1
–
15
1
–
3
2
–
5
1
–
n
156 Egyptian multiplication
4
–
13
own symbol. Thus the Egyptians only dealt with frac-
tions of the form (with the exception of two-thirds).
Fractions with unit numerators are known today as
E
GYPTIAN FRACTIONS
. All other fractional quantities
were expressed as sums of distinct Egyptian fractions.
For example, , which equals + , was written
, and as .
The Egyptian’s ability to compute such expressions
is impressive. The Rhind papyrus provides reference
lists of such expressions, and the first 23 problems in
the document are exercises in working with such frac-
tional representations.
The ancient Egyptians were adept at solving
LIN
-
EAR EQUATION
s. They used a method called false posi-
tion to attain solutions. This involves guessing an
answer, observing the outcome from the guess, and
adjusting the guess accordingly. As an example, prob-
lem 24 of the Rhind papyrus asks:
Find the quantity so that when 1/7 of itself is
added to it, the total is 19.
To demonstrate the solution, the author suggests a
guess of 7. That plus its one-seventh is 8, by far too
small, but multiplying the outcome by 19/8 produces
the answer of 19 that we need. Thus 7 ×(19/8) must be
the quantity we desire.
The majority of problems in the Rhind papyrus are
practical in nature, dealing with issues of area (of rect-
angles, trapezoids, triangles, circles), volume (of cylin-
ders, for example), slopes and altitudes of pyramids
(which were built 1,000 years before the text was writ-
ten), and number theoretic problems about sharing
goods under certain constraints. Some problems, how-
ever, indicate a delight in mathematical thinking for its
own sake. For example, problem 79 asks:
If there are seven houses, each house with
seven cats, seven mice for each cat, seven ears
of grain for each mouse, and each ear of grain
would produce seven measures of grain if
planted, how many items are there altogether?
This problem appears in F
IBONACCI
’s Liber abaci, writ-
ten 600 years before the Rhind papyrus was discovered.
A version of this problem also appears as a familiar
nursery-rhyme and riddle, “As I Was Going to St. Ives.”
Egyptian multiplication The R
HIND PAPYRUS
indi-
cates that the ancient Egyptians of around 2000
B
.
C
.
E
.
used a process of “successive doubling” to multiply
numbers. They computed 19 ×35, for example, by
repeatedly doubling 35:
Since 19 = 16 + 2 + 1, summing 560 + 70 + 35 = 665
gives the product. This method shows that knowledge
of the two-times table is all that is needed to compute
multiplications. R
USSIAN MULTIPLICATION
follows an
approach similar to this method.
See also E
GYPTIAN MATHEMATICS
; E
LIZABETHAN
MULTIPLICATION
;
FINGER MULTIPLICATION
;
MULTIPLICA
-
TION
; N
APIER
’
S BONES
; R
USSIAN MULTIPLICATION
.
eigenvalue (e-value, latent root) See
EIGENVECTOR
.
eigenvector (e-vector, latent vector, characteristic vec-
tor, proper vector) For a square n×n
MATRIX
A, we
say a nonzero
VECTOR
xis an eigenvector for Aif there
is a number λsuch that Ax= λx. The number λis
called the eigenvalue associated with that eigenvector. If
xis an eigenvector of A, then we have that (A– λI)x=
0, where Iis the
IDENTITY MATRIX
. This shows that the
matrix A– λIis not invertible, and so must have zero
determinant: det(A– λI) = 0. This is a polynomial equa-
tion in λof degree n, called the “characteristic polyno-
mial” of A. As there can only be at most nsolutions to
such an equation, we have that an n×nmatrix Ahas at
most ndistinct eigenvalues. Mathematicians have
proved that associated with each possible eigenvalue
there is at least one corresponding eigenvector. More-
over, it has been established that eigenvectors associated
with distinct eigenvalues are linearly independent.
The study of eigenvectors and eigenvalues greatly
simplifies matrix manipulations. Suppose, for example,
a square 3 ×3 matrix Ahas three distinct eigenvalues
λ1, λ2, and λ3. Set Dto be the diagonal matrix
135
270
4 140
8 280
16 560
4 18 468
•• •
++
315
••
+