Solutions = Roots = Zeros...WHY?

mathdad

mathdad

Full Member
Joined
Apr 24, 2015
Messages
737
Why do we call the solution(s) to quadratic equations roots and/or zeros?
 
A zero of a function is a value of the input for which the function value is 0. Properly speaking, we use this term only of functions, not of equations. That is, we talk about the zeros of f, not of f(x) = 0 (which would be redundant). Similarly, we shouldn't use the word "root" to mean a zero of a function; it applies only to equations.

A root of an equation means the same thing as solution, probably because it was thought of as the number from which the equation "grew", metaphorically. (It makes a little more sense to me in the case of square roots of numbers, from which squares grow. We even talk of "extracting the root" as if we were pulling up a radish -- which comes from the same "root" as "radical".)

I suspect that this usage is a lot older than the term solution, which has the same meaning when applied to an equation, but is more general. We tend to hold on to old words long after they could have been replaced.

See this page that I wrote a long time ago.
 
Dr.Peterson said:
A zero of a function is a value of the input for which the function value is 0. Properly speaking, we use this term only of functions, not of equations. That is, we talk about the zeros of f, not of f(x) = 0 (which would be redundant). Similarly, we shouldn't use the word "root" to mean a zero of a function; it applies only to equations.

A root of an equation means the same thing as solution, probably because it was thought of as the number from which the equation "grew", metaphorically. (It makes a little more sense to me in the case of square roots of numbers, from which squares grow. We even talk of "extracting the root" as if we were pulling up a radish -- which comes from the same "root" as "radical".)

I suspect that this usage is a lot older than the term solution, which has the same meaning when applied to an equation, but is more general. We tend to hold on to old words long after they could have been replaced.

See this page that I wrote a long time ago.
Click to expand...

More study notes for me. Thank you very much.
 
During my first year at university, I was taught that polynomials have roots, functions have zeros and equations have solutions. Since then, I've seen the words 'root', 'zero' and 'solution' interchanged to such an extent that the distinction hardly matters anymore (to me).

In other words, definitions in math can be flexible. Therefore, people who study math ought to be flexible, too. It took me a while to realize this because in my day mathematics was initially drilled into me as something factual, reliable and rigorous. That's fine, but not all "facts" are absolute, so be flexible. (If you research the words you asked about, you'll find each of them used both differently and synonymously.)

Here's another example (arbitrary student and institutions).

Facts from high school: 0^0 has no definition, and zero is neither positive nor negative.

Later, in college: 0^0 = 1, and zero may be defined as either positive, negative or neither.

?
 
Otis said:
During my first year at university, I was taught that polynomials have roots, functions have zeros and equations have solutions. Since then, I've seen the words 'root', 'zero' and 'solution' interchanged to such an extent that the distinction hardly matters anymore (to me).

In other words, definitions in math can be flexible. Therefore, people who study math ought to be flexible, too. It took me a while to realize this because in my day mathematics was initially drilled into me as something factual, reliable and rigorous. That's fine, but not all "facts" are absolute, so be flexible. (If you research the words you asked about, you'll find each of them used both differently and synonymously.)

Here's another example.

Facts from high school: 0^0 has no definition, and zero is neither positive or negative.

Later, in college: 0^0 = 1, and zero may be defined as either positive, negative or neither.

?
Click to expand...

Thanks for sharing.
 
Sometimes roots are used in place of zeros. Authors may say "the roots of a polynomial function...". The definitions are not universal. Pay attention to your own materials and just remember they are generally interchangeable.
 
Otis said:
During my first year at university, I was taught that polynomials have roots, functions have zeros and equations have solutions. Since then, I've seen the words 'root', 'zero' and 'solution' interchanged to such an extent that the distinction hardly matters anymore (to me).

In other words, definitions in math can be flexible. Therefore, people who study math ought to be flexible, too. It took me a while to realize this because in my day mathematics was initially drilled into me as something factual, reliable and rigorous. That's fine, but not all "facts" are absolute, so be flexible. (If you research the words you asked about, you'll find each of them used both differently and synonymously.)

Here's another example.

Facts from high school: 0^0 has no definition, and zero is neither positive or negative.

Later, in college: 0^0 = 1, and zero may be defined as either positive, negative or neither.

?
Click to expand...
Were you really told that 0 can be defined as neg or pos?!! Wow!
 
daon2 said:
Sometimes roots are used in place of zeros. Authors may say "the roots of a polynomial function...". The definitions are not universal. Pay attention to your own materials and just remember they are generally interchangeable.
Click to expand...

My own material? Am I a student at age 54?
 
Jomo said:
Were you really told that 0 can be defined as neg or pos?!! Wow!
Click to expand...

I feel that professors are no different than representatives hired by different companies. Person A tells you that the company does not accept online resumes. Person B is absolutely sure that only online resumes are accepted for employment.
 
Math people agree more that you think. There are some (silly) times where some definitions are different.
One book may start with a definition and then prove a theorem while another book will start off with theorem as their definition and then prove the definition. All mathematicians will be happy with this.
There are some theorems in real analysis that some mathematicians will not accept.
There are a set of mathematicians who do not believe that, for example, if the light swith is not on then must be is off. They will argue that there is a third option. These mathematicians (I think ) are called constructionist. They are a small breed and are not liked by (some) other mathematicians.
 
Jomo said:
Were you really told that 0 can be defined as neg or pos? ...
Click to expand...
I wasn't posting about me, in that second example, but I can see how it reads that way. I'll fix that, thank you.

Yes, you can find college textbooks where 0 has a sign, as some methods make use of that convention.

?
 
mathdad said:
My own material? Am I a student at age 54?
Click to expand...
I believe that "your own materials" means whatever you are learning from. I think you mentioned using a textbook; that is your "materials". Use the definitions in that book. (By the way, this is a reason I recommend working primarily from one textbook, rather than trying to get little bits from all over the web: They may confuse you by using different definitions or names for things, or a different order of presentation.)

And, yes, if you are studying something, even if it is on your own, then you are a student. That's all the word really means. (And, in fact, I have had students in community college who are your age.)
 
Otis said:
I wasn't posting about me, in that second example, but I can see how it reads that way. I'll fix that, thank you.

Yes, you can find college textbooks where 0 has a sign (certain methods make use of that convention).

?
Click to expand...
After re-reading your post it is clear that it is an example. I still can't believe would say that 0 is positive or negative.
 
Jomo said:
Math people agree more that you think. There are some (silly) times where some definitions are different.
One book may start with a definition and then prove a theorem while another book will start off with theorem as their definition and then prove the definition. All mathematicians will be happy with this.
There are some theorems in real analysis that some mathematicians will not accept.
There are a set of mathematicians who do not believe that, for example, if the light swith is not on then must be is off. They will argue that there is a third option. These mathematicians (I think ) are called constructionist. They are a small breed and are not liked by (some) other mathematicians.
Click to expand...

I have not found a better set of math books than the "Math For Dummies" series. Too bad there is no Calculus 3 For Dummies. The series does not go beyond Calculus 2 since the last time I checked amazon (maybe 2 years ago).

For the most part, the books do not derail from its teaching method. It is taught in step by step fashion and directly to the questions most often seen in the classroom. Lastly for tonight, most math teachers may agree with content where you live but in NYC every teacher has a "better" way or method for teaching at the level of students.
 
Last edited:
mathdad said:
I feel that professors are no different than representatives hired by different companies. Person A tells you [one thing, and person B tells you something else.]
Click to expand...
Yes, the situation is similar. Just as we need to speak with persons C,D,E and so on (until a consensus forms), so too we need experience with multiple viewpoints before our own perception gels (in general).

?
 
Dr.Peterson said:
I believe that "your own materials" means whatever you are learning from. I think you mentioned using a textbook; that is your "materials". Use the definitions in that book. (By the way, this is a reason I recommend working primarily from one textbook, rather than trying to get little bits from all over the web: They may confuse you by using different definitions or names for things, or a different order of presentation.)

And, yes, if you are studying something, even if it is on your own, then you are a student. That's all the word really means. (And, in fact, I have had students in community college who are your age.)
Click to expand...

1. I study from one book at a time.

2. I guess I am a student and did not know it.

3. As crazy as my family and friends think I am for "wasting time learning material that I will never teach for a living" I will never stop trying to better myself in terms of numbers.

4. In NYC, there are students 50 and over. Unfortunately for me, I exhausted my PELL & TAP Financial Awards back in the 1990s to get my A.A. and B.A. degrees. I do not have money to return to college for a third diploma. With the MHF and other free math help sites and youtube, there is no need to return to the campus life.

5. In fact, at 54, I better start saving for my future grave (no pun intended). I plan to be buried with at least one math book and the Bible, of course. More math tomorrow. Good night.
 
mathdad said:
4. In NYC, there are students 50 and over. Unfortunately for me, I exhausted my PELL & TAP Financial Awards back in the 1990s to get my A.A. and B.A. degrees. I do not have money to return to college for a third diploma. With the MHF and other free math help sites and youtube, there is no need to return to the campus life.
Click to expand...


Since you a NY resident you may qualify for the Excelsior Scholorship: https://www.hesc.ny.gov/pay-for-col...arships-awards/the-excelsior-scholarship.html

It has pretty hefty requirements though.
 
Otis said:
Facts from high school: 0^0 has no definition, and zero is neither positive nor negative.

Later, in college: 0^0 = 1, and zero may be defined as either positive, negative or neither.

?
Click to expand...

Your high-school teachers are "correct". Has "no definition" is misleading though because i don't doubt many sources define it that way. The debate will probably never be settled.
 
daon2 said:
Your high-school teachers are "correct". Has "no definition" is misleading though because i don't doubt many sources define it that way. The debate will probably never be settled.
Click to expand...

Back in my high school days, I never took regular math classes. I was in modified mathematics for most of my teen academic journey. Thus, I never came across the idea of 0^0 or (infinity)^0 or 0^(infinity), etc.

I will now disclose something very private about me. I am learning disabled. I have trouble learning new "stuff" but once I pick up on it, I become a champ. Because I have a learning disability, job training has been a nightmare for most of my adult life. It is a miracle that I graduated from two CUNY colleges, the Family Radio School of the Bible and served proudly in the United States Navy.
 
You must log in or register to reply here.
Top