Counting

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Counting Computation through the Ages

• Lynda Colgan, PhD
• Faculty of Education
• Lynda.Colgan_at_queensu.ca
• May 23, 2009

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Welcome

• Finger Computation
• Counting Boards
• Napiers Bones
• Genaille-Lucas Rulers
• Soroban

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Morning Calc-isthenics Digit-y Wizardry
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Fingerin out the 9 times table
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Fingerin out the 9 times table
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Fingerin out the 9 times table
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Fingerin out the 9 times table
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Digithmetic...Multiplying by 6 n UP
7 X 8
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Digithmetic...Multiplying by 6 n UP
8 X 6
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Digithmetic...Multiplying by 6 n UP
6 X 6
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(No Transcript)
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Roman Numerals

• I 1
• V 5
• X 10
• L 50
• C 100
• D 500
• M 1000

Early Roman numerals were purely
additive. III 3 IIII 4 VII 7 VIIII 9 DCCCCI 90
1 MMDLXIII 2563
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Subtractive Form

• IV 4
• IX 9
• XC 90
• CD 400
• MCM 1900
• Never is more than one letter on the left used,
  e.g., we write VIII, not IIX for 8.

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Counting Boards
One of the earliest counting boards is the
Salamis tablet, dating to about 400 BC.
Counting boards were used in Europe until about
1500 AD.
Hindu-Arabic vs Counting Board
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• Robert Recorde
• Welsh mathematician
• Inventor of
• Author of Grounde of Artes or(Arithmetick) (1543)

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The lines represent 1, 10, 100, and 1000. The
spaces between the lines represent 5, 50, and 500.
M (1000)
D
C (100)
L
X (10)
V
I (1)
Pebbles were used for calculations.
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Representing Numbers
MMMDCCCLXXIIII (3874)
M
D
C
L
X
V
I
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Addition MMDCCXXXVII MMMDCCCLXXIIII
MMMDCCCLXXIIII (3874)
MMDCCXXXVII (2737)
M
D
C
L
X
V
I
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Subtraction MMMMCCCCXXI MMDCXXXVIII
M
D
C
L
X
V
I
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Multiplication XXVI LXXXIII (26 X 83)
M
D
C
L
X
V
I
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Napiers Bones
A box of Napiers bones, one of the oldest
calculating machines. Invented by the Scottish
mathematician James Napier in 1617. The strips
with 4, 7, 9 give partial products of any digit
from 1 to 9 times 479.
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Napiers Bones
The Napiers bones consist of vertical strips of
the table. Each entry is the product of index
number and strip number, e.g., 7 x 8 56, with 5
at the upper left half of square and 6 on lower
right.
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Genaille-Lucas Rulers
Similar to Napiers bones, but without the need
to mentally calculate the partial sum. Just
follow the arrows and read off the answer
backwards(least significant to most significant
digits). The device was invented by Genaille in
1885.
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7 X 286
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3271 X 4
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47 X 2207
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96 X 5937
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7431 4
7431 7
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please open part 2