√a+ b

√a+ b

Cramer, Gabriel 107

counterexample An example that demonstrates that

a claim made is not true or, at the very least, not always

true is called a counterexample. For instance, although

the assertion = √

–

a+ √

–

bhappens to be true for

a= 1 and b= 0, this identity is false in general, as the

counterexample a= 9 and b= 16 demonstrates. We can

thus say that the claim = √

–

a+ √

–

bis an invalid

statement.

In the mid-1800s French mathematician Alphonse

de Polignac (1817–90) made the assertion:

Every odd number can be written as the sum of

a power of two and a

PRIME

number.

For instance, the number 17 can be written as 17 = 22+

13, 37 as 37 = 23+ 29, and 1,065 as 1,065 = 210 + 41.

De Polignac claimed to have checked his assertion for all

odd numbers up to 3 million, and consequently was con-

vinced that his claim was true. Unfortunately it is not,

for the number 127 provides a counterexample to this

assertion, as none of the following differences are prime:

127 – 2 = 125 = 5 ×25

127 – 4 = 123 = 3 ×41

127 – 8 = 119 = 7 ×17

127 – 16 = 111 = 3 ×37

127 – 32 = 95 = 5 ×19

127 – 64 = 63 = 3 ×21

(We need not go further, since the next power of two,

128, is larger than 127.) De Polignac overlooked this

simple counterexample.

No counterexamples have yet been found to the

famous G

OLDBACH

’

S CONJECTURE

and C

OLLATZ

’

S

CONJECTURE

.

covariance If a scientific study records numerical

information about two features of the individuals or

events under examination (such as the height and shoe

size of participating adults, or seasonal rainfall and

crop yield from year to year), then the

DATA

obtained

from the study is appropriately recorded as pairs of

values. If a study has Nparticipants, with data pairs

(x1, y1),…,(xN, yN), then the covariance of the sample is

the quantity:

where –

xis the

MEAN

x-value and –

ythe mean y-value.

An exercise in algebra shows that this formula can also

be written:

The covariance is used to calculate

CORRELATION

COEFFICIENT

s and

REGRESSION

lines such as for the

LEAST SQUARES METHOD

, for instance. Pearson’s corre-

lation coefficient shows that two variables with covari-

ance zero are independent, that is, the value of one

variable has no effect on the value of the other.

See also

STATISTICS

:

DESCRIPTIVE

.

Cramer, Gabriel (1704–1752) Swiss Algebra, Geom-

etry, Probability theory Born on July 31, 1704, in

Geneva, Switzerland, scholar Gabriel Cramer is remem-

bered for his 1750 text Introduction à l’analyse des

lignes courbes algébriques (Introduction to the analysis

of algebraic curves), in which he classifies certain types

of algebraic curves and presents an efficient method,

today called C

RAMER

’

S RULE

, for solving systems of

linear equations. Cramer never claimed to have dis-

covered the rule. It was, in fact, established decades

earlier by Scottish mathematician C

OLIN

M

ACLAURIN

(1698–1746).

Cramer earned a doctorate degree at the young age

of 18 after completing a thesis on the theory of sound,

and two years later he was awarded a joint position as

chair of mathematics at the Académie de Clavin in

Geneva. After sharing the position for 10 years with

young mathematician Giovanni Ludovico Calandrini

(senior to him by just one year), Cramer was eventually

awarded the full chairmanship.

The full position gave Cramer much opportunity to

travel and collaborate with other mathematicians

across Europe, such as L

EONHARD

E

ULER

, Johann

Bernoulli and Daniel Bernoulli of the famous

B

ERNOULLI FAMILY

, and A

BRAHAM

D

E

M

OIVRE

.

Cramer wrote many articles in mathematics covering

topics as diverse as geometry and algebra, the history

of mathematics, and a mathematical analysis of the

dates on which Easter falls.

His 1750 text Introduction à l’analyse des lignes

courbes algébriques was Cramer’s most famous work.

Beginning chapter one with a discussion on the types of