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I am wondering about the difference between the following demands:

Prove that P(x) has at least one root.

Prove that P(x) has at least one solution.

Are they the same?

The background to my question:

Let $c \in \Bbb R$. Prove that the equation: $$\frac{1}{\ln x} - \frac{1}{x-1}$$

Has only one solution in $(0,1)$.

Here is what I have in mind:

  1. Show that if $f'$ has no roots, then $f$ has one root at most.
  2. Calculate the derivative.
  3. Show that when $\frac{1}{2}<x<1$, $f'(x)<0$
  4. Show that $f'(1)=0$. This means that 0 is a local minimum point in $(0,1)$
  5. Therefore, $f$ has at most one root.
  6. Show from Intermediate value theorem that $f$ has a root in $(0,1)$.
Travis Willse
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Alan
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    Equations (expressions containing an equal sign) have zero or more solutions. Functions have roots that solve the equation $f(x)=0$. As written your given 'equation' is not an equation. – John McGee Jul 11 '15 at 15:51
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    To be precise (or maybe just pedantic) a "root" of P(x) is the same as a "solution" to the equation P(x) = 0. People use the terms pretty much interchangeably. – lulu Jul 11 '15 at 15:51
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    Confusion is made even more likely, by the fact that $-$ and $=$ are adjacent on most keyboards and the roots of $f(x) - g(x)$ are the solutions of $f(x) = g(x)$ $\ddot{\smile}$. Has a typo of this sort happened in your example. By the way your solution looks to be along the right lines. – Rob Arthan Jul 11 '15 at 16:27
  • @RobArthan thank you for the feedback. – Alan Jul 11 '15 at 16:35
  • In the light of the comments above your second line should be stated as: Proof that $P(x)=0$ has at least one solution – Markus Scheuer Jul 12 '15 at 06:43
  • The question is currently mangled, because what follows "Prove that the equation:" is not an equation (no equals sign). – Daniel R. Collins Oct 02 '15 at 11:35

3 Answers3

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Yes, a root is basically the same as a solution. We often use the work "root" when we talk about polynomials.

Now, this doesn't mean that the words are interchangeable. We can say

The roots of the polynomial are ...

We can't say

The solutions to the polynomial are ...

Solutions are always to equations. Roots are for/of polynomials (functions).

You write that the equation $$ \frac{1}{\ln(x)} - \frac{1}{x-1} $$ has a solution, but what you have written isn't an equation because you don't have an equal sign. Instead you probably want to say that the equation $$ \frac{1}{\ln(x)} - \frac{1}{x-1} =0 $$ has a solution ...

Example: Let $f(x) = x^2 -3x + 2$. Then $f$ is a polynomial. We can say the following

$1$ and $2$ are the two roots of the polynomial $f$.

$1$ and $2$ are the two solutions to the equation $f(x) = 0$.

Thomas
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Functions have roots, but equations have solutions.

So, we'd say that

the function $P$ has at least one root,

but not that "$P$ has at least one solution".

Conversely, we'd say that

the equation $P(x) = 0$ has at least one solution

but not that "$P(x) = 0$ has at least one root".

Of course the two notions are related, in that, by definition, $x_0$ is a root of $P$ iff $P(x_0) = 0$, or equivalently, if $x = x_0$ is a solution of $P(x) = 0$.

Travis Willse
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  • Thank you. Do you think the way I was going to solve the question above is correct? – Alan Jul 11 '15 at 15:57
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    @Alan Very likely there are very similar questions already answered on the site, I suggest you look there. If you can't find one, or the answers you find don't help you, I suggest you pose it as a new question. (In general, it's a good idea not to post two rather different questions in the same question here, as that sort of separation will certainly make it easier for others to find your question if they have a similar one.) – Travis Willse Jul 11 '15 at 16:00
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    I agree with this answer as the correct use of mathematical English, but I would bet that if you look through most mathematician's writing (including mine) you'll find they get this wrong on a regular basis. I'd say this distinction is about as honored as the less/fewer distinction. – David E Speyer Jul 11 '15 at 17:31
  • Can you say “expressions have roots” too? Would it be correct to say: the roots of the expression $(x-p)(x-q)$ are $x=p,q$? I am thinking that an expression is essentially what defines a function, therefore you could talk about the roots of an expression, as we do with polynomials. – lukejanicke Jan 10 '19 at 06:17
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Prove that P(x) has at least one root.

The solution to the above will involve radicals, like square roots. This means you'll find a zero, but not a solution. You will solve your problem, however.

Here's an example $x^2-x-1=0$

Prove that P(x) has at least one solution.

The solution can take any form, and instead of showing one side of the equation has zeros you show that there are values where both sides of the equation are equal.

For example $e^x=7$

Zach466920
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