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I am studying the history of complex numbers, and I don't understand the part on the screenshots. In particular, I don't understand why a cubic always has at least one real root.

I don't see why the reason is "since $y^3 − py − q$ is positive for sufficiently large positive $y$ and negative for sufficiently large negative $y$)."

Secondly, I don't see how this fact led to the need to introduce complex numbers. It seems like the book is hinting at this. But maybe I am wrong. I don't see why it was necessary to introduce complex numbers due to cubic equations if they were shown to have at least one real solution. Then the real solution could be enough, right?

Thank you for your explanation!

14.2 Quadratic Equations

The usual way to introduce complex numbers in a mathematics course is to point out that they are needed to solve certain quadratic equations, such as the equation $x^2 + 1 = 0$. However, this did not happen when quadratic equations first appeared, since at that time there was no need for all quadratic equations to have solutions. Many quadratic equations are implicit in Greek geometry, as one would expect when circles, parabolas, and the like are being investigated, but one does not demand that every geometric problem have a solution. If one asks whether a particular circle and line intersect, say, then the answer can be yes or no. If yes, the quadratic equation for the intersection has a solution; if no, it has no solution. An "imaginary solution" is uncalled for in this context.


14.3 Cubic Equations

The del Ferro-Tartaglia-Cardano solution of the cubic equation $$ y^3 = py + q $$ is $$ y = \root{3}\of{ \frac{q}{2} + \sqrt{ \left(\frac{q}{2}\right)^2 - \left(\frac{p}{3}\right)^3 } } + \root{3}\of{ \frac{q}{2} - \sqrt{ \left(\frac{q}{2}\right)^2 - \left(\frac{p}{3}\right)^3 } } $$ as we saw in Section 6.5. The formula involves complex numbers when $(q/2)^2 - (p/3)^3 < 0$. However, it is not possible to dismiss this as a case with no solution, because a cubic always has at least one real root (since $y^3 - py - q$ is positive for sufficiently large positive $y$ and negative for sufficiently large negative $y$). Thus the Cardano formula raises the problem of reconciling a real value, found by inspection, say, with an expression of the form $$ \root{3}\of{ a + b\sqrt{ -1 } } + \root{3}\of{ a - b\sqrt{ -1 } } $$

The quote is from the Mathematics and its history book pp 276-277.

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    $\begingroup$ First of all, we have to correct some arguments here: "a cubic (WITH REAL COEFICIENTS) always has at least one (REAL) root. $\endgroup$
    – DiegoMath
    Commented 2 days ago
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    $\begingroup$ A (3rd) degree polynomial, $~f(x),~$ is a continuous function. If you can establish that there exists $~x_1, ~x_2, ~$ such that $~f(x_1) < 0,~$ and $~f(x_2) > 0,~$ then, because $~f(x)~$ is continuous, $~f(x)~$ must cross the $~x~$ axis, for some value $~x_3~$ that is between $~x_1~$ and $~x_2.$ $\endgroup$ Commented 2 days ago
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    $\begingroup$ Since all real polynomial is a continuous function you can apply the mean value theorem. Since the polynomial has positive and negative values, there exists some zero value. $\endgroup$
    – DiegoMath
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    $\begingroup$ cubic polynomial is continuous and tends to negative infinity at one end and to positive infinity at the other end so there has to be a zero somewhere according to the IVT $\endgroup$
    – Vasili
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    $\begingroup$ @DiegoMath Intermediate Value Theorem! $\endgroup$ Commented 2 days ago

6 Answers 6

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You ask in a comment

I am not sure, because if it was a real root, then I don't understand why this is used as an argument for introducing complex numbers? Having a real root feels unrelated to having a complex root, no? Maybe that's my real misunderstanding. Even if I saw why it always has a real root, I don't see how this leads to need for introducing complex numbers.

Even though the cubic has real coefficients the formula for the roots requires complex numbers.

That seemingly contradictory fact is probably what delayed discovering the formula. The book on the history of mathematics you quote from should discuss that.

Read about the casus irreducibilis on wikipedia.

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    $\begingroup$ Thank you, this is incredibly helpful! So if I understand correctly, the thought sequence is 1. We are sure this is the form the roots have. (And that they always exist). 2. That means that there is something "unreal" (complex) in the form of the roots when that one expression is < 0. (Which we are sure can happen, so we can't just avoid this case and avoid existence of complex numbers..)? $\endgroup$ Commented 2 days ago
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    $\begingroup$ @TerezaTizkova Yes, that's a good summary. $\endgroup$ Commented 2 days ago
  • $\begingroup$ @TerezaTizkova, also, the story explains why we call some numbers "real" and some numbers "imaginary." They guy who figured it out said something like, "of course, we all know that in reality, there is no number whose square equals -1, but if I imagine that there is such a number, and if I pretend that it obeys all the rules of algebra, then..., etc., and so forth." $\endgroup$ Commented yesterday
  • $\begingroup$ @SolomonSlow: It probably helps that the complex numbers are so well-behaved. It would have been much harder to justify the quaternions or octonions on such a basis, specifically because you can't do "regular" algebra with them. Pretty much the only "obvious" anomaly in the complex numbers is the existence of a branch cut in the square root function, which is rather unfortunate if you're writing $\sqrt{-1}$ everywhere instead of $i$, but it's not that difficult to avoid otherwise. $\endgroup$
    – Kevin
    Commented yesterday
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I don't understand why a cubic always has at least one real root.

I don't see why the reason is "since $y^3 − py − q$ is positive for sufficiently large positive $y$ and negative for sufficiently large negative $y$)."

It is known that the graph of a cubic function is a single, unbroken curve (i.e. the function is continuous). If it is negative for some value and positive for another, then it must pass through zero. This is because any unbroken curve that goes from below the $x$-axis to above it must pass through it. Hence, every cubic of the form $y^3 - py - q$ has at least one real root. Since any cubic can be transformed into one of that form by a simple horizontal translation -- which doesn't add or remove any crossings of the axis -- we can extend that result to all cubics.

I don't see why it was necessary to introduce complex numbers due to cubic equations if they were shown to have at least one real solution. Then the real solution could be enough, right?

This is logical, but there's a step between putting in $p$ and $q$ and getting out that root. That step is evaluating the expression for the root given at the top of 14.3. The problem is that the evaluation of the inner square roots is not possible (exclusively in reals) for all $p$ and $q$, but as argued above, there is a root for all $p$ and $q$ and this expression should evaluate to that root in all cases. If you try some actual values of $p$ and $q$, using your knowledge of complex numbers, you will quickly see what's happening: the two terms of the expression are complex conjugates, and so their sum is always real. But if you don't have a theory of complex numbers, you have a gap in the evaluation of the expression and thus you cannot find the solution that you know is there.

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    $\begingroup$ The proof for "it must pass through zero" is the Intermediate Value Theorem. $\endgroup$
    – qwr
    Commented 12 hours ago
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    $\begingroup$ @qwr This is true, but to my understanding, that wasn't the justification at this point in history, so I didn't want to reference it. $\endgroup$
    – Alex Jones
    Commented 10 hours ago
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(this is just a simple historical explanation, without delving in the theory of cubic equations)

The trouble with the solution of the cubic equation is that when the equation has three real solutions the formula found by Tartaglia required to take square roots of negative numbers. Eventually he noticed that if he supposed that such number existed and the usual rules for sum and multiplication could be applied, at the end of the calculation those number disappeared and an actual (real) root was found, so Tartaglia did not worry too much. Of course, this was not a sound reasoning, but at that time it was enough as long as it worked.

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  • $\begingroup$ Just to clarify: At the time, this wasn't considered sound reasoning because equation-solving was still viewed as a geometric pursuit based on the lengths of line segments and the areas and volumes formed from them. In classical Euclidean geometry, there are no line segments such that the area of a square formed with those sides is negative. $\endgroup$ Commented yesterday
  • $\begingroup$ actually the usual rules don't apply, notably sqrt(x) * sqrt(y) != sqrt(xy) $\endgroup$
    – qwr
    Commented yesterday
  • $\begingroup$ @ChrisBouchard: I am not really very sure about this. When Ludovico Ferrari exploited the rules for solving a cubic in order to solve a quartic, no geometrical meaning could be attached to this. $\endgroup$
    – mau
    Commented 48 mins ago
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First you were asking about: I don't see why the reason is "since $y^3 − py − q$ is positive for sufficiently large positive $y$ and negative for sufficiently large negative $y$)."

Answer for this question: If $f(x) = x^3 +ax^2+bx+c$, then it is a continuous function and observe that $\displaystyle \lim_{x \to \infty} f(x) = \infty$ and $\displaystyle \lim_{x \to -\infty} f(x) = -\infty$. As it is continuous, it can't always be positive or negative. So there exists $x_1,x_2 \in \Bbb R$ such that $f(x_1) < 0$ and $f(x_2) > 0$, so by IVP there exists $c$ between $x_1$ and $x_2$ such that $f(c) = 0$.

Also, by the Fundamental Theorem of Algebra, we know that any $3$ degree polynomial has exactly $3$ roots. Also, if $z \in \Bbb C$ is a root of $f(x)$, then $\bar{z}$ is also a root of $f(x)$(as $f(\bar{z}) = \bar{f}(z)$). These two situations tell us

  1. $f(x)$ can have one real and two complex roots.
  2. $f(x)$ have three real roots.

Your second query was: Secondly, I don't see how this fact led to the need to introduce complex numbers.

Answer: Each polynomial in $\Bbb R[x]$(set of all polynomials with real coefficients) doesn't necessarily have a real root. A great example is the polynomial $x^2+1$, which has no real roots. So, we have to extend the field where we can find the roots of this polynomial. This field extension is also called a splitting field for a polynomial, which can be split into linear factors.

Edit: You are right because extension fields were made to introduce the subject for studying the roots of polynomials. A cubic equation always has at least one real solution. Cardan discovered that to obtain real solutions to a cubic equation, you need to make sense of the square roots of negative numbers. So they faced a dilemma: you either say that square roots of negative numbers do not exist and then lose these legitimate real solutions for cubics, or else introduce a theory for square roots of negative numbers so that we can keep these legitimate real solutions. So, solving polynomials is the actual motivation. Galois made this generalization. See this.

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    $\begingroup$ Thank you! Regarding your second answer, the 𝑥^2+1 is exactly what is argumented in the book as not having to have any solutions, so it still wouldn't be enough motivation for introducing new kind of numbers. $\endgroup$ Commented 2 days ago
  • $\begingroup$ Yes, it's better to say in terms of polynomials because to find the roots of that polynomial, you have to extend the field. $\endgroup$
    – Afntu
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With tools from Calculus the intermediate value theorem would do the trick.

If calculus is not part of the tool box, the Descartes' sign change criteria works well for real polynomials of the form $x^3-px-p$.

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A cubic function $y = Ax^3 + Bx^2 + Cx$ can have two turning values. If you plot it out, it can have a maximum and a minimum. Some cubic functions such as $y = x^3 + x$ have no turning values.

A graphical solution for $D = Ax^3 + Bx^2 + Cx$ would be the points where $y = Ax^3 + Bx^2 + Cx$ crosses the horizontal line $y=D$. If D lies between the two turning values then the line will cross the curve three times. If D is not between the two turning values, or there are no turning values then you will get only one intersection. If you have two intersections then your curve has a single turning value, and so it is a quadratic, and not a cubic.

You do not have to use complex numbers. You don't have to use negative numbers, or zero if it comes to that. But it does help understanding.

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    $\begingroup$ It can have a local (or relative) maximum and minimum, but obviously the graph is unbounded. $\endgroup$ Commented 17 hours ago

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