Arithmetic by Computer and Arithmetic by Human Tony … · and Arithmetic by Computer Three...
Transcript of Arithmetic by Computer and Arithmetic by Human Tony … · and Arithmetic by Computer Three...
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Arithmetic by Computer and
Arithmetic by Human
Tony Mann
4 February 2013
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Gresham College
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15 April: Proof by Computer and Proof by Human
4 March: How Computers get it Wrong - 2+2 = 5
4 February: Arithmetic by Human and Arithmetic by Computer
Three lectures
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Pierre Pica
When I come back from Amazonia I lose sense of time and sense of number, and perhaps sense of space. I have extreme difficulty
adjusting to Paris again, with its angles and straight lines.
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Children and Animals
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121
213
363
121
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25773
121 times 213
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8 times 7
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121 times 213
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121 times 213
121 213
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121 times 213
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121 times 213
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121 times 213
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121 times 213
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121 times 213
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121 times 213
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121 times 213
121 213
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121 times 213
121 213
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121 times 213
121 213
60 426
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1
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121 times 213
121 213
60 426
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1 13632
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121 times 213
121 213
60 426
30 852
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1 13632
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121 times 213
121 213
60 426
30 852
15 1704
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3 6816
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121 times 213
121 213
60 426
30 852
15 1704
7 3408
3 6816
1 13632 25773
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Binary Notation
1910 = 1x101 + 9x100 = 10+9
100112 = 1x24+0x23+0x22+1x21+1x20
= 16 + 2 + 1
= 1910
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121 times 213
121 = 11110012
= (64 + 32 + 16 + 8 + 1)
121 x 213 = (64 + 32 + 16 + 8 + 1)x 213
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121 times 213
121 213
60 426
30 852
15 1704
7 3408
3 6816
1 13632 25773
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Roman Numerals
Twice CXXI = CCXXXXII
Half XXXII = XVI
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XIII times XXXI
XIII XXXI
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XIII times XXXI
XIII XXXI
VI
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XIII times XXXI
XIII XXXI
VI
III
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XIII times XXXI
XIII XXXI
VI
III
I
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XIII times XXXI
XIII XXXI
VI XXXXXXII
III
I
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XIII times XXXI
XIII XXXI
VI XXXXXXII = LXII
III
I
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XIII times XXXI
XIII XXXI
VI XXXXXXII = LXII
III LLXXIIII
I
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XIII times XXXI
XIII XXXI
VI XXXXXXII = LXII
III LLXXIIII = CXXIIII
I
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XIII times XXXI
XIII XXXI
VI XXXXXXII = LXII
III LLXXIIII = CXXIIII
I
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XIII times XXXI
XIII XXXI
VI XXXXXXII = LXII
III LLXXIIII = CXXIIII
I CCXXXXIIIIIIII
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XIII times XXXI
XIII XXXI
VI XXXXXXII = LXII
III LLXXIIII = CXXIIII
I CCXXXXIIIIIIII =
CCXXXXVIII
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XIII times XXXI
XIII XXXI
VI XXXXXXII = LXII
III LLXXIIII = CXXIIII
I CCXXXXIIIIIIII =
CCXXXXVIII
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XIII times XXXI
XIII XXXI
VI XXXXXXII = LXII
III LLXXIIII = CXXIIII
I CCXXXXIIIIIIII =
CCXXXXVIII
CCC XXXXXXXXX V IIIIIIII
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XIII times XXXI
XIII XXXI
VI XXXXXXII = LXII
III LLXXIIII = CXXIIII
I CCXXXXIIIIIIII =
CCXXXXVIII
CCC XXXXXXXXX V IIIIIIII
= CCC L XXXX X III
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XIII times XXXI
XIII XXXI
VI XXXXXXII = LXII
III LLXXIIII = CXXIIII
I CCXXXXIIIIIIII =
CCXXXXVIII
CCC XXXXXXXXX V IIIIIIII
= CCC L XXXX X III
= CCCC III
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1000
Let’s do this one together
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+ 40
Let’s do this one together
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+ 1000
Let’s do this one together
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+ 30
Let’s do this one together
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+ 1000
Let’s do this one together
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+ 20
Let’s do this one together
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+ 1000
Let’s do this one together
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+ 10
Let’s do this one together
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Can I have a volunteer please?
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4567
4+5+6+7 = 22
2+2 = 4
4567 = 9x507 + 4
Casting out nines
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Counting on your fingers
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Abacus and Counting-Table
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Abacus
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Abacus
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How does this work?
1 + 3 + 5 + … + 2n-1 = n2
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John Napier (1550 – 1617)
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Napier’s Bones
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Logarithms
10a x 10b = 10a+b
To find A times B:
If A = 10a and B = 10b
Then AxB = 10a+b
a = log (A)
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2 times 3
2 ≈ 100.301
3 ≈ 100.477
2x3 ≈ 100.301+0.477
= 100.778
≈ 5.998
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Newton by GBS
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John Napier (1550 – 1617)
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Henry Savile to Edmund Gunter
"This is mere showing of tricks,
man!"
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The Slide Rule
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Machines
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Leslie Comrie
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Consul the Educated Monkey
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Binary arithmetic
1910 = 100112 = 24+21+20
10 x 25 = 250
2 x 1112 = 11102
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Binary arithmetic – 19x10
10011 x 4 = 1001100
so 10011 x 5 = 1001100
+ 10011
1011111
x 2 = 10111110
(= 128+32+16+8+4+2 = 190)
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Computer Multiplication uses the ancient
Egyptian method!
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Our courses Comment at
www.tonysmaths.blogspot.com
Thank you for listening!
[email protected] @Tony_Mann
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Acknowledgments and picture credits
Thanks to Noel-Ann Bradshaw and James Smith, and everyone at Gresham College
Picture credits
Gresham College: George Vertue (1740), Wikimedia Commons Pierre Pica: Pierre Pica, Wikimedia Commons Baby: Microsoft Clipart Crow: Walter Siegmund (2006), Wikimedia Commons Stephen Byers: BBC News website Abacus and counting-table: from Gregor Reisch, Margarita Philosophica, 1508. Wikimedia Commons “Volunteer” photo – Microsoft Clipart Finger counting: Luca Pacioli, Summa, 1494, Wikiimedia Commons Napier – 1616 portrait by unknown artist, Wikimedia Commons Napier Memorial Stone – Ad Meskens, Wikimedia Commons Isaac Newton by Godfrey Kneller, Wikimedia Commons George Bernard Shaw, anonymous photo (1936), Wikimedia Commons Newton using logarithms, from George Bernard Shaw, In Good King Charles's Golden Days (1939) Napier title page: Archive.com, issued under Creative Commons Licence Henry Savile, possibly by Marcus Gheeraerts the Younger. Wikimedia Commons Pascaline: David Monniaux (2005), Wikimedia Commons Leibniz’s Stepped Reckoner: User:Kolossos, Wikimedia Commons Walther Calculator: Hannes Grobe (2006), Wikimedia Commons Comrie: from “L.J. Comrie”, http://www.columbia.edu/cu/computinghistory/comrie.html Brunsviga: Wernerroth at de.wikipedia, Wikimedia Commons auf Deutsch