History of Mathematics

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History of Mathematics
Egyptian Mathematics
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In the beginning there was counting.
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In the beginning there was counting. Of what?
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In the beginning there was counting.
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In the beginning there was counting.
tallying
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Mathematics beyond tallying arises from the
demands of civilization the culture of people
who live in permanent cities.
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Mathematics beyond tallying arises from the
demands of civilization the culture of people
who live in permanent cities. Where did the
oldest civilizations develop?
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Mathematics beyond tallying arises from the
demands of civilization the culture of people
who live in permanent cities. Where did the
oldest civilizations develop? Around the Nile
River (Egyptian) and the Tigris and Euphrates
Rivers (Mesopotamian).
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How would civilization create demands for better
ways of tallying? How would tallying lead
naturally to counting? What other needs for
mathematics might arise with civilization?
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Working with large numbers creates the need to
count tally marks in groups.
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Working with large numbers creates the need to
count tally marks in groups.
Quick! How many tally marks are there?
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Working with large numbers creates the need to
count tally marks in groups.
Quick! How many tally marks are there?
Now how many tally marks are there?
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Egyptians grouped tally marks like this
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Egyptians grouped tally marks like this
How might they write the number 123?
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Egyptians grouped tally marks like this
How might they write the number 123? They used
additive notation.
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Egyptians grouped tally marks like this
How would ancient Egyptians write 999?
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Egyptians grouped tally marks like this
How would ancient Egyptians write 999? The
hieratic writing developed from hierogylphics
simplified notation for large numbers.
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In order to appreciate the importance of the
invention of positional notation for numbers, the
author of our textbook would like us to try doing
mathematics using the additive notation of the
ancient Egyptians. He doesnt, however, want us
to waste time memorizing and learning to write
new symbols for numbers. For this reason he
suggests we use pseudohieratic numerals. Our
number 123 in pseudohieratic is 100, 20, 3.
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Lets try two of the homework problems for
Section 1.1 1ab 3ab Several of the assigned
homework problems asks you to write fractions in
hieroglyphics and hieratic. The reading for
Section 1.1 explains how to do this.
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The next section in the text introduces the Rhind
papyrus. What is it?
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The next section in the text introduces the Rhind
papyrus. What is it? It is the most
comprehensive piece of ancient Egyptian
mathematics that still exists as a complete work.

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What kind of mathematics appears in the Rhind
papyrus? We see examples showing methods for
multiplication, division, solving equations, and
computing areas and volumes. The papyrus is
written in hieratic, so we will follow what it
contains in pseudohieratic.
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Multiplication Multiply 10,3 by 20,7. \
1 20,7 2 50,4 \ 4 100,8 \ 8 200,10,6 The
result is 300,50,1.
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Multiplication Multiply 10,3 by 20,7. \
1 10,3 \ 2 20,6 4 50,2 \ 8 100,4 \
16 200,8 The result is 300,50,1.
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Division Divide 300,50,1 by 20,7. \ 1 20, 7
2 50, 4 \ 4 100, 8 \ 8 200,10,6 The result is
10,3.
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Division Divide 300,50,1 by 10,3. \ 1 10,3 \
2 20,6 4 50,2 \ 8 100,4 \ 16 200,8 The result
is 20,7.
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Lets try doing multiplication and division using
these methods for some of the homework problems
in Section 1.2. 1ab 2ab
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• a) Multiply 40,3 by 10,8.
• 1 40,3
• \ 2 80,6
• 4 100,70,2
• 8 300,40,4
• \ 16 600,80,8
• The result is 700,70,4.

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• b) Multiply 10,2 by 90,7.
• 1 90,7
• 2 100,90,4
• \ 4 300,80,8
• \ 8 700,70,6
• The result is 1000,100,60,4.

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2. a) Divide 40,8 by 10,2. 1 10,2 2 20,4 \
4 40,8 The result is 4.
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2. b) Divide 90,6 by 8. 1 8 2 10,6 \
4 30,2 \ 8 60,6 The result is 10,2.
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What do we do when a number doesnt divide evenly
into another?
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What do we do when a number doesnt divide evenly
into another? Divide 80,8 by 6. 1 6 \
8 40,8 \ 2 10,2 \ 2/3 4 \ 4 20,4 The result is
10,4,2/3.
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Divide 50,6 by 9. 1 9 \ 2 10,8 \ 4 30,6
1/9 1 \ 2/9 2 The result is 10,2/9.
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Ancient Egyptians usually preferred to work with
unit fractions (those of the form 1/n), but they
made an exception for 2/3. For values of n other
than 3, they created tables listing numbers
equivalent to 2/n involving only unit fractions
(see page 11). n 2/n n 2/n 5 1/3,
1/15 11 1/6, 1/66 7 1/4, 1/28 13 1/8, 1/52,
1/104 9 1/6, 1/18 15 1/10, 1/30
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Where did the values for 2/n come from? How were
they used?
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Where did the values for 2/n come from? How were
they used? Lets see why 2/9 is equivalent to
1/6, 1/18. We want to divide 2 by 9. How many
copies of 9 make 2?
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Divide 2 by 9.
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Divide 2 by 9. Create some easy fractions of 9
to get started.
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Divide 2 by 9. 1 9 1/3 3 1/2 4, 1/2
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Divide 2 by 9. 1 9 1/3 3 1/2 4,
1/2 1/4 2, 1/4
1/4 is half of 1/2
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Divide 2 by 9. 1 9 1/3 3 1/2 4,
1/2 1/4 2, 1/4 1/8 1, 1/8
1/8 is half of 1/4
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Divide 2 by 9. 1 9 1/3 3 1/2 4, 1/2
1/6 1, 1/2 1/4 2, 1/4 1/8 1, 1/8
1/6 is half of 1/3
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Divide 2 by 9. 1 9 1/3 3 1/2 4, 1/2
1/6 1, 1/2 1/4 2, 1/4 1/8 1, 1/8
1/12 1/2, 1/4
1/12 is half of 1/6
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Divide 2 by 9. 1 9 1/3 3 1/2 4, 1/2
1/6 1, 1/2 1/4 2, 1/4 1/9 1 1/8 1,
1/8 1/12, 1/2, 1/4
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Divide 2 by 9. 1 9 1/3 3 1/2 4, 1/2
1/6 1, 1/2 1/4 2, 1/4 1/9 1 1/8 1,
1/8 1/12, 1/2, 1/4 There are various
ways to proceed, but it is often easiest to find
the largest number smaller than the target 2 and
work from there.
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Divide 2 by 9. 1 9 1/3 3 1/2 4,
1/2 \ 1/6 1, 1/2 1/4 2, 1/4 1/9 1 1/8
1, 1/8 1/12, 1/2, 1/4 There are various
ways to proceed, but it is often easiest to find
the largest number smaller than the target 2 and
work from there.
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Divide 2 by 9. 1 9 1/3 3 1/2 4,
1/2 \ 1/6 1, 1/2 1/4 2, 1/4 1/9 1 1/8
1, 1/8 1/12, 1/2, 1/4 1/6 of 9 gives us
1, 1/2. We now need 1/2 more. How can we get
it?
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Divide 2 by 9. 1 9 1/3 3 1/2 4,
1/2 \ 1/6 1, 1/2 1/4 2, 1/4 1/9 1 1/8
1, 1/8 1/12, 1/2, 1/4 ??? 1/2 1/6 of
9 gives us 1, 1/2. We now need 1/2 more. How
can we get it?
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Divide 2 by 9. 1 9 1/3 3 1/2 4,
1/2 \ 1/6 1, 1/2 1/4 2, 1/4 1/9 1 1/8
1, 1/8 1/12, 1/2, 1/4 ??? 1/2 1/6 of
9 gives us 1, 1/2. We now need 1/2 more. How
can we get it?
Half of 1 is 1/2.
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Divide 2 by 9. 1 9 1/3 3 1/2 4,
1/2 \ 1/6 1, 1/2 1/4 2, 1/4 1/9 1 1/8
1, 1/8 1/12, 1/2, 1/4 ???
1/2 1/6 of 9 gives us 1, 1/2. We now need 1/2
more. How can we get it?
Half of 1 is 1/2.
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Divide 2 by 9. 1 9 1/3 3 1/2 4,
1/2 \ 1/6 1, 1/2 1/4 2, 1/4 1/9 1 1/8
1, 1/8 1/12, 1/2, 1/4 ??? 1/2 1/6 of
9 gives us 1, 1/2. We now need 1/2 more. How
can we get it?
Half of 1/9 is 1/18.
Half of 1 is 1/2.
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Divide 2 by 9. 1 9 1/3 3 1/2 4,
1/2 \ 1/6 1, 1/2 1/4 2, 1/4 1/9 1 1/8
1, 1/8 1/12, 1/2, 1/4 1/18 1/2 1/6 of
9 gives us 1, 1/2. We now need 1/2 more. How
can we get it?
Half of 1/9 is 1/18.
Half of 1 is 1/2.
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Divide 2 by 9. 1 9 1/3 3 1/2 4,
1/2 \ 1/6 1, 1/2 1/4 2, 1/4 1/9 1 1/8
1, 1/8 1/12, 1/2, 1/4 \ 1/18 1/2 1/6 of
9 gives us 1, 1/2. We now need 1/2 more. How
can we get it?
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Divide 2 by 9. 1 9 1/3 3 1/2 4,
1/2 \ 1/6 1, 1/2 1/4 2, 1/4 1/9 1 1/8
1, 1/8 1/12, 1/2, 1/4 \ 1/18 1/2
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Divide 2 by 9. 1 9 1/3 3 1/2 4,
1/2 \ 1/6 1, 1/2 1/4 2, 1/4 1/9 1 1/8
1, 1/8 1/12, 1/2, 1/4 \ 1/18 1/2 The
result is 1/6, 1/18.
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Divide 2 by 9. 1 9 1/3 3 1/2 4,
1/2 \ 1/6 1, 1/2 1/4 2, 1/4 1/9 1 1/8
1, 1/8 1/12, 1/2, 1/4 \ 1/18 1/2 The
result is 1/6, 1/18. Check 1/6 1/18 4/18
2/9.
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Divide 2 by 10,5. 1 10,5 1/3 5
1/2 7, 1/2 1/5 3 1/4 3, 1,2,
1/4 \ 1/10 1, 1/2 1/8 1, 1/2, 1/4, 1/8
1/15 1 \ 1/30 1/2 The result is
1/10, 1/30.
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How was this table used? n 2/n n
2/n 5 1/3, 1/15 11 1/6, 1/66 7 1/4,
1/28 13 1/8, 1/52, 1/104 9 1/6, 1/18 15 1/10,
1/30
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Divide 10,2 by 5.
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Divide 10,2 by 5. 1 5 \ 2 10 1/5 1 \
2/5 2 The result is 2,2/5 which is 2,1/3,1/15
according to the table.
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Divide 10,3 by 5. 1 5 \ 2 10 \ 1/5 1 \
2/5 2 The result is 2,1/5,2/5 which is
2,1/5,1/3,1/15. We rearrange our answer as
2,1/3,1/5,1/15.
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Divide 60,9 by 7. \ 1 7 1/7 1 2
10,4 \ 2/7 2 4 20,8 \ 4/7 4 \ 8
50,6 The result is 9, 2/7, 4/7. From the
table we know 2/7 is 1/4, 1/28. Since 4/7 is
twice 2/7, it must be 2/4, 2/28, which we rewrite
as 1/2, 1/14. The final answer is 9, 1/4, 1/28,
1/2, 1/14 which we rearrange as 9, 1/2, 1/4,
1/14, 1/28.
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Divide 60,9 by 7. \ 1 7 1/7 1 2
10,4 \ 2/7 2 4 20,8 \ 4/7 4 \ 8
50,6 The result is 9, 2/7, 4/7. From the
table we know 2/7 is 1/4, 1/28. Since 4/7 is
twice 2/7, it must be 2/4, 2/28, which we rewrite
as 1/2, 1/14. The final answer is 9, 1/4, 1/28,
1/2, 1/14 which we rearrange as 9, 1/2, 1/4,
1/14, 1/28. Check 91/21/41/141/28 276/28
69/7
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Why does the table include only fractions of the
form 2/n? Why exclude more general fractions
such as the ones below? 7/12 19/21 15/23
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Why does the table include only fractions of the
form 2/n? Why exclude more general fractions
such as the ones below? 7/12 6/12 1/12 1/2
1/12 19/21 15/23
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Why does the table include only fractions of the
form 2/n? Why exclude more general fractions
such as the ones below? 7/12 6/12 1/12 1/2
1/12 19/21 15/23
6 and 1 are divisors of 12.
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Why does the table include only fractions of the
form 2/n? Why exclude more general fractions
such as the ones below? 7/12 6/12 1/12 1/2
1/12 19/21 15/23
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Why does the table include only fractions of the
form 2/n? Why exclude more general fractions
such as the ones below? 7/12 6/12 1/12 1/2
1/12 19/21 7/21 7/21 3/21 1/21 1/21
1/3 1/3 1/7 2/21
2/3 1/7 1/14 1/42
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Why does the table include only fractions of the
form 2/n? Why exclude more general fractions
such as the ones below? 13/23 1/23 12/23
1/23 6(2/23) 1/23 6(1/12
1/276) 1/23 6/12 6/276
1/23 1/3 1/46 1/3 1/23 1/46
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We study the history of mathematics by working
through the mathematics of history in part to
appreciate discoveries such as positional
notation and the usual multiplication algorithm.
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• We also study the mathematics of history to
• obtain a flexibility with numbers and
  mathematical concepts that helps us evaluate
  student work when it is presented in a
  nonstandard way,
• remind ourselves what mathematics is about (e.g.
  studying nonstandard multiplication algorithms
  reminds us what multiplication really does, and
  encourages us to examine why the standard
  algorithm works), and
• remind ourselves how difficult it is to learn
  mathematics. We are so accustomed to modern
  notation and algorithms that we forget how hard
  it was to deal with mathematical ideas when we
  first learned them. (For example, the additive
  notation for numbers used in ancient Egypt is
  much more natural than positional notation, yet
  we struggle with it when we try to use it.)

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In addition we study the history of mathematics
to know the order in which human minds invented
different mathematical ideas. The later an idea
arrived, the more difficult it most likely was.
Knowing which ideas caused humanity as a whole
the most trouble will help us identify those
concepts likely to cause our students the most
concern.
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Given an ancient document such as the Rhind
papyrus, how do we know what is says when the
writing is no longer in use?
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On page 24 of the textbook, we find a portion of
a tablet carved with cuneiform writing.
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On page 24 of the textbook, we find a portion of
a tablet carved with cuneiform writing. Lets
try to decipher it by looking for patterns and
thinking about the types of patterns we might
expect to find.
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Now we return to the Rhind papyrus and the
mathematics of ancient Egypt. In addition to
algorithms for multiplication and division, the
papyrus also contains examples of solving linear
equations.
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A number and its seventh make 10,9. What is the
number?
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A number and its seventh make 10,9. What is the
number? First lets restate this problem using
modern notation.
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A number and its seventh make 10,9. What is the
number? First lets restate this problem using
modern notation. Solve for x x (1/7)x 19
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A number and its seventh make 10,9. What is the
number?
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A number and its seventh make 10,9. What is the
number? Here is the solution found on the
Rhind papyrus Suppose the number is 7. Then 7
and its seventh make 8 \ 1 7 \ 1/7 1
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If 1,1/7 multiplied by 7 gives 8, then 1,1/7
multiplied by what number will give 10,9?
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If 1,1/7 multiplied by 7 gives 8, then 1,1/7
multiplied by what number will give 10,9? We
must divide 10,9 by 8.
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If 1,1/7 multiplied by 7 gives 8, then 1,1/7
multiplied by what number will give 10,9? We
must divide 10,9 by 8. 1 8 \ 2 10,6 1/2 4 \
1/4 2 \ 1/8 1 8 multiplied by 2,1/4,1/8 gives
19. So 7 multiplied by 2,1/4,1/8 must give the
answer.
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7 multiplied by 2,1/4,1/8 is 10,6,1/2,1/8.
1 7 \ 2 10,4 1/2 3, 1/2 \ 1/4 1, 1/2, 1/4 \
1/8 1/2, 1/4, 1/8 7 times 2,1/4,1/8 is
10,4,1,1/2,1/4,1/2,1/4,1/8 which simplifies to
10,6,1/2,1/8.
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We should have 1,1/7 times 10,6,1/2,1/8 equal to
10,9.
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We should have 1,1/7 times 10,6,1/2,1/8 equal to
10,9. Check 8/7 16 5/8 8/7 133/8
133/7 19
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A number and its third are 5. What is the number?
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A number and its third are 5. What is the
number? The number is 3,1/2,1/4.
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A number and its third are 5. What is the
number? The number is 3,1/2,1/4. Check that
1,1/3 multiplied by 3,1/2,1/4 is 5.
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Although the Rhind papyrus includes no nonlinear
equations, other papyri from ancient Egypt do
contain evidence of the consideration of
nonlinear equations. An example appears as
Problem 1.2 on page 15. Restate the problem
using modern notation.
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The Rhind papyrus also contains geometric
results. Problem 1.3 on page 16 concerns finding
the seked of a pyramid. Give a modern formula
for the seked of a pyramid. What does the seked
tell you?
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The Rhind papyrus also contains geometric
results. Problem 1.3 on page 16 concerns finding
the seked of a pyramid. Give a modern formula
for the seked of a pyramid. What does the seked
tell you? The solution to Problem 1.4 on page 17
shows how to find the area of a circle. What
formula does this method suggest?
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The solution to Problem 1.7 on page 18 shows how
to find the volume of a truncated pyramid. What
formula does this solution suggest?
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Homework Assignment Section 1.1 1ace, 2, 3ac,
4ac, 5ac, 6ace, 7, 8 Section 1.2 1ac, 2ace, 3,
5ac, 6ac, 7ab, 10 Section 1.3 3, 5 Section 1.4
1ab, 2ac, 3ab, 4, 6