Assignment 1 Solutions — LibEd3100
1. Write each of the following numbers from our system in Egyptian hieroglyphics:
a) 531 in Egyptian is 5 scrolls plus 3 heel-bones plus 1 staff, written right to
left.
b) 2011 in Egyptian is 2 lotus flowers plus 1 heel-bone plus 1 staff, written
right to left.
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c) 23 23
is 2 heel-bones plus 3 staffs followed by an oval over 2 heel-bones plus
3 staffs, written right to left.
2. Write each of the following numbers in Egyptian hieratic form:
a) 32 is the symbol for 30 followed by the symbol for 2, written right to left.
b) 78 is the symbol for 70 followed by the symbol for 8, written right to left.
c) 110 is the symbol for 100 followed by the symbol for 10, written right to
left.
3. Write the numbers of the following sums and differences in Egyptian hieroglyphics, and carry out the operation (in hieroglyphics only):
a) 2202 + 11,403: Using PF as an abbreviation for “pointing finger,” we
have
2 staffs plus 0 heel-bones plus 2 scrolls plus 2 lotusflowers
PLUS 3 staffs plus 0 heel-bones plus 4 scrolls plus 1 lotusflower plus 1 PF
EQUALS 5 staffs plus 0 heel-bones plus 6 scrolls plus 3 lotusflowers plus 1 PF
b) 242 − 126
2 staffs + 4 heel-bones + 2 scrolls
MINUS
6 staffs + 2 heel-bones + 1 scroll
EQUALS 6 staffs + 1 heel-bone + 1 scroll.
Notice that we have to “borrow” a heel-bone from the 4 heel-bones, convert it
to 10 staffs, and then take a total of 12 − 6 staffs
4. Do the following multiplications by the repeated doubling method used by the
Egyptians:
a) 9 × 21
→
→
1 21 ←
2 42
4 84
8 168 ←
9 189
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Here the arrow means that we need to use this row. Since 1 + 8 = 9, we use
the entries in the right column for the 1st and 4th rows, adding 21 and 168 to
get an answer of 189.
b) 13 × 17
→
→
→
1 17 ←
2 34
4 68 ←
8 136 ←
9 189
Here the arrow means that we need to use this row. Since 1 + 4 + 8 = 13, we
use the entries in the right column for the 1st, 3rd and 4th rows, adding 17, 68
and 136 to get an answer of 221.
5. a) In Question 1a) you wrote the number 531 in Egyptian hieroglyphics. Now
write the number 5310 in hieroglyphics.
531 is 5 scrolls, 3 heel-bones and 1 staff, written right to left; the number
5310 is 5 lotus flowers, 3 scrolls, 1 heelbone and 0 staffs written right to left.
b) When you multiply a number in our number system by 10, how is a resulting
number related to the original number?
In our number system, when we multiply a number by 10 we simply have to
put a zero on the end of the number!
c) In the Egyptian hieroglyphic system, how is the number you get when you
multiply a given number by 10 related to the given number? That is, describe
in words a rule for multiplying any Egyptian hieroglyphic number by 10.
Starting on the right side of the list of symbols, simply change each symbol
to the one after it in the list of symbols for 1, 10, 100, 1000, etc.
6. Solve the Rhind Mathematical Papyrus Problem 28, using the method of false
position: “A quantity and its 2/3 are added together and from the sum 1/3 of
the sum is subtracted and 10 remains. What is the quantity?”
Solution: Letting x be the variable, we have the equation (x + 2x/3) − 1/3(x +
2x/3) = 10.
By false position, try x = 3 (since fractions in thirds are present): this gives
(x + 2x/3) − 1/3(x + 2x/3) = 5 − 1/3(5) = 10/3. We want an answer of 10, so
we need to multiply by a factor of 3. This gives us an answer of x = 3(3) = 9.
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7. On p. 52 of the Coursepack is found a translation of the Rhind Mathematical
Papyrus (RMP) problem 6 showing a multiplication involving fractions. Convert the fractions in the translation into our familiar representation of fractions
and verify the steps shown in RMP6 and that, therefore,
1 ¯3̄ 10 30 + 7 5̄ = 9.
Solution: The translation has the multiplication
¯3̄ 5̄ 30
1 ¯3̄ 10 30
3 2̄ 10
7 5̄
.
\ 2
4
\ 8
so we need to convert the Egyptian fractions into our notation:
¯3̄ 5̄ 30 = 2 + 1 + 1 = 20 + 6 + 1 = 27 = 9
3
5
30
30
30
30
30
10
1
1
20
3
1
54
+ 30
= 30
+
+
+
= 30
=
1 ¯3̄ 10 30 = = 1 + 23 + 10
30
30
30
30
1
1
30
5
1
36
18
3 2̄ 10 = 3 + 2 + 10 = 10 + 10 + 10 = 10 = 5
+ 51 = 36
.
7 5̄ = 7 + 15 = 35
5
5
9
5
We can rewrite the multiplication as
1
\ 2
4
\ 8
9/10
9/5
18/5
36/5
so the doubling that occurs at each step is now apparent. Using the marked
lines (since 2 + 8 = 10) gives
10 × ¯3̄ 5̄ 30 = 1 ¯3̄ 10 30 + 7 5̄ = 9 + 36 = 45 = 9,
5
5
5
which is the answer the scribe claims.
8. A quantity and its 3/4 are added together, and when 1 is added the answer is
15. Use the Egyptian method of false position to find the quantity.
Solution: Letting x be the variable, we have the equation x + (3/4)x + 1 = 15.
Before we can use the false position, we need to adjust this to x + (3/4)x = 14.
Now trying a guess of x = 4, to get rid of fractions, would lead to x + (3/4)x =
4 + (3/4)4 = 7. We need to double this, to go from 7 to 14, so we should double
our original guess from 4 to 8. So our answer is 8.
9. Translate the Rhind papyrus passages from the distributed Plates A and B
labelled 9 and 15 (Plate A) and 37 (Plate B). Hieroglyphs that aren’t numbers
and transcription annotations (such as “sic”) should just be recopied in your
translations. Can you identify what type of computation is being performed in
each case?
This was done in detail in the Oct 1 class.
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10. We saw that most Egyptian fractions were sums of unit fractions, but sometimes
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these sums are not unique. Show that 40
has the two representations
3̄ 8̄ 60 and 4̄ 8̄ 10.
The second choice is what an Egyptian scribe would choose. Why do you think
that would be the choice?
Solution: Converting to our notation we find
3̄ 8̄ 60 =
=
4̄ 8̄ 10 =
1
1
1
+ 18 + 60
= 11
+ 60
3
24
11
1
2·1
3·19
+ 5·12
= 5·11
+ 120
= 55+2
= 3·40
= 19
2·12
120
120
40
1
1
1
3
1
3·5
4·1
15+4
+
+
=
+
=
+
=
= 19
,
4
8
10
8
10
8·5
4·10
40
40
as required.
The second of these unit fractions is what the scribe would have used since the
denominators in the second fraction are (overall) smaller (and so the hieroglyphs
in forming the fractions would involve fewer symbols overall).
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