Proof of multiplicative commutativity for all real numbers
I have seen proofs for commutativity for all integers, and these can be extended to rationals easily because a rational number is just the ratio of two integers. However, I have yet to see a proof that multiplication of real numbers is commutative. How would you prove this one?
analysis
add a comment |
I have seen proofs for commutativity for all integers, and these can be extended to rationals easily because a rational number is just the ratio of two integers. However, I have yet to see a proof that multiplication of real numbers is commutative. How would you prove this one?
analysis
Is multiplication of real numbers defined via limits of multiplication of rationals?
– Mark
May 13 '17 at 22:10
"these can be extended to rationals easily because a rational number is just the ratio of two integers." and these can be extended just as easily to the reals as real is just a limit of a sequence of rational numbers.
– fleablood
May 13 '17 at 22:17
Can you point me to the “proofs for [multiplicative] commutativity for all integers”? I am looking for them but can only find proofs for multiplicative commutativity of natural numbers.
– lukejanicke
11 hours ago
add a comment |
I have seen proofs for commutativity for all integers, and these can be extended to rationals easily because a rational number is just the ratio of two integers. However, I have yet to see a proof that multiplication of real numbers is commutative. How would you prove this one?
analysis
I have seen proofs for commutativity for all integers, and these can be extended to rationals easily because a rational number is just the ratio of two integers. However, I have yet to see a proof that multiplication of real numbers is commutative. How would you prove this one?
analysis
analysis
asked May 13 '17 at 22:04
u8y7541
373314
373314
Is multiplication of real numbers defined via limits of multiplication of rationals?
– Mark
May 13 '17 at 22:10
"these can be extended to rationals easily because a rational number is just the ratio of two integers." and these can be extended just as easily to the reals as real is just a limit of a sequence of rational numbers.
– fleablood
May 13 '17 at 22:17
Can you point me to the “proofs for [multiplicative] commutativity for all integers”? I am looking for them but can only find proofs for multiplicative commutativity of natural numbers.
– lukejanicke
11 hours ago
add a comment |
Is multiplication of real numbers defined via limits of multiplication of rationals?
– Mark
May 13 '17 at 22:10
"these can be extended to rationals easily because a rational number is just the ratio of two integers." and these can be extended just as easily to the reals as real is just a limit of a sequence of rational numbers.
– fleablood
May 13 '17 at 22:17
Can you point me to the “proofs for [multiplicative] commutativity for all integers”? I am looking for them but can only find proofs for multiplicative commutativity of natural numbers.
– lukejanicke
11 hours ago
Is multiplication of real numbers defined via limits of multiplication of rationals?
– Mark
May 13 '17 at 22:10
Is multiplication of real numbers defined via limits of multiplication of rationals?
– Mark
May 13 '17 at 22:10
"these can be extended to rationals easily because a rational number is just the ratio of two integers." and these can be extended just as easily to the reals as real is just a limit of a sequence of rational numbers.
– fleablood
May 13 '17 at 22:17
"these can be extended to rationals easily because a rational number is just the ratio of two integers." and these can be extended just as easily to the reals as real is just a limit of a sequence of rational numbers.
– fleablood
May 13 '17 at 22:17
Can you point me to the “proofs for [multiplicative] commutativity for all integers”? I am looking for them but can only find proofs for multiplicative commutativity of natural numbers.
– lukejanicke
11 hours ago
Can you point me to the “proofs for [multiplicative] commutativity for all integers”? I am looking for them but can only find proofs for multiplicative commutativity of natural numbers.
– lukejanicke
11 hours ago
add a comment |
3 Answers
3
active
oldest
votes
Let $x, y in mathbb R$. Then there exist two sequences of rational numbers ${q_n} rightarrow x$ and ${p_n}rightarrow y$.
It's a standard exercise to prove that if $lim p_n = x$ and $lim q_n = y$ then then $lim p_n*q_n = lim q_n*p_n = x*y$
....
$p_nq_n - xy = (p_n -y)(q_n - x) + y(q_n - x) + x(p_n - y)$
$(p_n -y)(q_n - x) = p_nq_n - xy - y(q_n-x) + x(p_n-y)$
For $epsilon > 0$ let $n > N$ imply $|p_n - y| < sqrt{epsilon}$
and $n > M$ imply $|q_n - x| < sqrt{epsilon}$.
So for $n > max(N,M) = K$ we have $|(x - q_n)(y-p_n)| < epsilon$
So $lim (q_n -x)(p_n -y) = 0$.
$lim p_nq_n - xy - lim y(q_n-x) + lim x(p_n-y) = 0$
So $xy = lim p_nq_n = lim q_np_n = yx$.
add a comment |
This is done very detailed in this paper for the construction of $mathbb R$ from the ring of integers via quasi-homomorphisms : Street : The efficient real numbers
So if you are ok with commutativity for integers, there is not even the need to go through rationals. All others properties of $+,times$ are also prooved in theorem 10.
Here is a quick summary if you don't want to read everything :
The motivation of this construction of $mathbb R$, is that if we define $f_x(n)=lfloor nxrfloor$ then $limlimits_{ntoinfty}frac{f_x(n)}n=x$ so we would like to map $f_x$ to the corresponding real $x$, but of course this would be biting our tail to do it via the floor function.
If $xneq y$ you can notice that $|f_x(n)-f_y(n)|to+infty$ this is the property of $mathbb R$ to be archimedian. So we will say that $x=yiff f_xsim f_y$ if their difference is bounded.
Also, notice that if $i$ is an integer then $f_i(n)+f_i(m)=f_i(m+n)$, so $f_i$ is a homomorphism.
So, instead we will say that if $|f(m+n)-f(m)-f(n)|le k$ is bounded then this quasi-homomorphism $f$ can be assimilated to a real.
The next step is to show three lemmas : $begin{cases}f(n)le a|n|+k\|f(mn)-mf(n)|le(|m|+1)k\|nf(m)-mf(n)|le(|m|+|n|+2)kend{cases}$
for any $(n,m)inmathbb Z^2$.
Now addition is defined ($f+g)(n)=f(n)+g(n)$ and multiplication is defined as $(fg)(n)=fcirc g(n)=f(g(n))$.
The sketch for commutativity of multiplication, then goes as the following :
$|nf(g(n))-ng(f(n))|=|nf(g(n))underbrace{-g(n)f(n)+f(n)g(n)}_{text{commutativity in } mathbb Z=0}-ng(f(n))|$
$le|nf(g(n))-g(n)f(n)|+|ng(f(n))-f(n)g(n)|quad$ [triangular inequality]
$le(g(n)+n)k_1+(f(n)+n)k_2quad$ [lemma 2]
$le nk_3+nk_4le nk_5quad $ [lemma 1]
So in the end we have $|f(g(n))-g(f(n))|le k_5$ which means that $fg=gf$ $(*)$
$(*)$ remember that quasi-homomorphisms are equal when they are in the same equivalence class, which is that their difference is bounded in this context.
add a comment |
The following proof is sketchy, but in the next section I test it out with a Python program.
Consider the set of positive real numbers $x gt 0$ where we apply the 'forgetful functor' and only know how to add; so we have to define multiplication in $left( ,(0,+infty),+ ,right)$ and then show that it commutative. But we certainly accept all the other axioms and laws of the real numbers.
Now even thought there is no multiplication, we have no problem 'multiplying' a real number by a positive integer, since that is just shorthand for 'repeated addition'.
Also, there is a real number, call it $2^{-1}$ with the property that
$tag 1 2^{-1} + 2^{-1} = 1$.
And so we can keep taking powers of our 'bisection operator',
$tag 2 2^{-(n + 1)} = 2^{-n} circ 2^{-1}$.
If $x gt 0$ and $n gt 0$, we can write
$tag 3 x = m_{x,n} 2^{-n} + r_{x,n} text{ with } r_{x,n} lt 2^{-n} text{ and } m_{x,n} in mathbb N$
If $y gt 0$ we also write
$tag 4 y = m_{y,n} 2^{-n} + r_{y,n} text{ with } r_{y,n} lt 2^{-n} text{ and } m_{y,n} in mathbb N$
We define the multiplication
$tag 5 x y = limlimits_{n to +∞} m_{x,n}, m_{y,n}, 2^{-2n} $
Since the multiplication of integers is commutative, so is the multiplication of real numbers.
Python Program:
a = 72.987654
b = 87.123456
print(a, b, a*b)
div = 1
for i in range(1,25):
div = div / 2
x = divmod(a, div)
m, g = x
y = divmod(b, div)
n, g = y
print(m*div, n*div, 'seq = ', i, m*n*div*div )
Output:
72.987654 87.123456 6358.936661812225
72.5 87.0 seq = 1 6307.5
72.75 87.0 seq = 2 6329.25
72.875 87.0 seq = 3 6340.125
72.9375 87.0625 seq = 4 6350.12109375
72.96875 87.09375 seq = 5 6355.1220703125
72.984375 87.109375 seq = 6 6357.623291015625
72.984375 87.1171875 seq = 7 6358.1934814453125
72.984375 87.12109375 seq = 8 6358.478576660156
72.986328125 87.123046875 seq = 9 6358.791286468506
72.9873046875 87.123046875 seq = 10 6358.87636756897
72.9873046875 87.123046875 seq = 11 6358.87636756897
72.987548828125 87.123291015625 seq = 12 6358.915457069874
72.987548828125 87.1234130859375 seq = 13 6358.924366682768
72.98760986328125 87.1234130859375 seq = 14 6358.929684273899
72.98764038085938 87.12344360351562 seq = 15 6358.934570475481
72.98764038085938 87.12344360351562 seq = 16 6358.934570475481
72.9876480102539 87.12345123291016 seq = 17 6358.935792026168
72.98765182495117 87.12345504760742 seq = 18 6358.936402801555
72.9876537322998 87.12345504760742 seq = 19 6358.936568976358
72.9876537322998 87.12345504760742 seq = 20 6358.936568976358
72.9876537322998 87.12345552444458 seq = 21 6358.9366037795835
72.98765397071838 87.12345576286316 seq = 22 6358.936641953047
72.98765397071838 87.12345588207245 seq = 23 6358.936650653853
72.98765397071838 87.1234559416771 seq = 24 6358.936655004256
add a comment |
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3 Answers
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3 Answers
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Let $x, y in mathbb R$. Then there exist two sequences of rational numbers ${q_n} rightarrow x$ and ${p_n}rightarrow y$.
It's a standard exercise to prove that if $lim p_n = x$ and $lim q_n = y$ then then $lim p_n*q_n = lim q_n*p_n = x*y$
....
$p_nq_n - xy = (p_n -y)(q_n - x) + y(q_n - x) + x(p_n - y)$
$(p_n -y)(q_n - x) = p_nq_n - xy - y(q_n-x) + x(p_n-y)$
For $epsilon > 0$ let $n > N$ imply $|p_n - y| < sqrt{epsilon}$
and $n > M$ imply $|q_n - x| < sqrt{epsilon}$.
So for $n > max(N,M) = K$ we have $|(x - q_n)(y-p_n)| < epsilon$
So $lim (q_n -x)(p_n -y) = 0$.
$lim p_nq_n - xy - lim y(q_n-x) + lim x(p_n-y) = 0$
So $xy = lim p_nq_n = lim q_np_n = yx$.
add a comment |
Let $x, y in mathbb R$. Then there exist two sequences of rational numbers ${q_n} rightarrow x$ and ${p_n}rightarrow y$.
It's a standard exercise to prove that if $lim p_n = x$ and $lim q_n = y$ then then $lim p_n*q_n = lim q_n*p_n = x*y$
....
$p_nq_n - xy = (p_n -y)(q_n - x) + y(q_n - x) + x(p_n - y)$
$(p_n -y)(q_n - x) = p_nq_n - xy - y(q_n-x) + x(p_n-y)$
For $epsilon > 0$ let $n > N$ imply $|p_n - y| < sqrt{epsilon}$
and $n > M$ imply $|q_n - x| < sqrt{epsilon}$.
So for $n > max(N,M) = K$ we have $|(x - q_n)(y-p_n)| < epsilon$
So $lim (q_n -x)(p_n -y) = 0$.
$lim p_nq_n - xy - lim y(q_n-x) + lim x(p_n-y) = 0$
So $xy = lim p_nq_n = lim q_np_n = yx$.
add a comment |
Let $x, y in mathbb R$. Then there exist two sequences of rational numbers ${q_n} rightarrow x$ and ${p_n}rightarrow y$.
It's a standard exercise to prove that if $lim p_n = x$ and $lim q_n = y$ then then $lim p_n*q_n = lim q_n*p_n = x*y$
....
$p_nq_n - xy = (p_n -y)(q_n - x) + y(q_n - x) + x(p_n - y)$
$(p_n -y)(q_n - x) = p_nq_n - xy - y(q_n-x) + x(p_n-y)$
For $epsilon > 0$ let $n > N$ imply $|p_n - y| < sqrt{epsilon}$
and $n > M$ imply $|q_n - x| < sqrt{epsilon}$.
So for $n > max(N,M) = K$ we have $|(x - q_n)(y-p_n)| < epsilon$
So $lim (q_n -x)(p_n -y) = 0$.
$lim p_nq_n - xy - lim y(q_n-x) + lim x(p_n-y) = 0$
So $xy = lim p_nq_n = lim q_np_n = yx$.
Let $x, y in mathbb R$. Then there exist two sequences of rational numbers ${q_n} rightarrow x$ and ${p_n}rightarrow y$.
It's a standard exercise to prove that if $lim p_n = x$ and $lim q_n = y$ then then $lim p_n*q_n = lim q_n*p_n = x*y$
....
$p_nq_n - xy = (p_n -y)(q_n - x) + y(q_n - x) + x(p_n - y)$
$(p_n -y)(q_n - x) = p_nq_n - xy - y(q_n-x) + x(p_n-y)$
For $epsilon > 0$ let $n > N$ imply $|p_n - y| < sqrt{epsilon}$
and $n > M$ imply $|q_n - x| < sqrt{epsilon}$.
So for $n > max(N,M) = K$ we have $|(x - q_n)(y-p_n)| < epsilon$
So $lim (q_n -x)(p_n -y) = 0$.
$lim p_nq_n - xy - lim y(q_n-x) + lim x(p_n-y) = 0$
So $xy = lim p_nq_n = lim q_np_n = yx$.
answered May 13 '17 at 22:53
fleablood
68.1k22684
68.1k22684
add a comment |
add a comment |
This is done very detailed in this paper for the construction of $mathbb R$ from the ring of integers via quasi-homomorphisms : Street : The efficient real numbers
So if you are ok with commutativity for integers, there is not even the need to go through rationals. All others properties of $+,times$ are also prooved in theorem 10.
Here is a quick summary if you don't want to read everything :
The motivation of this construction of $mathbb R$, is that if we define $f_x(n)=lfloor nxrfloor$ then $limlimits_{ntoinfty}frac{f_x(n)}n=x$ so we would like to map $f_x$ to the corresponding real $x$, but of course this would be biting our tail to do it via the floor function.
If $xneq y$ you can notice that $|f_x(n)-f_y(n)|to+infty$ this is the property of $mathbb R$ to be archimedian. So we will say that $x=yiff f_xsim f_y$ if their difference is bounded.
Also, notice that if $i$ is an integer then $f_i(n)+f_i(m)=f_i(m+n)$, so $f_i$ is a homomorphism.
So, instead we will say that if $|f(m+n)-f(m)-f(n)|le k$ is bounded then this quasi-homomorphism $f$ can be assimilated to a real.
The next step is to show three lemmas : $begin{cases}f(n)le a|n|+k\|f(mn)-mf(n)|le(|m|+1)k\|nf(m)-mf(n)|le(|m|+|n|+2)kend{cases}$
for any $(n,m)inmathbb Z^2$.
Now addition is defined ($f+g)(n)=f(n)+g(n)$ and multiplication is defined as $(fg)(n)=fcirc g(n)=f(g(n))$.
The sketch for commutativity of multiplication, then goes as the following :
$|nf(g(n))-ng(f(n))|=|nf(g(n))underbrace{-g(n)f(n)+f(n)g(n)}_{text{commutativity in } mathbb Z=0}-ng(f(n))|$
$le|nf(g(n))-g(n)f(n)|+|ng(f(n))-f(n)g(n)|quad$ [triangular inequality]
$le(g(n)+n)k_1+(f(n)+n)k_2quad$ [lemma 2]
$le nk_3+nk_4le nk_5quad $ [lemma 1]
So in the end we have $|f(g(n))-g(f(n))|le k_5$ which means that $fg=gf$ $(*)$
$(*)$ remember that quasi-homomorphisms are equal when they are in the same equivalence class, which is that their difference is bounded in this context.
add a comment |
This is done very detailed in this paper for the construction of $mathbb R$ from the ring of integers via quasi-homomorphisms : Street : The efficient real numbers
So if you are ok with commutativity for integers, there is not even the need to go through rationals. All others properties of $+,times$ are also prooved in theorem 10.
Here is a quick summary if you don't want to read everything :
The motivation of this construction of $mathbb R$, is that if we define $f_x(n)=lfloor nxrfloor$ then $limlimits_{ntoinfty}frac{f_x(n)}n=x$ so we would like to map $f_x$ to the corresponding real $x$, but of course this would be biting our tail to do it via the floor function.
If $xneq y$ you can notice that $|f_x(n)-f_y(n)|to+infty$ this is the property of $mathbb R$ to be archimedian. So we will say that $x=yiff f_xsim f_y$ if their difference is bounded.
Also, notice that if $i$ is an integer then $f_i(n)+f_i(m)=f_i(m+n)$, so $f_i$ is a homomorphism.
So, instead we will say that if $|f(m+n)-f(m)-f(n)|le k$ is bounded then this quasi-homomorphism $f$ can be assimilated to a real.
The next step is to show three lemmas : $begin{cases}f(n)le a|n|+k\|f(mn)-mf(n)|le(|m|+1)k\|nf(m)-mf(n)|le(|m|+|n|+2)kend{cases}$
for any $(n,m)inmathbb Z^2$.
Now addition is defined ($f+g)(n)=f(n)+g(n)$ and multiplication is defined as $(fg)(n)=fcirc g(n)=f(g(n))$.
The sketch for commutativity of multiplication, then goes as the following :
$|nf(g(n))-ng(f(n))|=|nf(g(n))underbrace{-g(n)f(n)+f(n)g(n)}_{text{commutativity in } mathbb Z=0}-ng(f(n))|$
$le|nf(g(n))-g(n)f(n)|+|ng(f(n))-f(n)g(n)|quad$ [triangular inequality]
$le(g(n)+n)k_1+(f(n)+n)k_2quad$ [lemma 2]
$le nk_3+nk_4le nk_5quad $ [lemma 1]
So in the end we have $|f(g(n))-g(f(n))|le k_5$ which means that $fg=gf$ $(*)$
$(*)$ remember that quasi-homomorphisms are equal when they are in the same equivalence class, which is that their difference is bounded in this context.
add a comment |
This is done very detailed in this paper for the construction of $mathbb R$ from the ring of integers via quasi-homomorphisms : Street : The efficient real numbers
So if you are ok with commutativity for integers, there is not even the need to go through rationals. All others properties of $+,times$ are also prooved in theorem 10.
Here is a quick summary if you don't want to read everything :
The motivation of this construction of $mathbb R$, is that if we define $f_x(n)=lfloor nxrfloor$ then $limlimits_{ntoinfty}frac{f_x(n)}n=x$ so we would like to map $f_x$ to the corresponding real $x$, but of course this would be biting our tail to do it via the floor function.
If $xneq y$ you can notice that $|f_x(n)-f_y(n)|to+infty$ this is the property of $mathbb R$ to be archimedian. So we will say that $x=yiff f_xsim f_y$ if their difference is bounded.
Also, notice that if $i$ is an integer then $f_i(n)+f_i(m)=f_i(m+n)$, so $f_i$ is a homomorphism.
So, instead we will say that if $|f(m+n)-f(m)-f(n)|le k$ is bounded then this quasi-homomorphism $f$ can be assimilated to a real.
The next step is to show three lemmas : $begin{cases}f(n)le a|n|+k\|f(mn)-mf(n)|le(|m|+1)k\|nf(m)-mf(n)|le(|m|+|n|+2)kend{cases}$
for any $(n,m)inmathbb Z^2$.
Now addition is defined ($f+g)(n)=f(n)+g(n)$ and multiplication is defined as $(fg)(n)=fcirc g(n)=f(g(n))$.
The sketch for commutativity of multiplication, then goes as the following :
$|nf(g(n))-ng(f(n))|=|nf(g(n))underbrace{-g(n)f(n)+f(n)g(n)}_{text{commutativity in } mathbb Z=0}-ng(f(n))|$
$le|nf(g(n))-g(n)f(n)|+|ng(f(n))-f(n)g(n)|quad$ [triangular inequality]
$le(g(n)+n)k_1+(f(n)+n)k_2quad$ [lemma 2]
$le nk_3+nk_4le nk_5quad $ [lemma 1]
So in the end we have $|f(g(n))-g(f(n))|le k_5$ which means that $fg=gf$ $(*)$
$(*)$ remember that quasi-homomorphisms are equal when they are in the same equivalence class, which is that their difference is bounded in this context.
This is done very detailed in this paper for the construction of $mathbb R$ from the ring of integers via quasi-homomorphisms : Street : The efficient real numbers
So if you are ok with commutativity for integers, there is not even the need to go through rationals. All others properties of $+,times$ are also prooved in theorem 10.
Here is a quick summary if you don't want to read everything :
The motivation of this construction of $mathbb R$, is that if we define $f_x(n)=lfloor nxrfloor$ then $limlimits_{ntoinfty}frac{f_x(n)}n=x$ so we would like to map $f_x$ to the corresponding real $x$, but of course this would be biting our tail to do it via the floor function.
If $xneq y$ you can notice that $|f_x(n)-f_y(n)|to+infty$ this is the property of $mathbb R$ to be archimedian. So we will say that $x=yiff f_xsim f_y$ if their difference is bounded.
Also, notice that if $i$ is an integer then $f_i(n)+f_i(m)=f_i(m+n)$, so $f_i$ is a homomorphism.
So, instead we will say that if $|f(m+n)-f(m)-f(n)|le k$ is bounded then this quasi-homomorphism $f$ can be assimilated to a real.
The next step is to show three lemmas : $begin{cases}f(n)le a|n|+k\|f(mn)-mf(n)|le(|m|+1)k\|nf(m)-mf(n)|le(|m|+|n|+2)kend{cases}$
for any $(n,m)inmathbb Z^2$.
Now addition is defined ($f+g)(n)=f(n)+g(n)$ and multiplication is defined as $(fg)(n)=fcirc g(n)=f(g(n))$.
The sketch for commutativity of multiplication, then goes as the following :
$|nf(g(n))-ng(f(n))|=|nf(g(n))underbrace{-g(n)f(n)+f(n)g(n)}_{text{commutativity in } mathbb Z=0}-ng(f(n))|$
$le|nf(g(n))-g(n)f(n)|+|ng(f(n))-f(n)g(n)|quad$ [triangular inequality]
$le(g(n)+n)k_1+(f(n)+n)k_2quad$ [lemma 2]
$le nk_3+nk_4le nk_5quad $ [lemma 1]
So in the end we have $|f(g(n))-g(f(n))|le k_5$ which means that $fg=gf$ $(*)$
$(*)$ remember that quasi-homomorphisms are equal when they are in the same equivalence class, which is that their difference is bounded in this context.
edited May 14 '17 at 0:45
answered May 14 '17 at 0:23
zwim
11.5k729
11.5k729
add a comment |
add a comment |
The following proof is sketchy, but in the next section I test it out with a Python program.
Consider the set of positive real numbers $x gt 0$ where we apply the 'forgetful functor' and only know how to add; so we have to define multiplication in $left( ,(0,+infty),+ ,right)$ and then show that it commutative. But we certainly accept all the other axioms and laws of the real numbers.
Now even thought there is no multiplication, we have no problem 'multiplying' a real number by a positive integer, since that is just shorthand for 'repeated addition'.
Also, there is a real number, call it $2^{-1}$ with the property that
$tag 1 2^{-1} + 2^{-1} = 1$.
And so we can keep taking powers of our 'bisection operator',
$tag 2 2^{-(n + 1)} = 2^{-n} circ 2^{-1}$.
If $x gt 0$ and $n gt 0$, we can write
$tag 3 x = m_{x,n} 2^{-n} + r_{x,n} text{ with } r_{x,n} lt 2^{-n} text{ and } m_{x,n} in mathbb N$
If $y gt 0$ we also write
$tag 4 y = m_{y,n} 2^{-n} + r_{y,n} text{ with } r_{y,n} lt 2^{-n} text{ and } m_{y,n} in mathbb N$
We define the multiplication
$tag 5 x y = limlimits_{n to +∞} m_{x,n}, m_{y,n}, 2^{-2n} $
Since the multiplication of integers is commutative, so is the multiplication of real numbers.
Python Program:
a = 72.987654
b = 87.123456
print(a, b, a*b)
div = 1
for i in range(1,25):
div = div / 2
x = divmod(a, div)
m, g = x
y = divmod(b, div)
n, g = y
print(m*div, n*div, 'seq = ', i, m*n*div*div )
Output:
72.987654 87.123456 6358.936661812225
72.5 87.0 seq = 1 6307.5
72.75 87.0 seq = 2 6329.25
72.875 87.0 seq = 3 6340.125
72.9375 87.0625 seq = 4 6350.12109375
72.96875 87.09375 seq = 5 6355.1220703125
72.984375 87.109375 seq = 6 6357.623291015625
72.984375 87.1171875 seq = 7 6358.1934814453125
72.984375 87.12109375 seq = 8 6358.478576660156
72.986328125 87.123046875 seq = 9 6358.791286468506
72.9873046875 87.123046875 seq = 10 6358.87636756897
72.9873046875 87.123046875 seq = 11 6358.87636756897
72.987548828125 87.123291015625 seq = 12 6358.915457069874
72.987548828125 87.1234130859375 seq = 13 6358.924366682768
72.98760986328125 87.1234130859375 seq = 14 6358.929684273899
72.98764038085938 87.12344360351562 seq = 15 6358.934570475481
72.98764038085938 87.12344360351562 seq = 16 6358.934570475481
72.9876480102539 87.12345123291016 seq = 17 6358.935792026168
72.98765182495117 87.12345504760742 seq = 18 6358.936402801555
72.9876537322998 87.12345504760742 seq = 19 6358.936568976358
72.9876537322998 87.12345504760742 seq = 20 6358.936568976358
72.9876537322998 87.12345552444458 seq = 21 6358.9366037795835
72.98765397071838 87.12345576286316 seq = 22 6358.936641953047
72.98765397071838 87.12345588207245 seq = 23 6358.936650653853
72.98765397071838 87.1234559416771 seq = 24 6358.936655004256
add a comment |
The following proof is sketchy, but in the next section I test it out with a Python program.
Consider the set of positive real numbers $x gt 0$ where we apply the 'forgetful functor' and only know how to add; so we have to define multiplication in $left( ,(0,+infty),+ ,right)$ and then show that it commutative. But we certainly accept all the other axioms and laws of the real numbers.
Now even thought there is no multiplication, we have no problem 'multiplying' a real number by a positive integer, since that is just shorthand for 'repeated addition'.
Also, there is a real number, call it $2^{-1}$ with the property that
$tag 1 2^{-1} + 2^{-1} = 1$.
And so we can keep taking powers of our 'bisection operator',
$tag 2 2^{-(n + 1)} = 2^{-n} circ 2^{-1}$.
If $x gt 0$ and $n gt 0$, we can write
$tag 3 x = m_{x,n} 2^{-n} + r_{x,n} text{ with } r_{x,n} lt 2^{-n} text{ and } m_{x,n} in mathbb N$
If $y gt 0$ we also write
$tag 4 y = m_{y,n} 2^{-n} + r_{y,n} text{ with } r_{y,n} lt 2^{-n} text{ and } m_{y,n} in mathbb N$
We define the multiplication
$tag 5 x y = limlimits_{n to +∞} m_{x,n}, m_{y,n}, 2^{-2n} $
Since the multiplication of integers is commutative, so is the multiplication of real numbers.
Python Program:
a = 72.987654
b = 87.123456
print(a, b, a*b)
div = 1
for i in range(1,25):
div = div / 2
x = divmod(a, div)
m, g = x
y = divmod(b, div)
n, g = y
print(m*div, n*div, 'seq = ', i, m*n*div*div )
Output:
72.987654 87.123456 6358.936661812225
72.5 87.0 seq = 1 6307.5
72.75 87.0 seq = 2 6329.25
72.875 87.0 seq = 3 6340.125
72.9375 87.0625 seq = 4 6350.12109375
72.96875 87.09375 seq = 5 6355.1220703125
72.984375 87.109375 seq = 6 6357.623291015625
72.984375 87.1171875 seq = 7 6358.1934814453125
72.984375 87.12109375 seq = 8 6358.478576660156
72.986328125 87.123046875 seq = 9 6358.791286468506
72.9873046875 87.123046875 seq = 10 6358.87636756897
72.9873046875 87.123046875 seq = 11 6358.87636756897
72.987548828125 87.123291015625 seq = 12 6358.915457069874
72.987548828125 87.1234130859375 seq = 13 6358.924366682768
72.98760986328125 87.1234130859375 seq = 14 6358.929684273899
72.98764038085938 87.12344360351562 seq = 15 6358.934570475481
72.98764038085938 87.12344360351562 seq = 16 6358.934570475481
72.9876480102539 87.12345123291016 seq = 17 6358.935792026168
72.98765182495117 87.12345504760742 seq = 18 6358.936402801555
72.9876537322998 87.12345504760742 seq = 19 6358.936568976358
72.9876537322998 87.12345504760742 seq = 20 6358.936568976358
72.9876537322998 87.12345552444458 seq = 21 6358.9366037795835
72.98765397071838 87.12345576286316 seq = 22 6358.936641953047
72.98765397071838 87.12345588207245 seq = 23 6358.936650653853
72.98765397071838 87.1234559416771 seq = 24 6358.936655004256
add a comment |
The following proof is sketchy, but in the next section I test it out with a Python program.
Consider the set of positive real numbers $x gt 0$ where we apply the 'forgetful functor' and only know how to add; so we have to define multiplication in $left( ,(0,+infty),+ ,right)$ and then show that it commutative. But we certainly accept all the other axioms and laws of the real numbers.
Now even thought there is no multiplication, we have no problem 'multiplying' a real number by a positive integer, since that is just shorthand for 'repeated addition'.
Also, there is a real number, call it $2^{-1}$ with the property that
$tag 1 2^{-1} + 2^{-1} = 1$.
And so we can keep taking powers of our 'bisection operator',
$tag 2 2^{-(n + 1)} = 2^{-n} circ 2^{-1}$.
If $x gt 0$ and $n gt 0$, we can write
$tag 3 x = m_{x,n} 2^{-n} + r_{x,n} text{ with } r_{x,n} lt 2^{-n} text{ and } m_{x,n} in mathbb N$
If $y gt 0$ we also write
$tag 4 y = m_{y,n} 2^{-n} + r_{y,n} text{ with } r_{y,n} lt 2^{-n} text{ and } m_{y,n} in mathbb N$
We define the multiplication
$tag 5 x y = limlimits_{n to +∞} m_{x,n}, m_{y,n}, 2^{-2n} $
Since the multiplication of integers is commutative, so is the multiplication of real numbers.
Python Program:
a = 72.987654
b = 87.123456
print(a, b, a*b)
div = 1
for i in range(1,25):
div = div / 2
x = divmod(a, div)
m, g = x
y = divmod(b, div)
n, g = y
print(m*div, n*div, 'seq = ', i, m*n*div*div )
Output:
72.987654 87.123456 6358.936661812225
72.5 87.0 seq = 1 6307.5
72.75 87.0 seq = 2 6329.25
72.875 87.0 seq = 3 6340.125
72.9375 87.0625 seq = 4 6350.12109375
72.96875 87.09375 seq = 5 6355.1220703125
72.984375 87.109375 seq = 6 6357.623291015625
72.984375 87.1171875 seq = 7 6358.1934814453125
72.984375 87.12109375 seq = 8 6358.478576660156
72.986328125 87.123046875 seq = 9 6358.791286468506
72.9873046875 87.123046875 seq = 10 6358.87636756897
72.9873046875 87.123046875 seq = 11 6358.87636756897
72.987548828125 87.123291015625 seq = 12 6358.915457069874
72.987548828125 87.1234130859375 seq = 13 6358.924366682768
72.98760986328125 87.1234130859375 seq = 14 6358.929684273899
72.98764038085938 87.12344360351562 seq = 15 6358.934570475481
72.98764038085938 87.12344360351562 seq = 16 6358.934570475481
72.9876480102539 87.12345123291016 seq = 17 6358.935792026168
72.98765182495117 87.12345504760742 seq = 18 6358.936402801555
72.9876537322998 87.12345504760742 seq = 19 6358.936568976358
72.9876537322998 87.12345504760742 seq = 20 6358.936568976358
72.9876537322998 87.12345552444458 seq = 21 6358.9366037795835
72.98765397071838 87.12345576286316 seq = 22 6358.936641953047
72.98765397071838 87.12345588207245 seq = 23 6358.936650653853
72.98765397071838 87.1234559416771 seq = 24 6358.936655004256
The following proof is sketchy, but in the next section I test it out with a Python program.
Consider the set of positive real numbers $x gt 0$ where we apply the 'forgetful functor' and only know how to add; so we have to define multiplication in $left( ,(0,+infty),+ ,right)$ and then show that it commutative. But we certainly accept all the other axioms and laws of the real numbers.
Now even thought there is no multiplication, we have no problem 'multiplying' a real number by a positive integer, since that is just shorthand for 'repeated addition'.
Also, there is a real number, call it $2^{-1}$ with the property that
$tag 1 2^{-1} + 2^{-1} = 1$.
And so we can keep taking powers of our 'bisection operator',
$tag 2 2^{-(n + 1)} = 2^{-n} circ 2^{-1}$.
If $x gt 0$ and $n gt 0$, we can write
$tag 3 x = m_{x,n} 2^{-n} + r_{x,n} text{ with } r_{x,n} lt 2^{-n} text{ and } m_{x,n} in mathbb N$
If $y gt 0$ we also write
$tag 4 y = m_{y,n} 2^{-n} + r_{y,n} text{ with } r_{y,n} lt 2^{-n} text{ and } m_{y,n} in mathbb N$
We define the multiplication
$tag 5 x y = limlimits_{n to +∞} m_{x,n}, m_{y,n}, 2^{-2n} $
Since the multiplication of integers is commutative, so is the multiplication of real numbers.
Python Program:
a = 72.987654
b = 87.123456
print(a, b, a*b)
div = 1
for i in range(1,25):
div = div / 2
x = divmod(a, div)
m, g = x
y = divmod(b, div)
n, g = y
print(m*div, n*div, 'seq = ', i, m*n*div*div )
Output:
72.987654 87.123456 6358.936661812225
72.5 87.0 seq = 1 6307.5
72.75 87.0 seq = 2 6329.25
72.875 87.0 seq = 3 6340.125
72.9375 87.0625 seq = 4 6350.12109375
72.96875 87.09375 seq = 5 6355.1220703125
72.984375 87.109375 seq = 6 6357.623291015625
72.984375 87.1171875 seq = 7 6358.1934814453125
72.984375 87.12109375 seq = 8 6358.478576660156
72.986328125 87.123046875 seq = 9 6358.791286468506
72.9873046875 87.123046875 seq = 10 6358.87636756897
72.9873046875 87.123046875 seq = 11 6358.87636756897
72.987548828125 87.123291015625 seq = 12 6358.915457069874
72.987548828125 87.1234130859375 seq = 13 6358.924366682768
72.98760986328125 87.1234130859375 seq = 14 6358.929684273899
72.98764038085938 87.12344360351562 seq = 15 6358.934570475481
72.98764038085938 87.12344360351562 seq = 16 6358.934570475481
72.9876480102539 87.12345123291016 seq = 17 6358.935792026168
72.98765182495117 87.12345504760742 seq = 18 6358.936402801555
72.9876537322998 87.12345504760742 seq = 19 6358.936568976358
72.9876537322998 87.12345504760742 seq = 20 6358.936568976358
72.9876537322998 87.12345552444458 seq = 21 6358.9366037795835
72.98765397071838 87.12345576286316 seq = 22 6358.936641953047
72.98765397071838 87.12345588207245 seq = 23 6358.936650653853
72.98765397071838 87.1234559416771 seq = 24 6358.936655004256
answered Nov 27 at 5:07
CopyPasteIt
4,0121627
4,0121627
add a comment |
add a comment |
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Is multiplication of real numbers defined via limits of multiplication of rationals?
– Mark
May 13 '17 at 22:10
"these can be extended to rationals easily because a rational number is just the ratio of two integers." and these can be extended just as easily to the reals as real is just a limit of a sequence of rational numbers.
– fleablood
May 13 '17 at 22:17
Can you point me to the “proofs for [multiplicative] commutativity for all integers”? I am looking for them but can only find proofs for multiplicative commutativity of natural numbers.
– lukejanicke
11 hours ago