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The terminology for an excluded solution
What is the proper verb for “doing” an integral?Using terminology for the different concepts of rational numberWhere does the word “roots” come from when talking about zerosDoes “factor” mean simply the multiplication (of any functions, numbers etc)Correct pronunciation of 'xth' (and workarounds for those who find it a tongue-twister)What is the term for the marks used to show congruence in geometric figures?Terminology: degree of coefficient?Why isn't the term *inequation* widely used in english?What is the best term for “probability measure” in an undergrad introduction to probability course?What is the value in creating distinguishing terminology between the $x$, $y$, and $(x, y)$ values of a possible point of extremum?
$begingroup$
I am not a native math teacher. I have a question related to a terminology when solving an algebraic equation.
Assume that we are solving some complicated equation like $x^{3}-sqrt{1-x^{2}}-frac{1}{x+1}=2$. After doing a bunch of algebraic manipulation, we come up with some finite possible solutions: $x=1$, $x=2$,.... Now, because I see that $x=1$ is not a solution (for example, by inserting into the equation to check). Then I might say that " $x=1$ does not satisfy the equation, so it is not a solution.". However I don't like this sentence. I would prefer to use " $x=1$ is excluded by not satisfying the equation". Is it ok with this?
Thanks.
terminology
asked 14 hours ago
AhmedAhmed
211
New contributor
Ahmed is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
I am not a native math teacher. I have a question related to a terminology when solving an algebraic equation.
Assume that we are solving some complicated equation like $x^{3}-sqrt{1-x^{2}}-frac{1}{x+1}=2$. After doing a bunch of algebraic manipulation, we come up with some finite possible solutions: $x=1$, $x=2$,.... Now, because I see that $x=1$ is not a solution (for example, by inserting into the equation to check). Then I might say that " $x=1$ does not satisfy the equation, so it is not a solution.". However I don't like this sentence. I would prefer to use " $x=1$ is excluded by not satisfying the equation". Is it ok with this?
Thanks.
terminology
asked 14 hours ago
AhmedAhmed
211
New contributor
Ahmed is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
3
$begingroup$
You do see the term "extraneous solution" sometimes.
$endgroup$
– Gerald Edgar
14 hours ago
$begingroup$
I saw this term couple of times but it appears to be confusing to me.
$endgroup$
– Ahmed
14 hours ago
add a comment |
$begingroup$
I am not a native math teacher. I have a question related to a terminology when solving an algebraic equation.
Assume that we are solving some complicated equation like $x^{3}-sqrt{1-x^{2}}-frac{1}{x+1}=2$. After doing a bunch of algebraic manipulation, we come up with some finite possible solutions: $x=1$, $x=2$,.... Now, because I see that $x=1$ is not a solution (for example, by inserting into the equation to check). Then I might say that " $x=1$ does not satisfy the equation, so it is not a solution.". However I don't like this sentence. I would prefer to use " $x=1$ is excluded by not satisfying the equation". Is it ok with this?
Thanks.
terminology
asked 14 hours ago
AhmedAhmed
211
New contributor
Ahmed is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
I am not a native math teacher. I have a question related to a terminology when solving an algebraic equation.
Assume that we are solving some complicated equation like $x^{3}-sqrt{1-x^{2}}-frac{1}{x+1}=2$. After doing a bunch of algebraic manipulation, we come up with some finite possible solutions: $x=1$, $x=2$,.... Now, because I see that $x=1$ is not a solution (for example, by inserting into the equation to check). Then I might say that " $x=1$ does not satisfy the equation, so it is not a solution.". However I don't like this sentence. I would prefer to use " $x=1$ is excluded by not satisfying the equation". Is it ok with this?
Thanks.
terminology
terminology
asked 14 hours ago
AhmedAhmed
211
New contributor
Ahmed is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 14 hours ago
AhmedAhmed
211
New contributor
Ahmed is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 14 hours ago
AhmedAhmed
211
New contributor
Ahmed is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 14 hours ago
AhmedAhmed
211
asked 14 hours ago
AhmedAhmed
211
211
New contributor
Ahmed is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Ahmed is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Ahmed is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
3
$begingroup$
You do see the term "extraneous solution" sometimes.
$endgroup$
– Gerald Edgar
14 hours ago
$begingroup$
I saw this term couple of times but it appears to be confusing to me.
$endgroup$
– Ahmed
14 hours ago
add a comment |
3
$begingroup$
You do see the term "extraneous solution" sometimes.
$endgroup$
– Gerald Edgar
14 hours ago
$begingroup$
I saw this term couple of times but it appears to be confusing to me.
$endgroup$
– Ahmed
14 hours ago
3
3
$begingroup$
You do see the term "extraneous solution" sometimes.
$endgroup$
– Gerald Edgar
14 hours ago
$begingroup$
You do see the term "extraneous solution" sometimes.
$endgroup$
– Gerald Edgar
14 hours ago
$begingroup$
I saw this term couple of times but it appears to be confusing to me.
$endgroup$
– Ahmed
14 hours ago
$begingroup$
I saw this term couple of times but it appears to be confusing to me.
$endgroup$
– Ahmed
14 hours ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
As noted in the comment, the word for this is "extraneous solution".
I explain it to high school students this way - When you are trying to remove the square root, i.e. by squaring both sides of an equation, you risk the issue of having started with the false 1 = -1 but after squaring, 1 = 1 is indeed true. I'm sure it can arise from other manipulations, but this one seems most common.
I'll add - I find that both for students and myself, that graphing almost always adds to an understanding of the answer involved. I can analyze your equation and see that for the fact that X=1 is the highest X can go (higher, and you have the square root of a negative number, same for -1, X cannot be lower). That immediately tells me that the equation can never be true. For some, this might be obvious, but for the fraction of students who are more visual, the graph really makes the point.
answered 14 hours ago
JoeTaxpayerJoeTaxpayer
5,2871743
$endgroup$
2
$begingroup$
I ask my students to use implication arrows correctly, and choose between a conditional or biconditional implication between each pair of lines. This forces them to think about which steps can create extraneous solutions.
$endgroup$
– Steven Gubkin
13 hours ago
$begingroup$
Nice idea, thanks. As I mentioned, the most common extraneous solutions that I see are the result of squaring. The other one is when multiplying both sides of an equation to cancel out a denominator. Any other obvious one you can tell me?
$endgroup$
– JoeTaxpayer
13 hours ago
2
$begingroup$
Those are the most common ones for sure. If they are applying a function to both sides (such as exponentiating both sides), I like them to point out that this is either biconditional because the function is injective, or merely conditional because the function is not injective. For the record, I am doing this in the context of teaching algebra to future middle school teachers, and they need to understand this stuff better than their students. But any non-injective function gives more examples (applying cosine to both sides for instance).
$endgroup$
– Steven Gubkin
13 hours ago
$begingroup$
@StevenGubkin At first I read "future middle school students" and I thought wow, those must be some really advanced elementary school students to talking about algebra and bijectivity/injectivity...
$endgroup$
– Quintec
7 hours ago
$begingroup$
@Quintec - these terms might be pretty advanced for high school level, but the concept itself is great. And easy to explain.
$endgroup$
– JoeTaxpayer
7 hours ago
add a comment |
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$begingroup$
As noted in the comment, the word for this is "extraneous solution".
I explain it to high school students this way - When you are trying to remove the square root, i.e. by squaring both sides of an equation, you risk the issue of having started with the false 1 = -1 but after squaring, 1 = 1 is indeed true. I'm sure it can arise from other manipulations, but this one seems most common.
I'll add - I find that both for students and myself, that graphing almost always adds to an understanding of the answer involved. I can analyze your equation and see that for the fact that X=1 is the highest X can go (higher, and you have the square root of a negative number, same for -1, X cannot be lower). That immediately tells me that the equation can never be true. For some, this might be obvious, but for the fraction of students who are more visual, the graph really makes the point.
answered 14 hours ago
JoeTaxpayerJoeTaxpayer
5,2871743
$endgroup$
2
$begingroup$
I ask my students to use implication arrows correctly, and choose between a conditional or biconditional implication between each pair of lines. This forces them to think about which steps can create extraneous solutions.
$endgroup$
– Steven Gubkin
13 hours ago
$begingroup$
Nice idea, thanks. As I mentioned, the most common extraneous solutions that I see are the result of squaring. The other one is when multiplying both sides of an equation to cancel out a denominator. Any other obvious one you can tell me?
$endgroup$
– JoeTaxpayer
13 hours ago
2
$begingroup$
Those are the most common ones for sure. If they are applying a function to both sides (such as exponentiating both sides), I like them to point out that this is either biconditional because the function is injective, or merely conditional because the function is not injective. For the record, I am doing this in the context of teaching algebra to future middle school teachers, and they need to understand this stuff better than their students. But any non-injective function gives more examples (applying cosine to both sides for instance).
$endgroup$
– Steven Gubkin
13 hours ago
$begingroup$
@StevenGubkin At first I read "future middle school students" and I thought wow, those must be some really advanced elementary school students to talking about algebra and bijectivity/injectivity...
$endgroup$
– Quintec
7 hours ago
$begingroup$
@Quintec - these terms might be pretty advanced for high school level, but the concept itself is great. And easy to explain.
$endgroup$
– JoeTaxpayer
7 hours ago
add a comment |
$begingroup$
As noted in the comment, the word for this is "extraneous solution".
I explain it to high school students this way - When you are trying to remove the square root, i.e. by squaring both sides of an equation, you risk the issue of having started with the false 1 = -1 but after squaring, 1 = 1 is indeed true. I'm sure it can arise from other manipulations, but this one seems most common.
I'll add - I find that both for students and myself, that graphing almost always adds to an understanding of the answer involved. I can analyze your equation and see that for the fact that X=1 is the highest X can go (higher, and you have the square root of a negative number, same for -1, X cannot be lower). That immediately tells me that the equation can never be true. For some, this might be obvious, but for the fraction of students who are more visual, the graph really makes the point.
answered 14 hours ago
JoeTaxpayerJoeTaxpayer
5,2871743
$endgroup$
2
$begingroup$
I ask my students to use implication arrows correctly, and choose between a conditional or biconditional implication between each pair of lines. This forces them to think about which steps can create extraneous solutions.
$endgroup$
– Steven Gubkin
13 hours ago
$begingroup$
Nice idea, thanks. As I mentioned, the most common extraneous solutions that I see are the result of squaring. The other one is when multiplying both sides of an equation to cancel out a denominator. Any other obvious one you can tell me?
$endgroup$
– JoeTaxpayer
13 hours ago
2
$begingroup$
Those are the most common ones for sure. If they are applying a function to both sides (such as exponentiating both sides), I like them to point out that this is either biconditional because the function is injective, or merely conditional because the function is not injective. For the record, I am doing this in the context of teaching algebra to future middle school teachers, and they need to understand this stuff better than their students. But any non-injective function gives more examples (applying cosine to both sides for instance).
$endgroup$
– Steven Gubkin
13 hours ago
$begingroup$
@StevenGubkin At first I read "future middle school students" and I thought wow, those must be some really advanced elementary school students to talking about algebra and bijectivity/injectivity...
$endgroup$
– Quintec
7 hours ago
$begingroup$
@Quintec - these terms might be pretty advanced for high school level, but the concept itself is great. And easy to explain.
$endgroup$
– JoeTaxpayer
7 hours ago
add a comment |
$begingroup$
As noted in the comment, the word for this is "extraneous solution".
I explain it to high school students this way - When you are trying to remove the square root, i.e. by squaring both sides of an equation, you risk the issue of having started with the false 1 = -1 but after squaring, 1 = 1 is indeed true. I'm sure it can arise from other manipulations, but this one seems most common.
I'll add - I find that both for students and myself, that graphing almost always adds to an understanding of the answer involved. I can analyze your equation and see that for the fact that X=1 is the highest X can go (higher, and you have the square root of a negative number, same for -1, X cannot be lower). That immediately tells me that the equation can never be true. For some, this might be obvious, but for the fraction of students who are more visual, the graph really makes the point.
answered 14 hours ago
JoeTaxpayerJoeTaxpayer
5,2871743
$endgroup$
As noted in the comment, the word for this is "extraneous solution".
I explain it to high school students this way - When you are trying to remove the square root, i.e. by squaring both sides of an equation, you risk the issue of having started with the false 1 = -1 but after squaring, 1 = 1 is indeed true. I'm sure it can arise from other manipulations, but this one seems most common.
I'll add - I find that both for students and myself, that graphing almost always adds to an understanding of the answer involved. I can analyze your equation and see that for the fact that X=1 is the highest X can go (higher, and you have the square root of a negative number, same for -1, X cannot be lower). That immediately tells me that the equation can never be true. For some, this might be obvious, but for the fraction of students who are more visual, the graph really makes the point.
answered 14 hours ago
JoeTaxpayerJoeTaxpayer
5,2871743
edited 14 hours ago
answered 14 hours ago
JoeTaxpayerJoeTaxpayer
5,2871743
answered 14 hours ago
JoeTaxpayerJoeTaxpayer
5,2871743
answered 14 hours ago
JoeTaxpayerJoeTaxpayer
5,2871743
5,2871743
2
$begingroup$
I ask my students to use implication arrows correctly, and choose between a conditional or biconditional implication between each pair of lines. This forces them to think about which steps can create extraneous solutions.
$endgroup$
– Steven Gubkin
13 hours ago
$begingroup$
Nice idea, thanks. As I mentioned, the most common extraneous solutions that I see are the result of squaring. The other one is when multiplying both sides of an equation to cancel out a denominator. Any other obvious one you can tell me?
$endgroup$
– JoeTaxpayer
13 hours ago
2
$begingroup$
Those are the most common ones for sure. If they are applying a function to both sides (such as exponentiating both sides), I like them to point out that this is either biconditional because the function is injective, or merely conditional because the function is not injective. For the record, I am doing this in the context of teaching algebra to future middle school teachers, and they need to understand this stuff better than their students. But any non-injective function gives more examples (applying cosine to both sides for instance).
$endgroup$
– Steven Gubkin
13 hours ago
$begingroup$
@StevenGubkin At first I read "future middle school students" and I thought wow, those must be some really advanced elementary school students to talking about algebra and bijectivity/injectivity...
$endgroup$
– Quintec
7 hours ago
$begingroup$
@Quintec - these terms might be pretty advanced for high school level, but the concept itself is great. And easy to explain.
$endgroup$
– JoeTaxpayer
7 hours ago
add a comment |
2
$begingroup$
I ask my students to use implication arrows correctly, and choose between a conditional or biconditional implication between each pair of lines. This forces them to think about which steps can create extraneous solutions.
$endgroup$
– Steven Gubkin
13 hours ago
$begingroup$
Nice idea, thanks. As I mentioned, the most common extraneous solutions that I see are the result of squaring. The other one is when multiplying both sides of an equation to cancel out a denominator. Any other obvious one you can tell me?
$endgroup$
– JoeTaxpayer
13 hours ago
2
$begingroup$
Those are the most common ones for sure. If they are applying a function to both sides (such as exponentiating both sides), I like them to point out that this is either biconditional because the function is injective, or merely conditional because the function is not injective. For the record, I am doing this in the context of teaching algebra to future middle school teachers, and they need to understand this stuff better than their students. But any non-injective function gives more examples (applying cosine to both sides for instance).
$endgroup$
– Steven Gubkin
13 hours ago
$begingroup$
@StevenGubkin At first I read "future middle school students" and I thought wow, those must be some really advanced elementary school students to talking about algebra and bijectivity/injectivity...
$endgroup$
– Quintec
7 hours ago
$begingroup$
@Quintec - these terms might be pretty advanced for high school level, but the concept itself is great. And easy to explain.
$endgroup$
– JoeTaxpayer
7 hours ago
2
2
$begingroup$
I ask my students to use implication arrows correctly, and choose between a conditional or biconditional implication between each pair of lines. This forces them to think about which steps can create extraneous solutions.
$endgroup$
– Steven Gubkin
13 hours ago
$begingroup$
I ask my students to use implication arrows correctly, and choose between a conditional or biconditional implication between each pair of lines. This forces them to think about which steps can create extraneous solutions.
$endgroup$
– Steven Gubkin
13 hours ago
$begingroup$
Nice idea, thanks. As I mentioned, the most common extraneous solutions that I see are the result of squaring. The other one is when multiplying both sides of an equation to cancel out a denominator. Any other obvious one you can tell me?
$endgroup$
– JoeTaxpayer
13 hours ago
$begingroup$
Nice idea, thanks. As I mentioned, the most common extraneous solutions that I see are the result of squaring. The other one is when multiplying both sides of an equation to cancel out a denominator. Any other obvious one you can tell me?
$endgroup$
– JoeTaxpayer
13 hours ago
2
2
$begingroup$
Those are the most common ones for sure. If they are applying a function to both sides (such as exponentiating both sides), I like them to point out that this is either biconditional because the function is injective, or merely conditional because the function is not injective. For the record, I am doing this in the context of teaching algebra to future middle school teachers, and they need to understand this stuff better than their students. But any non-injective function gives more examples (applying cosine to both sides for instance).
$endgroup$
– Steven Gubkin
13 hours ago
$begingroup$
Those are the most common ones for sure. If they are applying a function to both sides (such as exponentiating both sides), I like them to point out that this is either biconditional because the function is injective, or merely conditional because the function is not injective. For the record, I am doing this in the context of teaching algebra to future middle school teachers, and they need to understand this stuff better than their students. But any non-injective function gives more examples (applying cosine to both sides for instance).
$endgroup$
– Steven Gubkin
13 hours ago
$begingroup$
@StevenGubkin At first I read "future middle school students" and I thought wow, those must be some really advanced elementary school students to talking about algebra and bijectivity/injectivity...
$endgroup$
– Quintec
7 hours ago
$begingroup$
@StevenGubkin At first I read "future middle school students" and I thought wow, those must be some really advanced elementary school students to talking about algebra and bijectivity/injectivity...
$endgroup$
– Quintec
7 hours ago
$begingroup$
@Quintec - these terms might be pretty advanced for high school level, but the concept itself is great. And easy to explain.
$endgroup$
– JoeTaxpayer
7 hours ago
$begingroup$
@Quintec - these terms might be pretty advanced for high school level, but the concept itself is great. And easy to explain.
$endgroup$
– JoeTaxpayer
7 hours ago
add a comment |
Ahmed is a new contributor. Be nice, and check out our Code of Conduct.
Ahmed is a new contributor. Be nice, and check out our Code of Conduct.
Ahmed is a new contributor. Be nice, and check out our Code of Conduct.
Ahmed is a new contributor. Be nice, and check out our Code of Conduct.
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3
$begingroup$
You do see the term "extraneous solution" sometimes.
$endgroup$
– Gerald Edgar
14 hours ago
$begingroup$
I saw this term couple of times but it appears to be confusing to me.
$endgroup$
– Ahmed
14 hours ago