Is 10 a polynomial?
$begingroup$
How 10 is a constant polynomial
since it can be written as $10+(2-2+2-2+2-2+2-2..........)$ and thus having infinitely many terms. Also from Wikipedia's definition a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
My expression of 10 also satisfying the definition but contains infinite terms
polynomials
edited Dec 27 '18 at 4:53
Key Flex
8,28261233
$endgroup$
|
show 6 more comments
$begingroup$
How 10 is a constant polynomial
since it can be written as $10+(2-2+2-2+2-2+2-2..........)$ and thus having infinitely many terms. Also from Wikipedia's definition a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
My expression of 10 also satisfying the definition but contains infinite terms
polynomials
edited Dec 27 '18 at 4:53
Key Flex
8,28261233
$endgroup$
1
$begingroup$
It can't be written that way, since the sum you've written doesn't converge. Also just because ten can be written as not a polynomial wouldn't mean that ten isn't still an integer and thus a constant polynomial with integer coefficients.
$endgroup$
– jgon
Dec 27 '18 at 3:53
3
$begingroup$
There are no non finite series involved in defining polynomials. Where is this coming from?
$endgroup$
– copper.hat
Dec 27 '18 at 4:02
2
$begingroup$
"5x+5x is also a polynomial and it also doesn't converge" What?
$endgroup$
– Noah Schweber
Dec 27 '18 at 4:06
1
$begingroup$
@user629353 "5x+5x is a polynomial and it doesn't converge" I don't think you know what "converge" means.
$endgroup$
– Noah Schweber
Dec 27 '18 at 4:10
2
$begingroup$
$$10=0x^3+0x^2+0x+10$$ $10$ is a polynomial
$endgroup$
– clathratus
Dec 27 '18 at 4:26
|
show 6 more comments
$begingroup$
How 10 is a constant polynomial
since it can be written as $10+(2-2+2-2+2-2+2-2..........)$ and thus having infinitely many terms. Also from Wikipedia's definition a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
My expression of 10 also satisfying the definition but contains infinite terms
polynomials
edited Dec 27 '18 at 4:53
Key Flex
8,28261233
$endgroup$
How 10 is a constant polynomial
since it can be written as $10+(2-2+2-2+2-2+2-2..........)$ and thus having infinitely many terms. Also from Wikipedia's definition a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
My expression of 10 also satisfying the definition but contains infinite terms
polynomials
polynomials
edited Dec 27 '18 at 4:53
Key Flex
8,28261233
edited Dec 27 '18 at 4:53
Key Flex
8,28261233
edited Dec 27 '18 at 4:53
Key Flex
8,28261233
edited Dec 27 '18 at 4:53
Key Flex
8,28261233
edited Dec 27 '18 at 4:53
Key Flex
8,28261233
8,28261233
asked Dec 27 '18 at 3:50
user629353
1
$begingroup$
It can't be written that way, since the sum you've written doesn't converge. Also just because ten can be written as not a polynomial wouldn't mean that ten isn't still an integer and thus a constant polynomial with integer coefficients.
$endgroup$
– jgon
Dec 27 '18 at 3:53
3
$begingroup$
There are no non finite series involved in defining polynomials. Where is this coming from?
$endgroup$
– copper.hat
Dec 27 '18 at 4:02
2
$begingroup$
"5x+5x is also a polynomial and it also doesn't converge" What?
$endgroup$
– Noah Schweber
Dec 27 '18 at 4:06
1
$begingroup$
@user629353 "5x+5x is a polynomial and it doesn't converge" I don't think you know what "converge" means.
$endgroup$
– Noah Schweber
Dec 27 '18 at 4:10
2
$begingroup$
$$10=0x^3+0x^2+0x+10$$ $10$ is a polynomial
$endgroup$
– clathratus
Dec 27 '18 at 4:26
|
show 6 more comments
1
$begingroup$
It can't be written that way, since the sum you've written doesn't converge. Also just because ten can be written as not a polynomial wouldn't mean that ten isn't still an integer and thus a constant polynomial with integer coefficients.
$endgroup$
– jgon
Dec 27 '18 at 3:53
3
$begingroup$
There are no non finite series involved in defining polynomials. Where is this coming from?
$endgroup$
– copper.hat
Dec 27 '18 at 4:02
2
$begingroup$
"5x+5x is also a polynomial and it also doesn't converge" What?
$endgroup$
– Noah Schweber
Dec 27 '18 at 4:06
1
$begingroup$
@user629353 "5x+5x is a polynomial and it doesn't converge" I don't think you know what "converge" means.
$endgroup$
– Noah Schweber
Dec 27 '18 at 4:10
2
$begingroup$
$$10=0x^3+0x^2+0x+10$$ $10$ is a polynomial
$endgroup$
– clathratus
Dec 27 '18 at 4:26
1
1
$begingroup$
It can't be written that way, since the sum you've written doesn't converge. Also just because ten can be written as not a polynomial wouldn't mean that ten isn't still an integer and thus a constant polynomial with integer coefficients.
$endgroup$
– jgon
Dec 27 '18 at 3:53
$begingroup$
It can't be written that way, since the sum you've written doesn't converge. Also just because ten can be written as not a polynomial wouldn't mean that ten isn't still an integer and thus a constant polynomial with integer coefficients.
$endgroup$
– jgon
Dec 27 '18 at 3:53
3
3
$begingroup$
There are no non finite series involved in defining polynomials. Where is this coming from?
$endgroup$
– copper.hat
Dec 27 '18 at 4:02
$begingroup$
There are no non finite series involved in defining polynomials. Where is this coming from?
$endgroup$
– copper.hat
Dec 27 '18 at 4:02
2
2
$begingroup$
"5x+5x is also a polynomial and it also doesn't converge" What?
$endgroup$
– Noah Schweber
Dec 27 '18 at 4:06
$begingroup$
"5x+5x is also a polynomial and it also doesn't converge" What?
$endgroup$
– Noah Schweber
Dec 27 '18 at 4:06
1
1
$begingroup$
@user629353 "5x+5x is a polynomial and it doesn't converge" I don't think you know what "converge" means.
$endgroup$
– Noah Schweber
Dec 27 '18 at 4:10
$begingroup$
@user629353 "5x+5x is a polynomial and it doesn't converge" I don't think you know what "converge" means.
$endgroup$
– Noah Schweber
Dec 27 '18 at 4:10
2
2
$begingroup$
$$10=0x^3+0x^2+0x+10$$ $10$ is a polynomial
$endgroup$
– clathratus
Dec 27 '18 at 4:26
$begingroup$
$$10=0x^3+0x^2+0x+10$$ $10$ is a polynomial
$endgroup$
– clathratus
Dec 27 '18 at 4:26
|
show 6 more comments
2 Answers
2
active
oldest
votes
$begingroup$
There are lots of silly ways to write $10$ (ignoring the fact that what you've written doesn't really mean anything). For example, $$10=5-sin(pi)+3!+e^{ipi}.$$ But none of this changes the fact that one of the ways to write $10$ is, well, "$10$" - and it's the fact that it can be written in such a way that makes it a polynomial. We don't care about the existence of other ways to write it. Similarly, an integer $a$ is even if $a$ can be written as $2cdot b$ for some integer $b$; we can write $12$ as both $2cdot 6$ and $3cdot 5-1-2!$, and the fact that the former works means that $12$ is even regardless of the silliness of the latter.
answered Dec 27 '18 at 4:10
Noah SchweberNoah Schweber
125k10150288
$endgroup$
1
$begingroup$
I hate to break it to you, but $3cdot 5-1-1=15-2=13 neq 12$
$endgroup$
– Mohammad Zuhair Khan
Dec 27 '18 at 4:41
$begingroup$
@MohammadZuhairKhan Bah ...
$endgroup$
– Noah Schweber
Dec 27 '18 at 16:09
add a comment |
$begingroup$
If we want to be very precise, we can use the precise definition of a polynomial. One definition is that a polynomial (with coefficients that are real numbers) is a sequence $(a_0, a_1, a_2, ldots)$ of real numbers such that $a_k = 0$ for all sufficiently large integers $k$.
By this definition, the number 10 is technically not a polynomial. However, people will often use the symbol 10 to denote the polynomial $(10,0,0,ldots)$. This is an example of a symbol being "overloaded", which happens sometimes in math. Hopefully the meaning will always be clear from the context.
answered Dec 27 '18 at 4:28
littleOlittleO
29.9k646109
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
There are lots of silly ways to write $10$ (ignoring the fact that what you've written doesn't really mean anything). For example, $$10=5-sin(pi)+3!+e^{ipi}.$$ But none of this changes the fact that one of the ways to write $10$ is, well, "$10$" - and it's the fact that it can be written in such a way that makes it a polynomial. We don't care about the existence of other ways to write it. Similarly, an integer $a$ is even if $a$ can be written as $2cdot b$ for some integer $b$; we can write $12$ as both $2cdot 6$ and $3cdot 5-1-2!$, and the fact that the former works means that $12$ is even regardless of the silliness of the latter.
answered Dec 27 '18 at 4:10
Noah SchweberNoah Schweber
125k10150288
$endgroup$
1
$begingroup$
I hate to break it to you, but $3cdot 5-1-1=15-2=13 neq 12$
$endgroup$
– Mohammad Zuhair Khan
Dec 27 '18 at 4:41
$begingroup$
@MohammadZuhairKhan Bah ...
$endgroup$
– Noah Schweber
Dec 27 '18 at 16:09
add a comment |
$begingroup$
There are lots of silly ways to write $10$ (ignoring the fact that what you've written doesn't really mean anything). For example, $$10=5-sin(pi)+3!+e^{ipi}.$$ But none of this changes the fact that one of the ways to write $10$ is, well, "$10$" - and it's the fact that it can be written in such a way that makes it a polynomial. We don't care about the existence of other ways to write it. Similarly, an integer $a$ is even if $a$ can be written as $2cdot b$ for some integer $b$; we can write $12$ as both $2cdot 6$ and $3cdot 5-1-2!$, and the fact that the former works means that $12$ is even regardless of the silliness of the latter.
answered Dec 27 '18 at 4:10
Noah SchweberNoah Schweber
125k10150288
$endgroup$
1
$begingroup$
I hate to break it to you, but $3cdot 5-1-1=15-2=13 neq 12$
$endgroup$
– Mohammad Zuhair Khan
Dec 27 '18 at 4:41
$begingroup$
@MohammadZuhairKhan Bah ...
$endgroup$
– Noah Schweber
Dec 27 '18 at 16:09
add a comment |
$begingroup$
There are lots of silly ways to write $10$ (ignoring the fact that what you've written doesn't really mean anything). For example, $$10=5-sin(pi)+3!+e^{ipi}.$$ But none of this changes the fact that one of the ways to write $10$ is, well, "$10$" - and it's the fact that it can be written in such a way that makes it a polynomial. We don't care about the existence of other ways to write it. Similarly, an integer $a$ is even if $a$ can be written as $2cdot b$ for some integer $b$; we can write $12$ as both $2cdot 6$ and $3cdot 5-1-2!$, and the fact that the former works means that $12$ is even regardless of the silliness of the latter.
answered Dec 27 '18 at 4:10
Noah SchweberNoah Schweber
125k10150288
$endgroup$
There are lots of silly ways to write $10$ (ignoring the fact that what you've written doesn't really mean anything). For example, $$10=5-sin(pi)+3!+e^{ipi}.$$ But none of this changes the fact that one of the ways to write $10$ is, well, "$10$" - and it's the fact that it can be written in such a way that makes it a polynomial. We don't care about the existence of other ways to write it. Similarly, an integer $a$ is even if $a$ can be written as $2cdot b$ for some integer $b$; we can write $12$ as both $2cdot 6$ and $3cdot 5-1-2!$, and the fact that the former works means that $12$ is even regardless of the silliness of the latter.
answered Dec 27 '18 at 4:10
Noah SchweberNoah Schweber
125k10150288
edited Dec 27 '18 at 16:09
answered Dec 27 '18 at 4:10
Noah SchweberNoah Schweber
125k10150288
answered Dec 27 '18 at 4:10
Noah SchweberNoah Schweber
125k10150288
answered Dec 27 '18 at 4:10
Noah SchweberNoah Schweber
125k10150288
125k10150288
1
$begingroup$
I hate to break it to you, but $3cdot 5-1-1=15-2=13 neq 12$
$endgroup$
– Mohammad Zuhair Khan
Dec 27 '18 at 4:41
$begingroup$
@MohammadZuhairKhan Bah ...
$endgroup$
– Noah Schweber
Dec 27 '18 at 16:09
add a comment |
1
$begingroup$
I hate to break it to you, but $3cdot 5-1-1=15-2=13 neq 12$
$endgroup$
– Mohammad Zuhair Khan
Dec 27 '18 at 4:41
$begingroup$
@MohammadZuhairKhan Bah ...
$endgroup$
– Noah Schweber
Dec 27 '18 at 16:09
1
1
$begingroup$
I hate to break it to you, but $3cdot 5-1-1=15-2=13 neq 12$
$endgroup$
– Mohammad Zuhair Khan
Dec 27 '18 at 4:41
$begingroup$
I hate to break it to you, but $3cdot 5-1-1=15-2=13 neq 12$
$endgroup$
– Mohammad Zuhair Khan
Dec 27 '18 at 4:41
$begingroup$
@MohammadZuhairKhan Bah ...
$endgroup$
– Noah Schweber
Dec 27 '18 at 16:09
$begingroup$
@MohammadZuhairKhan Bah ...
$endgroup$
– Noah Schweber
Dec 27 '18 at 16:09
add a comment |
$begingroup$
If we want to be very precise, we can use the precise definition of a polynomial. One definition is that a polynomial (with coefficients that are real numbers) is a sequence $(a_0, a_1, a_2, ldots)$ of real numbers such that $a_k = 0$ for all sufficiently large integers $k$.
By this definition, the number 10 is technically not a polynomial. However, people will often use the symbol 10 to denote the polynomial $(10,0,0,ldots)$. This is an example of a symbol being "overloaded", which happens sometimes in math. Hopefully the meaning will always be clear from the context.
answered Dec 27 '18 at 4:28
littleOlittleO
29.9k646109
$endgroup$
add a comment |
$begingroup$
If we want to be very precise, we can use the precise definition of a polynomial. One definition is that a polynomial (with coefficients that are real numbers) is a sequence $(a_0, a_1, a_2, ldots)$ of real numbers such that $a_k = 0$ for all sufficiently large integers $k$.
By this definition, the number 10 is technically not a polynomial. However, people will often use the symbol 10 to denote the polynomial $(10,0,0,ldots)$. This is an example of a symbol being "overloaded", which happens sometimes in math. Hopefully the meaning will always be clear from the context.
answered Dec 27 '18 at 4:28
littleOlittleO
29.9k646109
$endgroup$
add a comment |
$begingroup$
If we want to be very precise, we can use the precise definition of a polynomial. One definition is that a polynomial (with coefficients that are real numbers) is a sequence $(a_0, a_1, a_2, ldots)$ of real numbers such that $a_k = 0$ for all sufficiently large integers $k$.
By this definition, the number 10 is technically not a polynomial. However, people will often use the symbol 10 to denote the polynomial $(10,0,0,ldots)$. This is an example of a symbol being "overloaded", which happens sometimes in math. Hopefully the meaning will always be clear from the context.
answered Dec 27 '18 at 4:28
littleOlittleO
29.9k646109
$endgroup$
If we want to be very precise, we can use the precise definition of a polynomial. One definition is that a polynomial (with coefficients that are real numbers) is a sequence $(a_0, a_1, a_2, ldots)$ of real numbers such that $a_k = 0$ for all sufficiently large integers $k$.
By this definition, the number 10 is technically not a polynomial. However, people will often use the symbol 10 to denote the polynomial $(10,0,0,ldots)$. This is an example of a symbol being "overloaded", which happens sometimes in math. Hopefully the meaning will always be clear from the context.
answered Dec 27 '18 at 4:28
littleOlittleO
29.9k646109
answered Dec 27 '18 at 4:28
littleOlittleO
29.9k646109
answered Dec 27 '18 at 4:28
littleOlittleO
29.9k646109
answered Dec 27 '18 at 4:28
littleOlittleO
29.9k646109
29.9k646109
add a comment |
add a comment |
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$begingroup$
It can't be written that way, since the sum you've written doesn't converge. Also just because ten can be written as not a polynomial wouldn't mean that ten isn't still an integer and thus a constant polynomial with integer coefficients.
$endgroup$
– jgon
Dec 27 '18 at 3:53
3
$begingroup$
There are no non finite series involved in defining polynomials. Where is this coming from?
$endgroup$
– copper.hat
Dec 27 '18 at 4:02
2
$begingroup$
"5x+5x is also a polynomial and it also doesn't converge" What?
$endgroup$
– Noah Schweber
Dec 27 '18 at 4:06
1
$begingroup$
@user629353 "5x+5x is a polynomial and it doesn't converge" I don't think you know what "converge" means.
$endgroup$
– Noah Schweber
Dec 27 '18 at 4:10
2
$begingroup$
$$10=0x^3+0x^2+0x+10$$ $10$ is a polynomial
$endgroup$
– clathratus
Dec 27 '18 at 4:26