# As it is appears here, how many constants do you see in $2x^3 + y^4 = \sqrt 5$?

My original phrasing was "how many constants are there in...". I am trying to determine if this is more clear. This is not intended for an exam, just a basic question to get intro-level algebra students thinking about the difference between variables and constants. Thank you.

• I would say 4 constants (although there is only 1 constant term). It of course depends on your definition of "constant". My answer 4 is from a programmer's perspective, where "constant" is roughly the same as "literal". That expression has 4 literal integers. – John Coleman Aug 23 '17 at 12:15
• If you are discussing the concept of a polynomial equation, you may say: the three constants in that equation are $2$, $1$, and $\sqrt{5}$. – Gerald Edgar Aug 23 '17 at 13:01
• Why not 3 and 4? – Michael Aug 23 '17 at 13:17
• Perhaps the word to use here is "coefficients", rather than "constants"? – mweiss Aug 23 '17 at 15:24
• "... a basic question to get intro-level algebra students thinking about the difference between variables and constants". I would say that this is a really bad example and it is not "basic". Such example/question is more confusing than enlightening. "Variables" and "constants" are meaningless without contexts. – Jack Aug 24 '17 at 14:08

The question

As it is appears here, how many constants do you see in $2x^3+y^4=\sqrt{5}$?

is quite vague. For example, are we to consider $x$ and $y$ as variables? Or perhaps they are also constants?

There is also a problem with "invisible" constants. For example, a quadratic equation is often defined as an equation of the form $ax^2+bx+c=0$, where $x$ is a variable and $a$, $b$, and $c$ are constants. $x^2-x=0$ is a quadratic equation, and one can argue that it has three constants: $a=1,b=-1,c=0$. (Or it could have four constants if we consider the $0$ at the right hand side of the equation.) Perhaps your equation has some "invisible" constants that should also be included in the count?

Let me also repeat John Coleman's comment:

I would say 4 constants (although there is only 1 constant term). It of course depends on your definition of "constant". My answer 4 is from a programmer's perspective, where "constant" is roughly the same as "literal". That expression has 4 literal integers.

In short, I feel that your question is too vague if it is to be intended to be asked by a teacher in, say, an exam.

• Such a question could be good to ask because it is vague, if the intention is to generate discussion which leads to the teacher pointing out the need to define your terms. – John Coleman Aug 23 '17 at 12:53
• Does any math teacher ever use x and y as constants? I realize that is possible, but in the United States I have never seen that. I just included this question on a first day handout. I thought it was an easy question. I was thinking the answer was 4. – Michael Aug 23 '17 at 13:16
• @JohnColeman, yes, of course. Thank you for pointing that out. – Joel Reyes Noche Aug 23 '17 at 13:16
• @Michael, some "letters" are usually treated as constants. For example, $e$ (the base of the natural logarithm) or $c$ (the speed of light in a vacuum). But I agree with you that $x$ and $y$ are seldom used as constants. – Joel Reyes Noche Aug 23 '17 at 13:19
• yes, "e" is an extremely well-known and universal constant. – Michael Aug 23 '17 at 13:23

The question is too ambiguous to be answered unequivocally, at almost all possible levels of sophistication. As in other answers and comments: from one viewpoint there are visibly 4 "literals", which might be what is intended by "constant". From another viewpoint, "x" and "y" might be names for constants, we have no idea. The fact that more than a single pair of possible values for x,y satisfy the equation does not give us any actual information about what they refer to. For that matter, for all we know the asserted equation is false... (this is possible because it is easily possible to make grammatically correct but factually false assertions). In a related vein, a problem is that the expression "$... = \sqrt{5}$" has two "free" "variables" (in a logic sense), $x$ and $y$. And so on.

• At no point ever in math education in America have I seen x and y used as constants. – Michael Aug 23 '17 at 21:52
• @Michael, ok, but how is one to know that they are "not constants"? As opposed to single-letter names earlier in the alphabet? Sure, there are conventions, but conventions are not "rules", and are not innate facts of mathematics. Hence, I'd not ask this kind of question, since it is far more about convention than mathematical realities. – paul garrett Aug 23 '17 at 21:54
• And, @Michael, in a situation like $x+1=3$, is $x$ a variable, or a constant? Or, if the answer is "it's an unknown", is it an unknown constant, or... what? :) Lotta potential trouble here, which has little to do with mathematics. – paul garrett Aug 23 '17 at 21:55
• @Michael: Regarding "At no point ever in math education in America have I seen x and y used as constants, what about labeling specific points in geometry as $x$ and $y?$ For example, the vertices of Triangle $T$ are $x,$ $y,$ and $z.$ – Dave L Renfro Aug 24 '17 at 14:59
• There is also an implied coefficient of $1$ for the $y^4$. – shoover Aug 24 '17 at 16:15

As the term "constant" is mostly used in polynomials, I would look at this equation as the polynomial $$2x^3 + y^4 - \sqrt{5} \in \mathbb{R}[x,y].$$ Thus, there is exactly one constant, the $\sqrt{5}$.
Depending on your definition, the number could of course differ.

However, I can't really see how this has something to do with mathematical education, care to explain?

• The idea of "what is a constant?" is one I want my students to understand. This is a question trying to get them to think about that. I'm not sure how this could not be related to math education. – Michael Aug 23 '17 at 10:54
• You are not asking about education, but rather about how many constants there are. If you are asking if this is a good question to allow your students to learn the concept of "constant", then this would be a question related to mathEdu (and my personal opinion would be "no"). As it is now, the question above is just an equation and a question that is missing a proper definition and clarification of the term "constant". – Dirk Aug 23 '17 at 10:57
• "If you are asking if this is a good question to allow your students to learn the concept of "constant " - this is more or less what I am trying to do, but I want to see how people respond to this question instead of asking them to evaluate the question itself. " a question related to mathEdu (and my personal opinion would be "no")." ---I don't get what you mean here. I don't know what MathEdu is referring to, or what question you are answering with your "no"- Maybe I am in the wrong place for this. Thank you for your input. – Michael Aug 23 '17 at 11:00
• @Michael "I want to see how people respond to this question" -- which people? If your interest is on how students would respond to this question, then you are asking people in the wrong population (math educators rather than students). Furthermore, only some math educators are members of this community and only some members of this community will respond. You have two layers of self-selection bias on what is possibly the wrong population. How will such an answer really help? – John Coleman Aug 23 '17 at 12:51
• Any people. I am trying to determine if this phrasing is more clear. Even a sample with self-selection bias could be useful in this regard. Most of life consists of collecting data in this manner. – Michael Aug 23 '17 at 13:37