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Is the constant term a coefficient?
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I'm a baby boomer who was taught that the constant term of a polynomial is a coefficient, being the constant factor for the x^0 term.
That's not what's taught today.
Current text books are vague on their definition of coefficient, but, when they ask for a student to list all coefficient in a given polynomial, they do not include the constant term in their answer keys.
The Kahn Academy video on the topic does include the constant term as a coefficient, as does the related Wikipedia article.
However, all modern text books written to the Common Core that I have seen imply that the constant term is not a coefficient. A lot of web sites created by today's teachers who are teaching to the Common Core explicitly state that the constant term is not a coefficient.
If you have a polynomial of one variable, aren't the coefficients supposed to uniquely determine that polynomial? Aren't they all a computer program needs to know to work with polynomials? Without the constant term, you get the related polynomial that represents a vertical displacement of the original polynomial by the opposite of the constant term. But, you don't get the original polynomial.
Further, if we are talking about natural number constants, isn't each digit, including the ones' place, a coefficient of a power of 10? We don't segregate that digit by calling it something else; it's a digit similar to all the other digits.
Isn't mathematics supposed to be non-arbitrary and consistent?
I would like to see this taught as: after combining like terms, all constant factors, including the constant term, are coefficients. Two of the coefficients are special. The coefficient of the term with the largest exponent is the leading coefficient. The coefficient of the term with the smallest exponent (aka zero) is the constant term. That shouldn't be too much for eighth graders who were taught exponent properties, including the zero exponent property, in seventh grade.
Is this going to be yet another contributor to generational divide?
Thanks for your thoughts,
- Tom
solving-polynomials
asked 3 hours ago
Thomas Martin
163
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Thomas Martin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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add a comment |Â
up vote
3
down vote
favorite
I'm a baby boomer who was taught that the constant term of a polynomial is a coefficient, being the constant factor for the x^0 term.
That's not what's taught today.
Current text books are vague on their definition of coefficient, but, when they ask for a student to list all coefficient in a given polynomial, they do not include the constant term in their answer keys.
The Kahn Academy video on the topic does include the constant term as a coefficient, as does the related Wikipedia article.
However, all modern text books written to the Common Core that I have seen imply that the constant term is not a coefficient. A lot of web sites created by today's teachers who are teaching to the Common Core explicitly state that the constant term is not a coefficient.
If you have a polynomial of one variable, aren't the coefficients supposed to uniquely determine that polynomial? Aren't they all a computer program needs to know to work with polynomials? Without the constant term, you get the related polynomial that represents a vertical displacement of the original polynomial by the opposite of the constant term. But, you don't get the original polynomial.
Further, if we are talking about natural number constants, isn't each digit, including the ones' place, a coefficient of a power of 10? We don't segregate that digit by calling it something else; it's a digit similar to all the other digits.
Isn't mathematics supposed to be non-arbitrary and consistent?
I would like to see this taught as: after combining like terms, all constant factors, including the constant term, are coefficients. Two of the coefficients are special. The coefficient of the term with the largest exponent is the leading coefficient. The coefficient of the term with the smallest exponent (aka zero) is the constant term. That shouldn't be too much for eighth graders who were taught exponent properties, including the zero exponent property, in seventh grade.
Is this going to be yet another contributor to generational divide?
Thanks for your thoughts,
- Tom
solving-polynomials
asked 3 hours ago
Thomas Martin
163
New contributor
Thomas Martin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I don't know why the constant term is not considered a coefficient, but perhaps it has something to do with the $x^0$ term---note that some mathematicians consider $0^0$ to be undefined. If so, and if we define a polynomial as containing an $x^0$ term, then all polynomials are undefined at $x=0$.
– Joel Reyes Noche
23 mins ago
Our algebra book uses the phrase "numerical coefficient" to describe what is often called a coefficient (e.g. $5$ in the expression $5x^2y$), but discusses general coefficients as belonging to particular variables. For example, in the expression $5x^2y$, it would say the coefficient of $y$ is $5x^2$, and the coefficient of $x^2$ is $5y$. In general, it describes a coefficient of a variable (or product/quotient/power/etc. of variables) as being the other factors attached to it.
– Nick C
15 mins ago
@JoelReyesNoche: That just gives another reason why $0^0$ has to be defined to be 1.
– Daniel Hast
11 mins ago
The limit of x^x as x -> 0 is 1. There's a great youtube video on this: youtube.com/watch?v=r0_mi8ngNnM.
– Thomas Martin
4 mins ago
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
I'm a baby boomer who was taught that the constant term of a polynomial is a coefficient, being the constant factor for the x^0 term.
That's not what's taught today.
Current text books are vague on their definition of coefficient, but, when they ask for a student to list all coefficient in a given polynomial, they do not include the constant term in their answer keys.
The Kahn Academy video on the topic does include the constant term as a coefficient, as does the related Wikipedia article.
However, all modern text books written to the Common Core that I have seen imply that the constant term is not a coefficient. A lot of web sites created by today's teachers who are teaching to the Common Core explicitly state that the constant term is not a coefficient.
If you have a polynomial of one variable, aren't the coefficients supposed to uniquely determine that polynomial? Aren't they all a computer program needs to know to work with polynomials? Without the constant term, you get the related polynomial that represents a vertical displacement of the original polynomial by the opposite of the constant term. But, you don't get the original polynomial.
Further, if we are talking about natural number constants, isn't each digit, including the ones' place, a coefficient of a power of 10? We don't segregate that digit by calling it something else; it's a digit similar to all the other digits.
Isn't mathematics supposed to be non-arbitrary and consistent?
I would like to see this taught as: after combining like terms, all constant factors, including the constant term, are coefficients. Two of the coefficients are special. The coefficient of the term with the largest exponent is the leading coefficient. The coefficient of the term with the smallest exponent (aka zero) is the constant term. That shouldn't be too much for eighth graders who were taught exponent properties, including the zero exponent property, in seventh grade.
Is this going to be yet another contributor to generational divide?
Thanks for your thoughts,
- Tom
solving-polynomials
asked 3 hours ago
Thomas Martin
163
New contributor
Thomas Martin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I'm a baby boomer who was taught that the constant term of a polynomial is a coefficient, being the constant factor for the x^0 term.
That's not what's taught today.
Current text books are vague on their definition of coefficient, but, when they ask for a student to list all coefficient in a given polynomial, they do not include the constant term in their answer keys.
The Kahn Academy video on the topic does include the constant term as a coefficient, as does the related Wikipedia article.
However, all modern text books written to the Common Core that I have seen imply that the constant term is not a coefficient. A lot of web sites created by today's teachers who are teaching to the Common Core explicitly state that the constant term is not a coefficient.
If you have a polynomial of one variable, aren't the coefficients supposed to uniquely determine that polynomial? Aren't they all a computer program needs to know to work with polynomials? Without the constant term, you get the related polynomial that represents a vertical displacement of the original polynomial by the opposite of the constant term. But, you don't get the original polynomial.
Further, if we are talking about natural number constants, isn't each digit, including the ones' place, a coefficient of a power of 10? We don't segregate that digit by calling it something else; it's a digit similar to all the other digits.
Isn't mathematics supposed to be non-arbitrary and consistent?
I would like to see this taught as: after combining like terms, all constant factors, including the constant term, are coefficients. Two of the coefficients are special. The coefficient of the term with the largest exponent is the leading coefficient. The coefficient of the term with the smallest exponent (aka zero) is the constant term. That shouldn't be too much for eighth graders who were taught exponent properties, including the zero exponent property, in seventh grade.
Is this going to be yet another contributor to generational divide?
Thanks for your thoughts,
- Tom
solving-polynomials
solving-polynomials
asked 3 hours ago
Thomas Martin
163
New contributor
Thomas Martin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 3 hours ago
Thomas Martin
163
New contributor
Thomas Martin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 3 hours ago
Thomas Martin
163
New contributor
Thomas Martin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 3 hours ago
Thomas Martin
163
asked 3 hours ago
Thomas Martin
163
163
New contributor
Thomas Martin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Thomas Martin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Thomas Martin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I don't know why the constant term is not considered a coefficient, but perhaps it has something to do with the $x^0$ term---note that some mathematicians consider $0^0$ to be undefined. If so, and if we define a polynomial as containing an $x^0$ term, then all polynomials are undefined at $x=0$.
– Joel Reyes Noche
23 mins ago
Our algebra book uses the phrase "numerical coefficient" to describe what is often called a coefficient (e.g. $5$ in the expression $5x^2y$), but discusses general coefficients as belonging to particular variables. For example, in the expression $5x^2y$, it would say the coefficient of $y$ is $5x^2$, and the coefficient of $x^2$ is $5y$. In general, it describes a coefficient of a variable (or product/quotient/power/etc. of variables) as being the other factors attached to it.
– Nick C
15 mins ago
@JoelReyesNoche: That just gives another reason why $0^0$ has to be defined to be 1.
– Daniel Hast
11 mins ago
The limit of x^x as x -> 0 is 1. There's a great youtube video on this: youtube.com/watch?v=r0_mi8ngNnM.
– Thomas Martin
4 mins ago
add a comment |Â
I don't know why the constant term is not considered a coefficient, but perhaps it has something to do with the $x^0$ term---note that some mathematicians consider $0^0$ to be undefined. If so, and if we define a polynomial as containing an $x^0$ term, then all polynomials are undefined at $x=0$.
– Joel Reyes Noche
23 mins ago
Our algebra book uses the phrase "numerical coefficient" to describe what is often called a coefficient (e.g. $5$ in the expression $5x^2y$), but discusses general coefficients as belonging to particular variables. For example, in the expression $5x^2y$, it would say the coefficient of $y$ is $5x^2$, and the coefficient of $x^2$ is $5y$. In general, it describes a coefficient of a variable (or product/quotient/power/etc. of variables) as being the other factors attached to it.
– Nick C
15 mins ago
@JoelReyesNoche: That just gives another reason why $0^0$ has to be defined to be 1.
– Daniel Hast
11 mins ago
The limit of x^x as x -> 0 is 1. There's a great youtube video on this: youtube.com/watch?v=r0_mi8ngNnM.
– Thomas Martin
4 mins ago
I don't know why the constant term is not considered a coefficient, but perhaps it has something to do with the $x^0$ term---note that some mathematicians consider $0^0$ to be undefined. If so, and if we define a polynomial as containing an $x^0$ term, then all polynomials are undefined at $x=0$.
– Joel Reyes Noche
23 mins ago
I don't know why the constant term is not considered a coefficient, but perhaps it has something to do with the $x^0$ term---note that some mathematicians consider $0^0$ to be undefined. If so, and if we define a polynomial as containing an $x^0$ term, then all polynomials are undefined at $x=0$.
– Joel Reyes Noche
23 mins ago
Our algebra book uses the phrase "numerical coefficient" to describe what is often called a coefficient (e.g. $5$ in the expression $5x^2y$), but discusses general coefficients as belonging to particular variables. For example, in the expression $5x^2y$, it would say the coefficient of $y$ is $5x^2$, and the coefficient of $x^2$ is $5y$. In general, it describes a coefficient of a variable (or product/quotient/power/etc. of variables) as being the other factors attached to it.
– Nick C
15 mins ago
Our algebra book uses the phrase "numerical coefficient" to describe what is often called a coefficient (e.g. $5$ in the expression $5x^2y$), but discusses general coefficients as belonging to particular variables. For example, in the expression $5x^2y$, it would say the coefficient of $y$ is $5x^2$, and the coefficient of $x^2$ is $5y$. In general, it describes a coefficient of a variable (or product/quotient/power/etc. of variables) as being the other factors attached to it.
– Nick C
15 mins ago
@JoelReyesNoche: That just gives another reason why $0^0$ has to be defined to be 1.
– Daniel Hast
11 mins ago
@JoelReyesNoche: That just gives another reason why $0^0$ has to be defined to be 1.
– Daniel Hast
11 mins ago
The limit of x^x as x -> 0 is 1. There's a great youtube video on this: youtube.com/watch?v=r0_mi8ngNnM.
– Thomas Martin
4 mins ago
The limit of x^x as x -> 0 is 1. There's a great youtube video on this: youtube.com/watch?v=r0_mi8ngNnM.
– Thomas Martin
4 mins ago
add a comment |Â
2 Answers
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up vote
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Your question is kind of two parts: one about a convention
Is the constant term a "coefficient"
and one about a philosophy, which I perhaps find to be a more important question to answer.
Isn't mathematics supposed to be non-arbitrary and consistent?
Different fields of mathematics have different conventions; this can lead to some mathematicians using "sigma" for "standard deviation" in statistics and other mathematicians using "sigma" for a map defining a simplex in algebraic topology, and others using it to represent a group element in a permutation group. Even more egregiously, some people consider "Natural numbers" to include zero, and others consider "natural numbers" to be only positive integers. Even in higher mathematics, the definition of a word, or the statement of a theorem, may differ slightly between two textbooks. What is important is that 1) their definitions are clearly stated when they are introduced, and 2) that when a single mathematician uses the word in one body of work, that they are consistent. We are capable of handling the fact that different texts have slightly different definitions of what, exactly, a "coefficient" is, because we know how to interpret mathematical definitions and classify objects according to whether or not they fit that definition. That skill -- determining whether an object satisfies certain properties, or classifying mathematical objects based on definition, and recognizing properties of objects satisfying a particular definition -- is often an unstated goal in teaching mathematical vocabulary.
In that sense, I do not think that the definition of "coefficient" needs a universal definition for schoolchildren. When we teach students the meaning of mathematical words, the intent is often to build in them the ability to think about different "parts" of what they are learning and clearly articulate their thinking mathematically. In this sense, teachers and students being consistent with themselves and their textbook matters. However, being consistent across different states, or even different school districts, seems unimportant. Ultimately, they may grow up and realize that what their school called a coefficient differs slightly from what their college called a coefficient. This realization may help them understand the importance of an "operational definition" in many contexts!
answered 2 hours ago
Opal E
1,225520
Thanks for your thoughtful reply! I guess my concern is that communication built upon very basic/low level definitions benefits greatly from consistency at that basic level. For example, I would love to be able to say, "The coefficients of a polynomial, together with their places in the polynomial, uniquely determine the polynomial, similar to the way the digits in natural number together with their places in the number uniquely determine the number." If some students are taught that the constant term is not a coefficient, while others are taught that they are, what do I say?
– Thomas Martin
1 hour ago
add a comment |Â
up vote
1
down vote
It sometimes happens that slightly different definitions of the same word each have advantages and disadvantages. In such cases, I wouldn't be surprised to see some people supporting one definition and other people supporting a different definition. In the case at hand, though, I can't think of any advantages for defining "coefficient" to exclude the constant term. It seems to me that, if one adopts this definition, one will repeatedly have to say "coefficients and constant term". In addition to your example, "a polyomial is determined by its coefficients and its constant term", we'd have "to multiply a polynomial by $7$, multiply all its coefficients and its constant term by $7$" and "to add two polynomials, add the corresponding coefficients and add the constant terms." All of these become simpler and clearer if we just include the constant terms among the coefficients instead of treating them separately. (And I refuse to even think about multiplying two polynomials if I'm expected to think about constant terms separately from the other coefficients.)
answered 18 mins ago
Andreas Blass
2,11211211
Thanks! Oh, and I fervently believe the natural numbers do not include 0. The natural numbers are so named because they come naturally to us (at least 1, 2, and 3 do). When we count, the first number we say is 1. We now know that before we say 1, we have 0--we start with nothing. But we start our count with 1. Formalizing 0 took centuries. I'm ok with educators creating the whole numbers to be the natural numbers plus 0. Educators can use that do clarify counting and distance on the number line without trying to redefine the foundational natural numbers.
– Thomas Martin
7 mins ago
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
Your question is kind of two parts: one about a convention
Is the constant term a "coefficient"
and one about a philosophy, which I perhaps find to be a more important question to answer.
Isn't mathematics supposed to be non-arbitrary and consistent?
Different fields of mathematics have different conventions; this can lead to some mathematicians using "sigma" for "standard deviation" in statistics and other mathematicians using "sigma" for a map defining a simplex in algebraic topology, and others using it to represent a group element in a permutation group. Even more egregiously, some people consider "Natural numbers" to include zero, and others consider "natural numbers" to be only positive integers. Even in higher mathematics, the definition of a word, or the statement of a theorem, may differ slightly between two textbooks. What is important is that 1) their definitions are clearly stated when they are introduced, and 2) that when a single mathematician uses the word in one body of work, that they are consistent. We are capable of handling the fact that different texts have slightly different definitions of what, exactly, a "coefficient" is, because we know how to interpret mathematical definitions and classify objects according to whether or not they fit that definition. That skill -- determining whether an object satisfies certain properties, or classifying mathematical objects based on definition, and recognizing properties of objects satisfying a particular definition -- is often an unstated goal in teaching mathematical vocabulary.
In that sense, I do not think that the definition of "coefficient" needs a universal definition for schoolchildren. When we teach students the meaning of mathematical words, the intent is often to build in them the ability to think about different "parts" of what they are learning and clearly articulate their thinking mathematically. In this sense, teachers and students being consistent with themselves and their textbook matters. However, being consistent across different states, or even different school districts, seems unimportant. Ultimately, they may grow up and realize that what their school called a coefficient differs slightly from what their college called a coefficient. This realization may help them understand the importance of an "operational definition" in many contexts!
answered 2 hours ago
Opal E
1,225520
Thanks for your thoughtful reply! I guess my concern is that communication built upon very basic/low level definitions benefits greatly from consistency at that basic level. For example, I would love to be able to say, "The coefficients of a polynomial, together with their places in the polynomial, uniquely determine the polynomial, similar to the way the digits in natural number together with their places in the number uniquely determine the number." If some students are taught that the constant term is not a coefficient, while others are taught that they are, what do I say?
– Thomas Martin
1 hour ago
add a comment |Â
up vote
3
down vote
Your question is kind of two parts: one about a convention
Is the constant term a "coefficient"
and one about a philosophy, which I perhaps find to be a more important question to answer.
Isn't mathematics supposed to be non-arbitrary and consistent?
Different fields of mathematics have different conventions; this can lead to some mathematicians using "sigma" for "standard deviation" in statistics and other mathematicians using "sigma" for a map defining a simplex in algebraic topology, and others using it to represent a group element in a permutation group. Even more egregiously, some people consider "Natural numbers" to include zero, and others consider "natural numbers" to be only positive integers. Even in higher mathematics, the definition of a word, or the statement of a theorem, may differ slightly between two textbooks. What is important is that 1) their definitions are clearly stated when they are introduced, and 2) that when a single mathematician uses the word in one body of work, that they are consistent. We are capable of handling the fact that different texts have slightly different definitions of what, exactly, a "coefficient" is, because we know how to interpret mathematical definitions and classify objects according to whether or not they fit that definition. That skill -- determining whether an object satisfies certain properties, or classifying mathematical objects based on definition, and recognizing properties of objects satisfying a particular definition -- is often an unstated goal in teaching mathematical vocabulary.
In that sense, I do not think that the definition of "coefficient" needs a universal definition for schoolchildren. When we teach students the meaning of mathematical words, the intent is often to build in them the ability to think about different "parts" of what they are learning and clearly articulate their thinking mathematically. In this sense, teachers and students being consistent with themselves and their textbook matters. However, being consistent across different states, or even different school districts, seems unimportant. Ultimately, they may grow up and realize that what their school called a coefficient differs slightly from what their college called a coefficient. This realization may help them understand the importance of an "operational definition" in many contexts!
answered 2 hours ago
Opal E
1,225520
Thanks for your thoughtful reply! I guess my concern is that communication built upon very basic/low level definitions benefits greatly from consistency at that basic level. For example, I would love to be able to say, "The coefficients of a polynomial, together with their places in the polynomial, uniquely determine the polynomial, similar to the way the digits in natural number together with their places in the number uniquely determine the number." If some students are taught that the constant term is not a coefficient, while others are taught that they are, what do I say?
– Thomas Martin
1 hour ago
add a comment |Â
up vote
3
down vote
up vote
3
down vote
Your question is kind of two parts: one about a convention
Is the constant term a "coefficient"
and one about a philosophy, which I perhaps find to be a more important question to answer.
Isn't mathematics supposed to be non-arbitrary and consistent?
Different fields of mathematics have different conventions; this can lead to some mathematicians using "sigma" for "standard deviation" in statistics and other mathematicians using "sigma" for a map defining a simplex in algebraic topology, and others using it to represent a group element in a permutation group. Even more egregiously, some people consider "Natural numbers" to include zero, and others consider "natural numbers" to be only positive integers. Even in higher mathematics, the definition of a word, or the statement of a theorem, may differ slightly between two textbooks. What is important is that 1) their definitions are clearly stated when they are introduced, and 2) that when a single mathematician uses the word in one body of work, that they are consistent. We are capable of handling the fact that different texts have slightly different definitions of what, exactly, a "coefficient" is, because we know how to interpret mathematical definitions and classify objects according to whether or not they fit that definition. That skill -- determining whether an object satisfies certain properties, or classifying mathematical objects based on definition, and recognizing properties of objects satisfying a particular definition -- is often an unstated goal in teaching mathematical vocabulary.
In that sense, I do not think that the definition of "coefficient" needs a universal definition for schoolchildren. When we teach students the meaning of mathematical words, the intent is often to build in them the ability to think about different "parts" of what they are learning and clearly articulate their thinking mathematically. In this sense, teachers and students being consistent with themselves and their textbook matters. However, being consistent across different states, or even different school districts, seems unimportant. Ultimately, they may grow up and realize that what their school called a coefficient differs slightly from what their college called a coefficient. This realization may help them understand the importance of an "operational definition" in many contexts!
answered 2 hours ago
Opal E
1,225520
Your question is kind of two parts: one about a convention
Is the constant term a "coefficient"
and one about a philosophy, which I perhaps find to be a more important question to answer.
Isn't mathematics supposed to be non-arbitrary and consistent?
Different fields of mathematics have different conventions; this can lead to some mathematicians using "sigma" for "standard deviation" in statistics and other mathematicians using "sigma" for a map defining a simplex in algebraic topology, and others using it to represent a group element in a permutation group. Even more egregiously, some people consider "Natural numbers" to include zero, and others consider "natural numbers" to be only positive integers. Even in higher mathematics, the definition of a word, or the statement of a theorem, may differ slightly between two textbooks. What is important is that 1) their definitions are clearly stated when they are introduced, and 2) that when a single mathematician uses the word in one body of work, that they are consistent. We are capable of handling the fact that different texts have slightly different definitions of what, exactly, a "coefficient" is, because we know how to interpret mathematical definitions and classify objects according to whether or not they fit that definition. That skill -- determining whether an object satisfies certain properties, or classifying mathematical objects based on definition, and recognizing properties of objects satisfying a particular definition -- is often an unstated goal in teaching mathematical vocabulary.
In that sense, I do not think that the definition of "coefficient" needs a universal definition for schoolchildren. When we teach students the meaning of mathematical words, the intent is often to build in them the ability to think about different "parts" of what they are learning and clearly articulate their thinking mathematically. In this sense, teachers and students being consistent with themselves and their textbook matters. However, being consistent across different states, or even different school districts, seems unimportant. Ultimately, they may grow up and realize that what their school called a coefficient differs slightly from what their college called a coefficient. This realization may help them understand the importance of an "operational definition" in many contexts!
answered 2 hours ago
Opal E
1,225520
answered 2 hours ago
Opal E
1,225520
answered 2 hours ago
Opal E
1,225520
answered 2 hours ago
Opal E
1,225520
1,225520
Thanks for your thoughtful reply! I guess my concern is that communication built upon very basic/low level definitions benefits greatly from consistency at that basic level. For example, I would love to be able to say, "The coefficients of a polynomial, together with their places in the polynomial, uniquely determine the polynomial, similar to the way the digits in natural number together with their places in the number uniquely determine the number." If some students are taught that the constant term is not a coefficient, while others are taught that they are, what do I say?
– Thomas Martin
1 hour ago
add a comment |Â
Thanks for your thoughtful reply! I guess my concern is that communication built upon very basic/low level definitions benefits greatly from consistency at that basic level. For example, I would love to be able to say, "The coefficients of a polynomial, together with their places in the polynomial, uniquely determine the polynomial, similar to the way the digits in natural number together with their places in the number uniquely determine the number." If some students are taught that the constant term is not a coefficient, while others are taught that they are, what do I say?
– Thomas Martin
1 hour ago
Thanks for your thoughtful reply! I guess my concern is that communication built upon very basic/low level definitions benefits greatly from consistency at that basic level. For example, I would love to be able to say, "The coefficients of a polynomial, together with their places in the polynomial, uniquely determine the polynomial, similar to the way the digits in natural number together with their places in the number uniquely determine the number." If some students are taught that the constant term is not a coefficient, while others are taught that they are, what do I say?
– Thomas Martin
1 hour ago
Thanks for your thoughtful reply! I guess my concern is that communication built upon very basic/low level definitions benefits greatly from consistency at that basic level. For example, I would love to be able to say, "The coefficients of a polynomial, together with their places in the polynomial, uniquely determine the polynomial, similar to the way the digits in natural number together with their places in the number uniquely determine the number." If some students are taught that the constant term is not a coefficient, while others are taught that they are, what do I say?
– Thomas Martin
1 hour ago
add a comment |Â
up vote
1
down vote
It sometimes happens that slightly different definitions of the same word each have advantages and disadvantages. In such cases, I wouldn't be surprised to see some people supporting one definition and other people supporting a different definition. In the case at hand, though, I can't think of any advantages for defining "coefficient" to exclude the constant term. It seems to me that, if one adopts this definition, one will repeatedly have to say "coefficients and constant term". In addition to your example, "a polyomial is determined by its coefficients and its constant term", we'd have "to multiply a polynomial by $7$, multiply all its coefficients and its constant term by $7$" and "to add two polynomials, add the corresponding coefficients and add the constant terms." All of these become simpler and clearer if we just include the constant terms among the coefficients instead of treating them separately. (And I refuse to even think about multiplying two polynomials if I'm expected to think about constant terms separately from the other coefficients.)
answered 18 mins ago
Andreas Blass
2,11211211
Thanks! Oh, and I fervently believe the natural numbers do not include 0. The natural numbers are so named because they come naturally to us (at least 1, 2, and 3 do). When we count, the first number we say is 1. We now know that before we say 1, we have 0--we start with nothing. But we start our count with 1. Formalizing 0 took centuries. I'm ok with educators creating the whole numbers to be the natural numbers plus 0. Educators can use that do clarify counting and distance on the number line without trying to redefine the foundational natural numbers.
– Thomas Martin
7 mins ago
add a comment |Â
up vote
1
down vote
It sometimes happens that slightly different definitions of the same word each have advantages and disadvantages. In such cases, I wouldn't be surprised to see some people supporting one definition and other people supporting a different definition. In the case at hand, though, I can't think of any advantages for defining "coefficient" to exclude the constant term. It seems to me that, if one adopts this definition, one will repeatedly have to say "coefficients and constant term". In addition to your example, "a polyomial is determined by its coefficients and its constant term", we'd have "to multiply a polynomial by $7$, multiply all its coefficients and its constant term by $7$" and "to add two polynomials, add the corresponding coefficients and add the constant terms." All of these become simpler and clearer if we just include the constant terms among the coefficients instead of treating them separately. (And I refuse to even think about multiplying two polynomials if I'm expected to think about constant terms separately from the other coefficients.)
answered 18 mins ago
Andreas Blass
2,11211211
Thanks! Oh, and I fervently believe the natural numbers do not include 0. The natural numbers are so named because they come naturally to us (at least 1, 2, and 3 do). When we count, the first number we say is 1. We now know that before we say 1, we have 0--we start with nothing. But we start our count with 1. Formalizing 0 took centuries. I'm ok with educators creating the whole numbers to be the natural numbers plus 0. Educators can use that do clarify counting and distance on the number line without trying to redefine the foundational natural numbers.
– Thomas Martin
7 mins ago
add a comment |Â
up vote
1
down vote
up vote
1
down vote
It sometimes happens that slightly different definitions of the same word each have advantages and disadvantages. In such cases, I wouldn't be surprised to see some people supporting one definition and other people supporting a different definition. In the case at hand, though, I can't think of any advantages for defining "coefficient" to exclude the constant term. It seems to me that, if one adopts this definition, one will repeatedly have to say "coefficients and constant term". In addition to your example, "a polyomial is determined by its coefficients and its constant term", we'd have "to multiply a polynomial by $7$, multiply all its coefficients and its constant term by $7$" and "to add two polynomials, add the corresponding coefficients and add the constant terms." All of these become simpler and clearer if we just include the constant terms among the coefficients instead of treating them separately. (And I refuse to even think about multiplying two polynomials if I'm expected to think about constant terms separately from the other coefficients.)
answered 18 mins ago
Andreas Blass
2,11211211
It sometimes happens that slightly different definitions of the same word each have advantages and disadvantages. In such cases, I wouldn't be surprised to see some people supporting one definition and other people supporting a different definition. In the case at hand, though, I can't think of any advantages for defining "coefficient" to exclude the constant term. It seems to me that, if one adopts this definition, one will repeatedly have to say "coefficients and constant term". In addition to your example, "a polyomial is determined by its coefficients and its constant term", we'd have "to multiply a polynomial by $7$, multiply all its coefficients and its constant term by $7$" and "to add two polynomials, add the corresponding coefficients and add the constant terms." All of these become simpler and clearer if we just include the constant terms among the coefficients instead of treating them separately. (And I refuse to even think about multiplying two polynomials if I'm expected to think about constant terms separately from the other coefficients.)
answered 18 mins ago
Andreas Blass
2,11211211
answered 18 mins ago
Andreas Blass
2,11211211
answered 18 mins ago
Andreas Blass
2,11211211
answered 18 mins ago
Andreas Blass
2,11211211
2,11211211
Thanks! Oh, and I fervently believe the natural numbers do not include 0. The natural numbers are so named because they come naturally to us (at least 1, 2, and 3 do). When we count, the first number we say is 1. We now know that before we say 1, we have 0--we start with nothing. But we start our count with 1. Formalizing 0 took centuries. I'm ok with educators creating the whole numbers to be the natural numbers plus 0. Educators can use that do clarify counting and distance on the number line without trying to redefine the foundational natural numbers.
– Thomas Martin
7 mins ago
add a comment |Â
Thanks! Oh, and I fervently believe the natural numbers do not include 0. The natural numbers are so named because they come naturally to us (at least 1, 2, and 3 do). When we count, the first number we say is 1. We now know that before we say 1, we have 0--we start with nothing. But we start our count with 1. Formalizing 0 took centuries. I'm ok with educators creating the whole numbers to be the natural numbers plus 0. Educators can use that do clarify counting and distance on the number line without trying to redefine the foundational natural numbers.
– Thomas Martin
7 mins ago
Thanks! Oh, and I fervently believe the natural numbers do not include 0. The natural numbers are so named because they come naturally to us (at least 1, 2, and 3 do). When we count, the first number we say is 1. We now know that before we say 1, we have 0--we start with nothing. But we start our count with 1. Formalizing 0 took centuries. I'm ok with educators creating the whole numbers to be the natural numbers plus 0. Educators can use that do clarify counting and distance on the number line without trying to redefine the foundational natural numbers.
– Thomas Martin
7 mins ago
Thanks! Oh, and I fervently believe the natural numbers do not include 0. The natural numbers are so named because they come naturally to us (at least 1, 2, and 3 do). When we count, the first number we say is 1. We now know that before we say 1, we have 0--we start with nothing. But we start our count with 1. Formalizing 0 took centuries. I'm ok with educators creating the whole numbers to be the natural numbers plus 0. Educators can use that do clarify counting and distance on the number line without trying to redefine the foundational natural numbers.
– Thomas Martin
7 mins ago
add a comment |Â
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I don't know why the constant term is not considered a coefficient, but perhaps it has something to do with the $x^0$ term---note that some mathematicians consider $0^0$ to be undefined. If so, and if we define a polynomial as containing an $x^0$ term, then all polynomials are undefined at $x=0$.
– Joel Reyes Noche
23 mins ago
Our algebra book uses the phrase "numerical coefficient" to describe what is often called a coefficient (e.g. $5$ in the expression $5x^2y$), but discusses general coefficients as belonging to particular variables. For example, in the expression $5x^2y$, it would say the coefficient of $y$ is $5x^2$, and the coefficient of $x^2$ is $5y$. In general, it describes a coefficient of a variable (or product/quotient/power/etc. of variables) as being the other factors attached to it.
– Nick C
15 mins ago
@JoelReyesNoche: That just gives another reason why $0^0$ has to be defined to be 1.
– Daniel Hast
11 mins ago
The limit of x^x as x -> 0 is 1. There's a great youtube video on this: youtube.com/watch?v=r0_mi8ngNnM.
– Thomas Martin
4 mins ago