AGE of the total measurement, then the error is called a
percentage error. For example, when 1.2 is used as an
approximation for the quantity 1.16, then the percentage
error is
× 100 ≈ 3.4%. If, in measuring the height
of a building as 62 feet, engineers use a device that measures to the nearest half a foot, then the height of the
building should be written: 62±0.5 feet. The percentage
error of this measurement is:
× 100 ≈ 0.81 percent.
See also RELATIVE ERROR.
percentile (centile, quartile) One of the 99 values
that divide a set of data arranged in numerical order
into 100 equal parts is called a percentile. For example,
the 90th percentile is the value such that 90 percent of
the data points are below that value. Often scores in
standardized tests are presented in terms of percentiles.
For example, if 525 students take an exam and 95 percent of the students receive a score of 73 or lower, then
the 95th percentile for that exam is 74.
It is often deemed convenient to divide data sets
into four equal parts. The lower (or first) quartile,
denoted Q 1 , is the 25th percentile. The middle (or second) quartile, Q 2 , the median, is the 50th percentile,
and upper (or third) quartile, Q 3 , is the 75th percentile.
perfect number A whole number that is equal to the
sum of its FACTORs—excluding the number itself—is
called a perfect number. For example, the numbers 6,
28, 496, and 8128 are each perfect.
6 = 1 + 2 + 3 = 2
1
(2
2 –1)
28 = 1 + 2 + 4 + 7 + 14 = 2
2 (2
3 –1)
496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
= 2
4
(2
5 –1)
8,128 = 1 + 2 + 4 + 8 + 32 + 64 + 127 + 254 + 508
+ 1016 + 2032 + 4064 = 2
6
(2
7 –1)
Perfect numbers were deemed to have important
numerological properties by ancient scholars, and were
extensively studied by the Greeks. In book IX of THE
ELEMENTS, written around 300 B.C.E., EUCLID proved
that if 2
n –1 is PRIME, then 2
n–1
(2
n – 1) is a perfect
number. At the turn of the 17th century, French
philosopher and mathematician MARIN MERSENNE
undertook a study to determine which numbers of the
form 2
n – 1 are prime. Any such number is today called
a MERSENNE PRIME.
Around 100 B.C.E. Syrian scholar Nicomachos of
Gerasa suggested that the converse of Euclid’s result is
true, that is, that every perfect number must be of the
form 2
n–1 (2
n – 1) with 2
n – 1 prime. (In particular, then,
there can be no odd perfect numbers.) In 1750 LEONHARD EULER proved that indeed every even perfect
number must be of this form, partially proving Nicomachos’s claim. Surprisingly, still to this day no one
knows whether or not examples of odd perfect numbers exist. This remains a famous unsolved problem. It
has been proved that if any odd perfect number exists,
it must be larger than a GOOGOL.
Euler’s result shows that every even perfect number
is a triangular FIGURATE NUMBER. Explicitly, 2 n–1 (2 n –1)
=
is the formula for the (2
n – 1)-th triangular
number.
It is also known that every even perfect number
must end with either a 6 or an 8 (this was conjectured by
Nicomachos) and that every even perfect number greater
than 6 is a sum of consecutive odd cube numbers:
28 = 1
3 + 3
3
496 = 1
3 + 3
3 + 5
3 + 7
3
8128 = 1
3 + 3
3 +…+ 15
3
for example, and that every even perfect is one more
than a multiple of 9.
The sum of the inverses of all the factors of a perfect
number is always 2. For example,
+
+
+
= 2
and
+
+
+
+
= 2. (This follows from the
fact that the sum of all the factors of a perfect number
N equals 2N. Dividing the sum through by N produces
the result.)
It is not known if there are infinitely many perfect
numbers. Fewer than 50 perfect numbers are known.
A number that is not perfect is classified as either
abundant or deficient according to whether the sum of
all its factors—excluding the whole number itself—is
greater than or less than the number. For example, 18
is abundant, since the sum of its factors is more than
18 (we have 1 + 2 + 3 + 6 + 9 = 21 > 18), whereas 25 is
deficient because its factors sum to less than 25. (We
have 1 + 5 = 6 < 25). All prime numbers are deficient.
See also AMICABLE NUMBERS; GREEK MATHEMATICS.
1
–
28
1
–
7
1
–
4
1
–
2
1
–
1
1
–
6
1
–
3
1
–
2
1
–
1
2
n
(2
n – 1)
———–
2
0.5
––
62
0.04
––
1.16
perfect number 389
percentage error. For example, when 1.2 is used as an
approximation for the quantity 1.16, then the percentage
error is
× 100 ≈ 3.4%. If, in measuring the height
of a building as 62 feet, engineers use a device that measures to the nearest half a foot, then the height of the
building should be written: 62±0.5 feet. The percentage
error of this measurement is:
× 100 ≈ 0.81 percent.
See also RELATIVE ERROR.
percentile (centile, quartile) One of the 99 values
that divide a set of data arranged in numerical order
into 100 equal parts is called a percentile. For example,
the 90th percentile is the value such that 90 percent of
the data points are below that value. Often scores in
standardized tests are presented in terms of percentiles.
For example, if 525 students take an exam and 95 percent of the students receive a score of 73 or lower, then
the 95th percentile for that exam is 74.
It is often deemed convenient to divide data sets
into four equal parts. The lower (or first) quartile,
denoted Q 1 , is the 25th percentile. The middle (or second) quartile, Q 2 , the median, is the 50th percentile,
and upper (or third) quartile, Q 3 , is the 75th percentile.
perfect number A whole number that is equal to the
sum of its FACTORs—excluding the number itself—is
called a perfect number. For example, the numbers 6,
28, 496, and 8128 are each perfect.
6 = 1 + 2 + 3 = 2
1
(2
2 –1)
28 = 1 + 2 + 4 + 7 + 14 = 2
2 (2
3 –1)
496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
= 2
4
(2
5 –1)
8,128 = 1 + 2 + 4 + 8 + 32 + 64 + 127 + 254 + 508
+ 1016 + 2032 + 4064 = 2
6
(2
7 –1)
Perfect numbers were deemed to have important
numerological properties by ancient scholars, and were
extensively studied by the Greeks. In book IX of THE
ELEMENTS, written around 300 B.C.E., EUCLID proved
that if 2
n –1 is PRIME, then 2
n–1
(2
n – 1) is a perfect
number. At the turn of the 17th century, French
philosopher and mathematician MARIN MERSENNE
undertook a study to determine which numbers of the
form 2
n – 1 are prime. Any such number is today called
a MERSENNE PRIME.
Around 100 B.C.E. Syrian scholar Nicomachos of
Gerasa suggested that the converse of Euclid’s result is
true, that is, that every perfect number must be of the
form 2
n–1 (2
n – 1) with 2
n – 1 prime. (In particular, then,
there can be no odd perfect numbers.) In 1750 LEONHARD EULER proved that indeed every even perfect
number must be of this form, partially proving Nicomachos’s claim. Surprisingly, still to this day no one
knows whether or not examples of odd perfect numbers exist. This remains a famous unsolved problem. It
has been proved that if any odd perfect number exists,
it must be larger than a GOOGOL.
Euler’s result shows that every even perfect number
is a triangular FIGURATE NUMBER. Explicitly, 2 n–1 (2 n –1)
=
is the formula for the (2
n – 1)-th triangular
number.
It is also known that every even perfect number
must end with either a 6 or an 8 (this was conjectured by
Nicomachos) and that every even perfect number greater
than 6 is a sum of consecutive odd cube numbers:
28 = 1
3 + 3
3
496 = 1
3 + 3
3 + 5
3 + 7
3
8128 = 1
3 + 3
3 +…+ 15
3
for example, and that every even perfect is one more
than a multiple of 9.
The sum of the inverses of all the factors of a perfect
number is always 2. For example,
+
+
+
= 2
and
+
+
+
+
= 2. (This follows from the
fact that the sum of all the factors of a perfect number
N equals 2N. Dividing the sum through by N produces
the result.)
It is not known if there are infinitely many perfect
numbers. Fewer than 50 perfect numbers are known.
A number that is not perfect is classified as either
abundant or deficient according to whether the sum of
all its factors—excluding the whole number itself—is
greater than or less than the number. For example, 18
is abundant, since the sum of its factors is more than
18 (we have 1 + 2 + 3 + 6 + 9 = 21 > 18), whereas 25 is
deficient because its factors sum to less than 25. (We
have 1 + 5 = 6 < 25). All prime numbers are deficient.
See also AMICABLE NUMBERS; GREEK MATHEMATICS.
1
–
28
1
–
7
1
–
4
1
–
2
1
–
1
1
–
6
1
–
3
1
–
2
1
–
1
2
n
(2
n – 1)
———–
2
0.5
––
62
0.04
––
1.16
perfect number 389