IRRATIONAL
AND OTHER CRAZY NUMBERS
Graeme Cohen
g.cohen@bigpond.net.au
Klein Conference
Castro Urdiales
2010
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
PART 1 — CRAZY NUMBERS?
I am interested in the names that have
been given to sets of numbers as they
were discovered.
Originally, there were just numbers:
1, 2, 3, . . .
There was no need to call these whole
numbers or integers since they were the
only numbers.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
PART 1 — CRAZY NUMBERS?
I am interested in the names that have
been given to sets of numbers as they
were discovered.
Originally, there were just numbers:
1, 2, 3, . . .
There was no need to call these whole
numbers or integers since they were the
only numbers.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
PART 1 — CRAZY NUMBERS?
I am interested in the names that have
been given to sets of numbers as they
were discovered.
Originally, there were just numbers:
1, 2, 3, . . .
There was no need to call these whole
numbers or integers since they were the
only numbers.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
PART 1 — CRAZY NUMBERS?
I am interested in the names that have
been given to sets of numbers as they
were discovered.
Originally, there were just numbers:
1, 2, 3, . . .
There was no need to call these whole
numbers or integers since they were the
only numbers.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
You could add them and multiply them
to give other numbers.
Sometimes, you could subtract them to
give other numbers.
17 − 3 = 14,
But what about
178 − 86 = 92
12 − 19 = ???
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
You could add them and multiply them
to give other numbers.
Sometimes, you could subtract them to
give other numbers.
17 − 3 = 14,
But what about
178 − 86 = 92
12 − 19 = ???
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
Equations like x + 6 = 0 were said to have
absurd or false solutions (Diophantus, 3rd
century). They were also called fictitious or
defective.
People settled on new names for two types of
numbers.
Positive numbers (assertive, strong, dogmatic)
Negative numbers (denying, contradictory)
These are very suggestive words for a very
ordinary idea, today.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
Equations like x + 6 = 0 were said to have
absurd or false solutions (Diophantus, 3rd
century). They were also called fictitious or
defective.
People settled on new names for two types of
numbers.
Positive numbers (assertive, strong, dogmatic)
Negative numbers (denying, contradictory)
These are very suggestive words for a very
ordinary idea, today.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
Equations like x + 6 = 0 were said to have
absurd or false solutions (Diophantus, 3rd
century). They were also called fictitious or
defective.
People settled on new names for two types of
numbers.
Positive numbers (assertive, strong, dogmatic)
Negative numbers (denying, contradictory)
These are very suggestive words for a very
ordinary idea, today.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
Equations like x + 6 = 0 were said to have
absurd or false solutions (Diophantus, 3rd
century). They were also called fictitious or
defective.
People settled on new names for two types of
numbers.
Positive numbers (assertive, strong, dogmatic)
Negative numbers (denying, contradictory)
These are very suggestive words for a very
ordinary idea, today.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
The language doesn’t matter much.
ENGLISH
ESPAÑOL
FRANÇAIS
DEUTSCH
ITALIANO
positive
positivo
positif
positiv
positivo
Graeme Cohen g.cohen@bigpond.net.au
negative
negativo
négatif
negativ
negativo
IRRATIONAL AND OTHER CRAZY NUMBERS
The language doesn’t matter much.
ENGLISH
ESPAÑOL
FRANÇAIS
DEUTSCH
ITALIANO
positive
positivo
positif
positiv
positivo
Graeme Cohen g.cohen@bigpond.net.au
negative
negativo
négatif
negativ
negativo
IRRATIONAL AND OTHER CRAZY NUMBERS
“Positive” and “negative”, for numbers,
have only been used since the 15th
century.
From John Ward, The Young
Mathematician’s Guide (1719):
“The Number out of which Subtraction is
required to be made, must be greater
than (or at least equal to) the Number to
be Subtracted.”
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
“Positive” and “negative”, for numbers,
have only been used since the 15th
century.
From John Ward, The Young
Mathematician’s Guide (1719):
“The Number out of which Subtraction is
required to be made, must be greater
than (or at least equal to) the Number to
be Subtracted.”
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
Division of numbers was also a problem
(in fact, a much earlier problem).
15
= 5,
3
But what about
323
= 17
19
23
= ???
7
Rational numbers, derived from Latin
“ratio” meaning “calculating”, were born.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
Division of numbers was also a problem
(in fact, a much earlier problem).
15
= 5,
3
But what about
323
= 17
19
23
= ???
7
Rational numbers, derived from Latin
“ratio” meaning “calculating”, were born.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
Division of numbers was also a problem
(in fact, a much earlier problem).
15
= 5,
3
But what about
323
= 17
19
23
= ???
7
Rational numbers, derived from Latin
“ratio” meaning “calculating”, were born.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
Not all numbers
√ are rational. Hippasus (5th century BC) could
not represent 2 as a fraction. Pythagoras could not accept
this so he sentenced Hippasus to death by drowning. (This is
probably not true.)
We now have (since the 14th century) another
distinction into two types of numbers.
Rational numbers (logical, clear-headed,
normal, reasonable)
Irrational numbers (illogical, goofy, stupid)
Which came first, the names of the numbers or
these very suggestive synonyms (based on
“ratio” also meaning “reasoning” or “thinking”)?
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
Not all numbers
√ are rational. Hippasus (5th century BC) could
not represent 2 as a fraction. Pythagoras could not accept
this so he sentenced Hippasus to death by drowning. (This is
probably not true.)
We now have (since the 14th century) another
distinction into two types of numbers.
Rational numbers (logical, clear-headed,
normal, reasonable)
Irrational numbers (illogical, goofy, stupid)
Which came first, the names of the numbers or
these very suggestive synonyms (based on
“ratio” also meaning “reasoning” or “thinking”)?
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
Not all numbers
√ are rational. Hippasus (5th century BC) could
not represent 2 as a fraction. Pythagoras could not accept
this so he sentenced Hippasus to death by drowning. (This is
probably not true.)
We now have (since the 14th century) another
distinction into two types of numbers.
Rational numbers (logical, clear-headed,
normal, reasonable)
Irrational numbers (illogical, goofy, stupid)
Which came first, the names of the numbers or
these very suggestive synonyms (based on
“ratio” also meaning “reasoning” or “thinking”)?
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
Not all numbers
√ are rational. Hippasus (5th century BC) could
not represent 2 as a fraction. Pythagoras could not accept
this so he sentenced Hippasus to death by drowning. (This is
probably not true.)
We now have (since the 14th century) another
distinction into two types of numbers.
Rational numbers (logical, clear-headed,
normal, reasonable)
Irrational numbers (illogical, goofy, stupid)
Which came first, the names of the numbers or
these very suggestive synonyms (based on
“ratio” also meaning “reasoning” or “thinking”)?
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
Not all numbers
√ are rational. Hippasus (5th century BC) could
not represent 2 as a fraction. Pythagoras could not accept
this so he sentenced Hippasus to death by drowning. (This is
probably not true.)
We now have (since the 14th century) another
distinction into two types of numbers.
Rational numbers (logical, clear-headed,
normal, reasonable)
Irrational numbers (illogical, goofy, stupid)
Which came first, the names of the numbers or
these very suggestive synonyms (based on
“ratio” also meaning “reasoning” or “thinking”)?
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
Not all numbers
√ are rational. Hippasus (5th century BC) could
not represent 2 as a fraction. Pythagoras could not accept
this so he sentenced Hippasus to death by drowning. (This is
probably not true.)
We now have (since the 14th century) another
distinction into two types of numbers.
Rational numbers (logical, clear-headed,
normal, reasonable)
Irrational numbers (illogical, goofy, stupid)
Which came first, the names of the numbers or
these very suggestive synonyms (based on
“ratio” also meaning “reasoning” or “thinking”)?
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
√ √
In
English,
irrational
numbers
like
3, 3 5,
√
√
3 + 2 3 5 (but not π or e) are called surds. I
thought this might be related to absurd (foolish,
illogical, impossible).
In fact “surd” is from Latin “surdus” meaning
“deaf” or “indistinct”.
Do other languages have surds?
Ward’s book calls expressions like
“surds”.
Graeme Cohen g.cohen@bigpond.net.au
√
aa + bb
IRRATIONAL AND OTHER CRAZY NUMBERS
√ √
In
English,
irrational
numbers
like
3, 3 5,
√
√
3 + 2 3 5 (but not π or e) are called surds. I
thought this might be related to absurd (foolish,
illogical, impossible).
In fact “surd” is from Latin “surdus” meaning
“deaf” or “indistinct”.
Do other languages have surds?
Ward’s book calls expressions like
“surds”.
Graeme Cohen g.cohen@bigpond.net.au
√
aa + bb
IRRATIONAL AND OTHER CRAZY NUMBERS
√ √
In
English,
irrational
numbers
like
3, 3 5,
√
√
3 + 2 3 5 (but not π or e) are called surds. I
thought this might be related to absurd (foolish,
illogical, impossible).
In fact “surd” is from Latin “surdus” meaning
“deaf” or “indistinct”.
Do other languages have surds?
Ward’s book calls expressions like
“surds”.
Graeme Cohen g.cohen@bigpond.net.au
√
aa + bb
IRRATIONAL AND OTHER CRAZY NUMBERS
√ √
In
English,
irrational
numbers
like
3, 3 5,
√
√
3 + 2 3 5 (but not π or e) are called surds. I
thought this might be related to absurd (foolish,
illogical, impossible).
In fact “surd” is from Latin “surdus” meaning
“deaf” or “indistinct”.
Do other languages have surds?
Ward’s book calls expressions like
“surds”.
Graeme Cohen g.cohen@bigpond.net.au
√
aa + bb
IRRATIONAL AND OTHER CRAZY NUMBERS
Then it was realized that it was useful for
x 2 + 1 = 0 to have two solutions.
But these “exist only in the mind” — they are
imaginary.
100 years ago they were also called impossible
numbers.
In contrast, now positive and negative numbers,
and rational and irrational numbers, are all real
(actual, concrete, tangible).
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
Then it was realized that it was useful for
x 2 + 1 = 0 to have two solutions.
But these “exist only in the mind” — they are
imaginary.
100 years ago they were also called impossible
numbers.
In contrast, now positive and negative numbers,
and rational and irrational numbers, are all real
(actual, concrete, tangible).
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
Then it was realized that it was useful for
x 2 + 1 = 0 to have two solutions.
But these “exist only in the mind” — they are
imaginary.
100 years ago they were also called impossible
numbers.
In contrast, now positive and negative numbers,
and rational and irrational numbers, are all real
(actual, concrete, tangible).
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
Then it was realized that it was useful for
x 2 + 1 = 0 to have two solutions.
But these “exist only in the mind” — they are
imaginary.
100 years ago they were also called impossible
numbers.
In contrast, now positive and negative numbers,
and rational and irrational numbers, are all real
(actual, concrete, tangible).
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
Again, it doesn’t matter what language
we are speaking here.
ENGLISH
ESPAÑOL
FRANÇAIS
DEUTSCH
ITALIANO
real
real
réel
reell
reale
Graeme Cohen g.cohen@bigpond.net.au
imaginary
imaginario
imaginaire
imaginär
immaginario
IRRATIONAL AND OTHER CRAZY NUMBERS
Again, it doesn’t matter what language
we are speaking here.
ENGLISH
ESPAÑOL
FRANÇAIS
DEUTSCH
ITALIANO
real
real
réel
reell
reale
Graeme Cohen g.cohen@bigpond.net.au
imaginary
imaginario
imaginaire
imaginär
immaginario
IRRATIONAL AND OTHER CRAZY NUMBERS
Finally, we have algebraic and
transcendental numbers.
This is a very suggestive word.
transcendental = fantastic, spiritual
What is wrong with “non-algebraic”?
Literally, transcendental numbers “climb
beyond” algebraic numbers.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
Finally, we have algebraic and
transcendental numbers.
This is a very suggestive word.
transcendental = fantastic, spiritual
What is wrong with “non-algebraic”?
Literally, transcendental numbers “climb
beyond” algebraic numbers.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
Finally, we have algebraic and
transcendental numbers.
This is a very suggestive word.
transcendental = fantastic, spiritual
What is wrong with “non-algebraic”?
Literally, transcendental numbers “climb
beyond” algebraic numbers.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
Finally, we have algebraic and
transcendental numbers.
This is a very suggestive word.
transcendental = fantastic, spiritual
What is wrong with “non-algebraic”?
Literally, transcendental numbers “climb
beyond” algebraic numbers.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
Finally, we have algebraic and
transcendental numbers.
This is a very suggestive word.
transcendental = fantastic, spiritual
What is wrong with “non-algebraic”?
Literally, transcendental numbers “climb
beyond” algebraic numbers.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
Reference for the above:
Steven Schwartzman, The Words of Mathematics: An
Etymological Dictionary of Mathematical Terms Used in
English, Mathematical Association of America,
Washington (1994)
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
PART 2 — IRRATIONAL NUMBERS
Three items that can be discussed with
students.
√
Proofs that 2 is irrational.
Irrational powers of irrational
numbers.
Decimal expansions of irrational
numbers.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
1 — PROOFS THAT
√
2 IS IRRATIONAL
In these, suppose there are integers√a, b with no common
factors
√ (except 1 and −1) such that 2 = a/b. Since
1 < 2 < 2, we can assume a > b > 1.
First proof — So 2b2 = a2 . Then a2 is even, so a is even.
Put a = 2c so 2b2 = 4c 2 , or 2c 2 = b2 . Then b2 is even, so
b is even. But a and b cannot both be even, since 2
cannot be a common factor. This contradiction proves the
theorem.
Second proof — Just look at 2b2 = a2 . There must be an
odd number of 2’s dividing the left-hand side and an even
number of 2’s (possibly none) dividing the right-hand side.
Impossible.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
1 — PROOFS THAT
√
2 IS IRRATIONAL
In these, suppose there are integers√a, b with no common
factors
√ (except 1 and −1) such that 2 = a/b. Since
1 < 2 < 2, we can assume a > b > 1.
First proof — So 2b2 = a2 . Then a2 is even, so a is even.
Put a = 2c so 2b2 = 4c 2 , or 2c 2 = b2 . Then b2 is even, so
b is even. But a and b cannot both be even, since 2
cannot be a common factor. This contradiction proves the
theorem.
Second proof — Just look at 2b2 = a2 . There must be an
odd number of 2’s dividing the left-hand side and an even
number of 2’s (possibly none) dividing the right-hand side.
Impossible.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
1 — PROOFS THAT
√
2 IS IRRATIONAL
In these, suppose there are integers√a, b with no common
factors
√ (except 1 and −1) such that 2 = a/b. Since
1 < 2 < 2, we can assume a > b > 1.
First proof — So 2b2 = a2 . Then a2 is even, so a is even.
Put a = 2c so 2b2 = 4c 2 , or 2c 2 = b2 . Then b2 is even, so
b is even. But a and b cannot both be even, since 2
cannot be a common factor. This contradiction proves the
theorem.
Second proof — Just look at 2b2 = a2 . There must be an
odd number of 2’s dividing the left-hand side and an even
number of 2’s (possibly none) dividing the right-hand side.
Impossible.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
The first proof is most complicated, but most common in
schools. It is not so easy to generalize. This is an
example of Felix Klein’s “double discontinuity” — the first
proof is learnt at school, teachers learn better proofs at
university, then teach the not-so-good proof themselves at
school.
The second proof looks easy but actually relies on the
fundamental theorem of arithmetic.
a2
Third proof — Just look at 2b = . The left-hand side is
b
an integer, so the right-hand side must be. Impossible,
since a and b have no common factors and b > 1.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
The first proof is most complicated, but most common in
schools. It is not so easy to generalize. This is an
example of Felix Klein’s “double discontinuity” — the first
proof is learnt at school, teachers learn better proofs at
university, then teach the not-so-good proof themselves at
school.
The second proof looks easy but actually relies on the
fundamental theorem of arithmetic.
a2
Third proof — Just look at 2b = . The left-hand side is
b
an integer, so the right-hand side must be. Impossible,
since a and b have no common factors and b > 1.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
The first proof is most complicated, but most common in
schools. It is not so easy to generalize. This is an
example of Felix Klein’s “double discontinuity” — the first
proof is learnt at school, teachers learn better proofs at
university, then teach the not-so-good proof themselves at
school.
The second proof looks easy but actually relies on the
fundamental theorem of arithmetic.
a2
Third proof — Just look at 2b = . The left-hand side is
b
an integer, so the right-hand side must be. Impossible,
since a and b have no common factors and b > 1.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
The third proof
is easy and easy to generalize to
√
m
prove that N is irrational (where N is a positive
integer) unless N = nm for some integer n:
√
m
Suppose N = a/b, where a and b are
integers with no common factors (except 1 and
−1). Assume a > b > 0. Then
Nb
m−1
am
=
.
b
The left-hand side is an integer, so the
right-hand side must be. Impossible, since a
and b have no common factors, unless b = 1.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
The third proof
is easy and easy to generalize to
√
m
prove that N is irrational (where N is a positive
integer) unless N = nm for some integer n:
√
m
Suppose N = a/b, where a and b are
integers with no common factors (except 1 and
−1). Assume a > b > 0. Then
Nb
m−1
am
=
.
b
The left-hand side is an integer, so the
right-hand side must be. Impossible, since a
and b have no common factors, unless b = 1.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
2 — A CUTE EXAMPLE OF A NON-CONSTRUCTIVE
PROOF
Question: Is it possible for a number ab to be rational
when both a and b are irrational?
Answer: Yes.
√ √2
Is c = 2 rational? If it is, we have our example. If it is
not, that is, if c is irrational, then
√
c
2
√ √2 √2 √ 2
=( 2 ) = 2 =2
is an example.
This is a non-constructive proof. The proof is correct, but
it leaves open whether or not c is rational.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
2 — A CUTE EXAMPLE OF A NON-CONSTRUCTIVE
PROOF
Question: Is it possible for a number ab to be rational
when both a and b are irrational?
Answer: Yes.
√ √2
Is c = 2 rational? If it is, we have our example. If it is
not, that is, if c is irrational, then
√
c
2
√ √2 √2 √ 2
=( 2 ) = 2 =2
is an example.
This is a non-constructive proof. The proof is correct, but
it leaves open whether or not c is rational.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
2 — A CUTE EXAMPLE OF A NON-CONSTRUCTIVE
PROOF
Question: Is it possible for a number ab to be rational
when both a and b are irrational?
Answer: Yes.
√ √2
Is c = 2 rational? If it is, we have our example. If it is
not, that is, if c is irrational, then
√
c
2
√ √2 √2 √ 2
=( 2 ) = 2 =2
is an example.
This is a non-constructive proof. The proof is correct, but
it leaves open whether or not c is rational.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
2 — A CUTE EXAMPLE OF A NON-CONSTRUCTIVE
PROOF
Question: Is it possible for a number ab to be rational
when both a and b are irrational?
Answer: Yes.
√ √2
Is c = 2 rational? If it is, we have our example. If it is
not, that is, if c is irrational, then
√
c
2
√ √2 √2 √ 2
=( 2 ) = 2 =2
is an example.
This is a non-constructive proof. The proof is correct, but
it leaves open whether or not c is rational.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
A much more general question was asked as Hilbert’s seventh
problem in 1900. It was not answered until 1934. Alexander
Gelfond and (a year later) Theodor Schneider gave
independent proofs of the Gelfond-Schneider Theorem:
If α and β are algebraic numbers (with α 6= 0 or 1), and if β is
not a real rational number, then any value of αβ is a
transcendental number.
√ √2 √
So 2 , 2 2 , eπ , for example, are all irrational. In fact they are
transcendental, so this exercise is a good starting point for a
discussion of algebraic and transcendental numbers.
√
Is there an “easy” proof that 2
Graeme Cohen g.cohen@bigpond.net.au
2
is irrational?
IRRATIONAL AND OTHER CRAZY NUMBERS
A much more general question was asked as Hilbert’s seventh
problem in 1900. It was not answered until 1934. Alexander
Gelfond and (a year later) Theodor Schneider gave
independent proofs of the Gelfond-Schneider Theorem:
If α and β are algebraic numbers (with α 6= 0 or 1), and if β is
not a real rational number, then any value of αβ is a
transcendental number.
√ √2 √
So 2 , 2 2 , eπ , for example, are all irrational. In fact they are
transcendental, so this exercise is a good starting point for a
discussion of algebraic and transcendental numbers.
√
Is there an “easy” proof that 2
Graeme Cohen g.cohen@bigpond.net.au
2
is irrational?
IRRATIONAL AND OTHER CRAZY NUMBERS
A much more general question was asked as Hilbert’s seventh
problem in 1900. It was not answered until 1934. Alexander
Gelfond and (a year later) Theodor Schneider gave
independent proofs of the Gelfond-Schneider Theorem:
If α and β are algebraic numbers (with α 6= 0 or 1), and if β is
not a real rational number, then any value of αβ is a
transcendental number.
√ √2 √
So 2 , 2 2 , eπ , for example, are all irrational. In fact they are
transcendental, so this exercise is a good starting point for a
discussion of algebraic and transcendental numbers.
√
Is there an “easy” proof that 2
Graeme Cohen g.cohen@bigpond.net.au
2
is irrational?
IRRATIONAL AND OTHER CRAZY NUMBERS
3 — ON DECIMAL EXPANSIONS
It is well known that the decimal expansion of a rational
number is either terminating, like
1
= 0.125
8
or periodic, like
1
= 0.090909 . . . = 0.0̇9̇
11
or
1
= 0.1249999 . . . = 0.1249̇.
8
It is less well known that:
the decimal expansion of a number may not be unique.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
3 — ON DECIMAL EXPANSIONS
It is well known that the decimal expansion of a rational
number is either terminating, like
1
= 0.125
8
or periodic, like
1
= 0.090909 . . . = 0.0̇9̇
11
or
1
= 0.1249999 . . . = 0.1249̇.
8
It is less well known that:
the decimal expansion of a number may not be unique.
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
Also, decimal expansions of irrational numbers are not
characterized by being (apparently) random, as in
√
2 = 1.414213562373095 . . .
π = 3.141592653589793 . . .
For example, the number
0.1010010001000010 . . . ,
with the number of 0’s between the 1’s increasing by one
each time, has a decimal expansion which is neither
terminating nor periodic. So this number, with a pleasing
pattern in its decimal representation, is irrational.
(The nth decimal place is 1 if n is a triangular number, 0
otherwise.)
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
Also, decimal expansions of irrational numbers are not
characterized by being (apparently) random, as in
√
2 = 1.414213562373095 . . .
π = 3.141592653589793 . . .
For example, the number
0.1010010001000010 . . . ,
with the number of 0’s between the 1’s increasing by one
each time, has a decimal expansion which is neither
terminating nor periodic. So this number, with a pleasing
pattern in its decimal representation, is irrational.
(The nth decimal place is 1 if n is a triangular number, 0
otherwise.)
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
Also, decimal expansions of irrational numbers are not
characterized by being (apparently) random, as in
√
2 = 1.414213562373095 . . .
π = 3.141592653589793 . . .
For example, the number
0.1010010001000010 . . . ,
with the number of 0’s between the 1’s increasing by one
each time, has a decimal expansion which is neither
terminating nor periodic. So this number, with a pleasing
pattern in its decimal representation, is irrational.
(The nth decimal place is 1 if n is a triangular number, 0
otherwise.)
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
Also, decimal expansions of irrational numbers are not
characterized by being (apparently) random, as in
√
2 = 1.414213562373095 . . .
π = 3.141592653589793 . . .
For example, the number
0.1010010001000010 . . . ,
with the number of 0’s between the 1’s increasing by one
each time, has a decimal expansion which is neither
terminating nor periodic. So this number, with a pleasing
pattern in its decimal representation, is irrational.
(The nth decimal place is 1 if n is a triangular number, 0
otherwise.)
Graeme Cohen g.cohen@bigpond.net.au
IRRATIONAL AND OTHER CRAZY NUMBERS
This talk was about straightforward ideas which would be
beyond most senior school syllabuses, but which readily
generate discussion with students.
THANK YOU
ENGLISH
ESPAÑOL
FRANÇAIS
DEUTSCH
ITALIANO
Graeme Cohen g.cohen@bigpond.net.au
Thank you
Gracias
Merci
Danke
Grazie
IRRATIONAL AND OTHER CRAZY NUMBERS
This talk was about straightforward ideas which would be
beyond most senior school syllabuses, but which readily
generate discussion with students.
THANK YOU
ENGLISH
ESPAÑOL
FRANÇAIS
DEUTSCH
ITALIANO
Graeme Cohen g.cohen@bigpond.net.au
Thank you
Gracias
Merci
Danke
Grazie
IRRATIONAL AND OTHER CRAZY NUMBERS
This talk was about straightforward ideas which would be
beyond most senior school syllabuses, but which readily
generate discussion with students.
THANK YOU
ENGLISH
ESPAÑOL
FRANÇAIS
DEUTSCH
ITALIANO
Graeme Cohen g.cohen@bigpond.net.au
Thank you
Gracias
Merci
Danke
Grazie
IRRATIONAL AND OTHER CRAZY NUMBERS