2 8 8

13. Consider Placeholders As Dimensions…

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

15. Three 128128

16. 128128

17. 128128

18. 128128

19. ones tens hundreds 128128

20. ones tens hundreds 128128

21. How many dimensions in a cube?

22. How many dimensions in a square?

23. How many dimensions in a line segment?

24. How many dimensions in a point?

25. 0 0

26. How many symbols in base one?

28. How do we count in base one?

29. | || |||| Base 1 – Counting

30. How do we add in base one?

31. + = ?

34. + =

36. Addition is equivalent to Grouping

37. Grouping is a powerful idea.

38. Addition is “closed” in base one.

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

40. How do we subtract In base one?

41. - = ?

52. - =

53. Subtraction is equivalent to Separating

54. Subtraction is not “closed” in base one.

55. The antistick requires another bit that makes base two!

56. - =

57. The antistick is a negative stick with the annihilation property:

58. + =

59. How do we multiply In base one?

60. x =

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

62. x =

63. And visa versa

64. x =

65. This demonstrates that multiplication in base one is commutative.

66. Multiplication is repeated addition.

67. Multiplication in base one is “closed”.

68. Base One Squares

69. Squaring is a special case of multiplication.

70. x =

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

72. x =

73. Or:

74. x =

76. How do we find the square root in base one?

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

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

79.  = ?

84.  =

86. How do we divide In base one?

87. Division is repeated subtraction.

88. = ?

101. =

102. Division, like subtraction is not closed in base one.

103. = ?

104. =

105. Base One Trigonometry

106. We must introduce some new symbols.

107. These symbols are not used for counting.

108. These symbols are containers for copies of our counting symbol.

109. x

110. x

111. x

112. x

113. x

114. x

115. x

116. x

117. x x

118. x x

119. x x

120. x x

121. x x

122. x x

123. y y x x

124. y y x x

125. x x y y

126. y y x x

127. x x y y

128. x x y y

129. x x y y

130. x x y y

131. x x y y

132. x x r r y y

133. x x r r y y

134. Name relationships.

135. x x r r   y y

136. r r sin(  ) = x x r r   y y y y

137. r r x x r r   y y y y

138. x x cos(  ) = x x r r   r r y y

139. x x r r   x x r r y y

140. tan(  ) = x x x x r r   y y y y

141. x x r r   y y x x y y

142. Base One Probability

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

144. Then plot the result in base one.

145. Number shown: Number of appearances

146. Questions?