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Naming numbers
History of Math: Ancient arithmetic
Paul Hewitt
University of Toledo
Math 3510, 31 August 2010
Anthyphairesis
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Naming numbers
Outline
Space and time
Anthyphairesis
Space and time
Naming numbers
Outline
Space and time
Naming numbers
Anthyphairesis
Space and time
Naming numbers
Outline
Space and time
Naming numbers
Anthyphairesis
Anthyphairesis
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Naming numbers
Outline
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Anthyphairesis
Anthyphairesis
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Rome: the beginning
The Roman Republic arose when Greek and Phoenician
colonies dominated the Mediterranean and Black Seas
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Rome: republican timeline
• Senate overthrows monarchy: −509
• Control of Italy: −265
• Punic wars:
• −264 to −241
• −218 to −201
• −149 to −146
• Control of Greece: −146
• Control of Turkey and the Levant: −62
• Conquest of France: −52
• Julius Caesar murdered: −44
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Rome: imperial timeline
• Caesar Augustus: −27
• Conquest of Britain: +47
• Destruction of Temple: +70
• Roman conquests at maximum: +117
• Age of Empires: −200 to +200
• Imperial crisis: +235 to +284
• Diocletian divides empire: +285
• Constantine adopts Christianity: +312
• Odoacer conquers Rome: +476
• Mehmet conquers Constantinople: +1453
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Outline
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Place value systems
• Hindu-Arabic
• Chinese counting board
• Iraqi sexagesimals
• Mayan vigesimals
• Egyptian binary arithmetic
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The decimal system
If we divide 41891 by ten then we obtain
A. a remainder of 1.
B. a quotient of 4189.
C. 4189 groups of ten, with one left over.
D. All of the above.
E. None of the above.
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Triskaidekaphobia
If we express 41891 in base thirteen then the last digit is
A. the remainder of 41891 ÷ 13.
B. the quotient of 41891 ÷ 13.
C. the number of groups of thirteen contained in 41891.
D. All of the above.
E. None of the above.
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Change of base
41891ten =?thirteen
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Change of base
41891ten =?thirteen
41891 = 13 × 3222 + 5
3222 = 13 × 247 + 11
247 = 13 × 19
19 = 13 × 1 + 6
1 < 13
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Change of base
41891ten =?thirteen
41891 = 13 × 3222 + 5
3222 = 13 × 247 + 11
247 = 13 × 19
19 = 13 × 1 + 6
1 < 13
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And the answer is. . .
Hence 41891ten
A. 5, 11, 0, 6, 1thirteen
B. 1, 6, 0, 11, 5thirteen
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Unit fractions
What is the ratio 41891 : 13?
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Unit fractions
What is the ratio 41891 : 13?
3222 and a little bit. How much extra?
Anthyphairesis
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Unit fractions
What is the ratio 41891 : 13?
3222 and a little bit. How much extra?
41891 : 13 is 3222 and 5 : 13.
Anthyphairesis
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Unit fractions
What is the ratio 41891 : 13?
3222 and a little bit. How much extra?
41891 : 13 is 3222 and 5 : 13. OK, so what is 5 : 13?
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Unit fractions
What is the ratio 41891 : 13?
3222 and a little bit. How much extra?
41891 : 13 is 3222 and 5 : 13. OK, so what is 5 : 13?
5 is a bit less than half of 13, but more than the third part:
5
1
2
1
1
1
= +
= +
+
.
13
3 39
3 20 780
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Anthyphairesis
Unit fractions
What is the ratio 41891 : 13?
3222 and a little bit. How much extra?
41891 : 13 is 3222 and 5 : 13. OK, so what is 5 : 13?
5 is a bit less than half of 13, but more than the third part:
5
1
2
1
1
1
= +
= +
+
.
13
3 39
3 20 780
Hence 5 is a third part of 13 plus a twentieth part plus another
seven-hundred-eightieth part.
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Outline
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What is π?
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What is π?
• Ratio of circumference to diameter
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What is π?
• Ratio of circumference to diameter
• A bit more than 3
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What is π?
• Ratio of circumference to diameter
• A bit more than 3
• Closer to 3 and 1/7
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What is π?
• Ratio of circumference to diameter
• A bit more than 3
• Closer to 3 and 1/7
• But not really 7. . . closer to 7 and 1/15
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What is π?
• Ratio of circumference to diameter
• A bit more than 3
• Closer to 3 and 1/7
• But not really 7. . . closer to 7 and 1/15
• Or perhaps 7 and 1/16 ?
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How’d we do?
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How’d we do?
1
22
C
≈3+ =
D
7
7
1
333
≈3+
=
7 + 1/15
106
1
355
≈3+
=
7 + 1/16
113
error: − 0.001264489 . . .
error: + 0.000083219 . . .
error: − 0.000000266 . . .
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Roots
• anthyphairesis = anti-hypo-hairesis = “reciprocal
subtraction”
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Roots
• anthyphairesis = anti-hypo-hairesis = “reciprocal
subtraction”
• “hairesis” can mean “taking” or “choosing”. So
“hypo-hairesis” is a direct cognate of “sub-traction”.
Anthyphairesis
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Roots
• anthyphairesis = anti-hypo-hairesis = “reciprocal
subtraction”
• “hairesis” can mean “taking” or “choosing”. So
“hypo-hairesis” is a direct cognate of “sub-traction”.
• Curiously, “hairesis” has the same root as “heresy”.
Anthyphairesis