multiplication using historical methods joanne peeples el paso community college [email protected]
TRANSCRIPT
Increasingly students are entering college unable to
multiply or divide integers without using a calculator
(hence having little number sense). Rather than
repeat multiplication methods they have learned,
why not try Egyptian arithmetic, multiplication per
gelosia (“by jealousy”) or Napier’s bones? History
plus math is often a winning strategy.
EGYPTIAN NUMBERS
EGYPTIAN NUMBERS
23 ∩∩│││ or │∩│∩│
Add 23 ∩∩│││ 23 ∩∩│││ ______________ ∩∩∩∩││││││
MULTIPLICATION
26 X 4
___26____________
1 4 2 8 4 16 8 32 16 64 32 128 Too Big! 32 is bigger than 26
26 = 16 + 10 = 16 + 8 + 2
Add: 64 + 32 + 8 = 104
WHY DOES THIS WORK?
424242 134
4222 34
104426
DIVISION (NO REMAINDER)
161 = 112 + 28 + 14 + 7 Therefore 161 7 = 16 +4+2+1 = 23
161 7
_____161___
1 7 2 14 4 28 8 5616 11232 224 Too Big! 224 is bigger than 161
DIVISION (WITH REMAINDER)
84 11
_________84___
1 11
2 22
4 44
8 88 Too Big! 88 is bigger than 84
84 = 44 + 22 + 11 + 7 Therefore 84 11 = 4 + 2 + 1 = 7
with a remainder of 7 We would write 84 11 = 7 + 7/11
EGYPTIAN FRACTIONS
││││ Four
││││ one-fourth
Write = =
3.14 = 3 + │││ ∩ ∩∩│││││
│││ ∩│││││
EXAMPLE
Is this approximately 4, 3, or 2?
1310
1
2
13
1310
4586
FRACTIONS DECREASE IN SIZE (SO THEY CAN BE USED FOR ESTIMATION.)
In a sense, Egyptian fractions are closer to “our” decimal system than “our” fractions.
141.325
1
10
13
FRACTIONS NEVER REPEAT.
Writing as is wrong, since would be a bad approximation.
Writing as is acceptable since is a good approximation.
FRACTIONS MUST NOT BE “CLOSE” TO EACH OTHER. Writing as is not acceptable, since and are
“too close” to each other.
Writing as is acceptable since an
approximation of is reasonable.
A FRACTIONAL REPRESENTATION MAY NOT BE UNIQUE. Note : 1.75 = 175% = = so our representation of numbers is not unique.
AND AS YOU MAY HAVE SEEN, ALL THE FRACTIONS ARE UNIT FRACTIONS.
HOW DO YOU FIND A FRACTIONAL REPRESENTATION? 4 ÷ 7. Think of something you can divide, like bread.
each part is of a whole
So
loaf left
loaves left
loaf left
of loaf
So
OR
45
45
Do not cut!
Whole loaf of loaf
Half loaf of loaf
So
WHAT IF THE AREA OF A TRIANGLE WHOSE BASE IS 37 UNITS AND HEIGHT IS 12 UNITS?
A = x base x height
x base
1 37 = 20 + 16 + 1
10 + 8 + = 18
2
22
(Write ││ as ) 2
2
2
( base) height
Area of triangle is 24 + 192 + 6 = 222 sq. units.
18
1 12
2 24
4 48
8 96
16 192
6
MULTIPLICATION PER GELOSIA 2,314 X 157 =
363,298 2 3 1 4
1
5
7
0 0 0 0
2 3 1
41 1 0 2
0 5 5
01 2 0 2
4 1 7
8
1 1
0
3
6
3 2 9 8
NAPIER’S BONES
BIBLIOGRAPHY
Reimer, David, Count Like an Egyptian, A Hands on Introduction to Ancient Mathematics. Published by the Princeton University Press, 2014.
Ifrah, Georges, The Universal History of Numbers. Published by John Wiley and Sons, Inc., 2000 Napier, John, Rabdology, translated by William Frank Richardson.Volume 15 in the Charles Babbage Institute Reprint Series for the History of Computing. Published by the Massachusetts Institute of Technology and Tomash Publishers, 1990
