Irrationality of 0.123456789101112 … and similar numbers The 2019 Stack Overflow Developer Survey Results Are InRainbow numbers: Can mapping digits to different bases produce different varieties of irrationality?Non-existence of irrational numbers?For each irrational number $b$, does there exist an irrational number $a$ such that $a^b$ is rational?Do the second-last-digits of the primes $ge 11$ form a transcendental number?Swapping the digits of an algebraic number (e.g. $sqrt 2$)Irrationality of a real number $b$ that is related to a real number $a$ by the umbral Taylor series.Can a change of basis modify irrationality/transcendence?Is the real number whose $n^rm th$ digit after the decimal point in decimal representation is the leading digit of $2^n$ a rational number?Are there transcendental numbers that cannot be reached?Defining number set sequences based on the fundamental theorem of arithmeticReal irrational algebraic numbers “never repeat”
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Irrationality of 0.123456789101112 … and similar numbers
The 2019 Stack Overflow Developer Survey Results Are InRainbow numbers: Can mapping digits to different bases produce different varieties of irrationality?Non-existence of irrational numbers?For each irrational number $b$, does there exist an irrational number $a$ such that $a^b$ is rational?Do the second-last-digits of the primes $ge 11$ form a transcendental number?Swapping the digits of an algebraic number (e.g. $sqrt 2$)Irrationality of a real number $b$ that is related to a real number $a$ by the umbral Taylor series.Can a change of basis modify irrationality/transcendence?Is the real number whose $n^rm th$ digit after the decimal point in decimal representation is the leading digit of $2^n$ a rational number?Are there transcendental numbers that cannot be reached?Defining number set sequences based on the fundamental theorem of arithmeticReal irrational algebraic numbers “never repeat”
$begingroup$
Consider four numbers in $(0,1)$:
$n_1$ in base $10$ is formed by listing the decimal digits $1,2,3,4,ldots$;
$b_1$ in binary is formed by $0$ and $1$ for each
even and odd digit of $n_1$:
$$
n_1 = 0.123456789101112131415161718192021 ldots
$$
$$
b_1 = 0.101010101101110111011101110110001 ldots
$$
$n_2$ and $b_2$ are formed similarly, but listing
the primes $2,3,5,7,ldots$:
$$
n_2 = 0.23571113171923293137414347535961 ldots
$$
$$
b_2 = 0.01111111111101011111010101111101 ldots
$$
Which of these numbers is known to be rational,
irrational, algebraic, transcendental?
I presume that all four are irrational.
irrational-numbers transcendental-numbers
asked Mar 30 at 10:02
Joseph O'RourkeJoseph O'Rourke
18.3k350113
$endgroup$
add a comment |
$begingroup$
Consider four numbers in $(0,1)$:
$n_1$ in base $10$ is formed by listing the decimal digits $1,2,3,4,ldots$;
$b_1$ in binary is formed by $0$ and $1$ for each
even and odd digit of $n_1$:
$$
n_1 = 0.123456789101112131415161718192021 ldots
$$
$$
b_1 = 0.101010101101110111011101110110001 ldots
$$
$n_2$ and $b_2$ are formed similarly, but listing
the primes $2,3,5,7,ldots$:
$$
n_2 = 0.23571113171923293137414347535961 ldots
$$
$$
b_2 = 0.01111111111101011111010101111101 ldots
$$
Which of these numbers is known to be rational,
irrational, algebraic, transcendental?
I presume that all four are irrational.
irrational-numbers transcendental-numbers
asked Mar 30 at 10:02
Joseph O'RourkeJoseph O'Rourke
18.3k350113
$endgroup$
2
$begingroup$
Irrationality isn't hard, for any of them. There are natural numbers with arbitrarily long strings of $1's$ or strings of $0's$, for example. And, similarly, there are primes with such strings (there are infinitely many primes that start with any fixed sequence).
$endgroup$
– lulu
Mar 30 at 10:11
3
$begingroup$
Well the first looks like Champernowne's constant: en.m.wikipedia.org/wiki/Champernowne_constant. Better than merely transcendental, it is normal.
$endgroup$
– badjohn
Mar 30 at 10:31
2
$begingroup$
@badjohn It's only known to be normal in base 10, and actually the same is known about $n_2$, which is the Copeland-Erdos constant. Also, normality is not "better than merely transcendental", all algebraic irrationals are expected to be normal.
$endgroup$
– Wojowu
Mar 30 at 10:43
$begingroup$
@Wojowu I considered mentioning that but it seems more than necessary for a comment. The "better than" was intended to be light hearted. Plenty of numbers are expected to be normal but relatively few are proven to be.
$endgroup$
– badjohn
Mar 30 at 10:55
$begingroup$
Thanks, badjohn & Wojowu, for identifying the constants as Champernowne and Copeland-Erdõs.
$endgroup$
– Joseph O'Rourke
Mar 30 at 11:57
add a comment |
$begingroup$
Consider four numbers in $(0,1)$:
$n_1$ in base $10$ is formed by listing the decimal digits $1,2,3,4,ldots$;
$b_1$ in binary is formed by $0$ and $1$ for each
even and odd digit of $n_1$:
$$
n_1 = 0.123456789101112131415161718192021 ldots
$$
$$
b_1 = 0.101010101101110111011101110110001 ldots
$$
$n_2$ and $b_2$ are formed similarly, but listing
the primes $2,3,5,7,ldots$:
$$
n_2 = 0.23571113171923293137414347535961 ldots
$$
$$
b_2 = 0.01111111111101011111010101111101 ldots
$$
Which of these numbers is known to be rational,
irrational, algebraic, transcendental?
I presume that all four are irrational.
irrational-numbers transcendental-numbers
asked Mar 30 at 10:02
Joseph O'RourkeJoseph O'Rourke
18.3k350113
$endgroup$
Consider four numbers in $(0,1)$:
$n_1$ in base $10$ is formed by listing the decimal digits $1,2,3,4,ldots$;
$b_1$ in binary is formed by $0$ and $1$ for each
even and odd digit of $n_1$:
$$
n_1 = 0.123456789101112131415161718192021 ldots
$$
$$
b_1 = 0.101010101101110111011101110110001 ldots
$$
$n_2$ and $b_2$ are formed similarly, but listing
the primes $2,3,5,7,ldots$:
$$
n_2 = 0.23571113171923293137414347535961 ldots
$$
$$
b_2 = 0.01111111111101011111010101111101 ldots
$$
Which of these numbers is known to be rational,
irrational, algebraic, transcendental?
I presume that all four are irrational.
irrational-numbers transcendental-numbers
irrational-numbers transcendental-numbers
asked Mar 30 at 10:02
Joseph O'RourkeJoseph O'Rourke
18.3k350113
asked Mar 30 at 10:02
Joseph O'RourkeJoseph O'Rourke
18.3k350113
asked Mar 30 at 10:02
Joseph O'RourkeJoseph O'Rourke
18.3k350113
asked Mar 30 at 10:02
Joseph O'RourkeJoseph O'Rourke
18.3k350113
asked Mar 30 at 10:02
Joseph O'RourkeJoseph O'Rourke
18.3k350113
18.3k350113
2
$begingroup$
Irrationality isn't hard, for any of them. There are natural numbers with arbitrarily long strings of $1's$ or strings of $0's$, for example. And, similarly, there are primes with such strings (there are infinitely many primes that start with any fixed sequence).
$endgroup$
– lulu
Mar 30 at 10:11
3
$begingroup$
Well the first looks like Champernowne's constant: en.m.wikipedia.org/wiki/Champernowne_constant. Better than merely transcendental, it is normal.
$endgroup$
– badjohn
Mar 30 at 10:31
2
$begingroup$
@badjohn It's only known to be normal in base 10, and actually the same is known about $n_2$, which is the Copeland-Erdos constant. Also, normality is not "better than merely transcendental", all algebraic irrationals are expected to be normal.
$endgroup$
– Wojowu
Mar 30 at 10:43
$begingroup$
@Wojowu I considered mentioning that but it seems more than necessary for a comment. The "better than" was intended to be light hearted. Plenty of numbers are expected to be normal but relatively few are proven to be.
$endgroup$
– badjohn
Mar 30 at 10:55
$begingroup$
Thanks, badjohn & Wojowu, for identifying the constants as Champernowne and Copeland-Erdõs.
$endgroup$
– Joseph O'Rourke
Mar 30 at 11:57
add a comment |
2
$begingroup$
Irrationality isn't hard, for any of them. There are natural numbers with arbitrarily long strings of $1's$ or strings of $0's$, for example. And, similarly, there are primes with such strings (there are infinitely many primes that start with any fixed sequence).
$endgroup$
– lulu
Mar 30 at 10:11
3
$begingroup$
Well the first looks like Champernowne's constant: en.m.wikipedia.org/wiki/Champernowne_constant. Better than merely transcendental, it is normal.
$endgroup$
– badjohn
Mar 30 at 10:31
2
$begingroup$
@badjohn It's only known to be normal in base 10, and actually the same is known about $n_2$, which is the Copeland-Erdos constant. Also, normality is not "better than merely transcendental", all algebraic irrationals are expected to be normal.
$endgroup$
– Wojowu
Mar 30 at 10:43
$begingroup$
@Wojowu I considered mentioning that but it seems more than necessary for a comment. The "better than" was intended to be light hearted. Plenty of numbers are expected to be normal but relatively few are proven to be.
$endgroup$
– badjohn
Mar 30 at 10:55
$begingroup$
Thanks, badjohn & Wojowu, for identifying the constants as Champernowne and Copeland-Erdõs.
$endgroup$
– Joseph O'Rourke
Mar 30 at 11:57
2
2
$begingroup$
Irrationality isn't hard, for any of them. There are natural numbers with arbitrarily long strings of $1's$ or strings of $0's$, for example. And, similarly, there are primes with such strings (there are infinitely many primes that start with any fixed sequence).
$endgroup$
– lulu
Mar 30 at 10:11
$begingroup$
Irrationality isn't hard, for any of them. There are natural numbers with arbitrarily long strings of $1's$ or strings of $0's$, for example. And, similarly, there are primes with such strings (there are infinitely many primes that start with any fixed sequence).
$endgroup$
– lulu
Mar 30 at 10:11
3
3
$begingroup$
Well the first looks like Champernowne's constant: en.m.wikipedia.org/wiki/Champernowne_constant. Better than merely transcendental, it is normal.
$endgroup$
– badjohn
Mar 30 at 10:31
$begingroup$
Well the first looks like Champernowne's constant: en.m.wikipedia.org/wiki/Champernowne_constant. Better than merely transcendental, it is normal.
$endgroup$
– badjohn
Mar 30 at 10:31
2
2
$begingroup$
@badjohn It's only known to be normal in base 10, and actually the same is known about $n_2$, which is the Copeland-Erdos constant. Also, normality is not "better than merely transcendental", all algebraic irrationals are expected to be normal.
$endgroup$
– Wojowu
Mar 30 at 10:43
$begingroup$
@badjohn It's only known to be normal in base 10, and actually the same is known about $n_2$, which is the Copeland-Erdos constant. Also, normality is not "better than merely transcendental", all algebraic irrationals are expected to be normal.
$endgroup$
– Wojowu
Mar 30 at 10:43
$begingroup$
@Wojowu I considered mentioning that but it seems more than necessary for a comment. The "better than" was intended to be light hearted. Plenty of numbers are expected to be normal but relatively few are proven to be.
$endgroup$
– badjohn
Mar 30 at 10:55
$begingroup$
@Wojowu I considered mentioning that but it seems more than necessary for a comment. The "better than" was intended to be light hearted. Plenty of numbers are expected to be normal but relatively few are proven to be.
$endgroup$
– badjohn
Mar 30 at 10:55
$begingroup$
Thanks, badjohn & Wojowu, for identifying the constants as Champernowne and Copeland-Erdõs.
$endgroup$
– Joseph O'Rourke
Mar 30 at 11:57
$begingroup$
Thanks, badjohn & Wojowu, for identifying the constants as Champernowne and Copeland-Erdõs.
$endgroup$
– Joseph O'Rourke
Mar 30 at 11:57
add a comment |
0
active
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$begingroup$
Irrationality isn't hard, for any of them. There are natural numbers with arbitrarily long strings of $1's$ or strings of $0's$, for example. And, similarly, there are primes with such strings (there are infinitely many primes that start with any fixed sequence).
$endgroup$
– lulu
Mar 30 at 10:11
3
$begingroup$
Well the first looks like Champernowne's constant: en.m.wikipedia.org/wiki/Champernowne_constant. Better than merely transcendental, it is normal.
$endgroup$
– badjohn
Mar 30 at 10:31
2
$begingroup$
@badjohn It's only known to be normal in base 10, and actually the same is known about $n_2$, which is the Copeland-Erdos constant. Also, normality is not "better than merely transcendental", all algebraic irrationals are expected to be normal.
$endgroup$
– Wojowu
Mar 30 at 10:43
$begingroup$
@Wojowu I considered mentioning that but it seems more than necessary for a comment. The "better than" was intended to be light hearted. Plenty of numbers are expected to be normal but relatively few are proven to be.
$endgroup$
– badjohn
Mar 30 at 10:55
$begingroup$
Thanks, badjohn & Wojowu, for identifying the constants as Champernowne and Copeland-Erdõs.
$endgroup$
– Joseph O'Rourke
Mar 30 at 11:57