How i Was Born |

In Memory of Johann Weilharter (1953-2021)

Girolamo Cardano (1501-76) was an Italian intellect whose interests and proficiencies ranged through those of mathematician, physician, biologist, physicist, chemist, astrologer, astronomer, philosopher, writer, and gambler.

While conducting research on solving algebraic equations, Cardano discovered that by means of a suitable substitution, the general cubic equation

$y^3+by^2+cy+d=0$

can be simplified. His substitution is $y = x-\frac{b}{3}$ , which yields

$(x-\frac{b}{3})^3 + b(x-\frac{b}{3})^2+c(x-\frac{b}{3})+d =0.$

Upon expanding and rearrange the terms, this becomes

$x^3+px=q,$

a depressed cubic (without the $x^2$ term) where $p = c-\frac{b^2}{3}, q = -d+\frac{bc}{3}-\frac{2b^3}{27}.$

Cadano applied this substitution in solving cubic equation $y^3-15y^2+81y-175=0:$

Substituting $y=x-(-15/3) = x+5$ into the cubic in $y$ , he obtained a depressed cubic in $x.$ namely,

$x^3+6x=20.$

Without a formula for this simplified equation, Cardano proceeded to solve it by way of ad hoc factoring:

$x^3+6x-20$

$= x^3-2x^2+2x^2+6x-20$

$= x^3-2x^2+2x^2-4x+10x-20$

$= x^2(x-2)+2x(x-2)+10(x-2)$

$= (x-2)(x^2+2x+10).$

Clearly, $x=2$ is a solution to $x^3+6x=20.$

Applying the quadratic formula to $x^2+2x+10=0$ gave $\frac{-2 \pm \sqrt{-36}}{2}.$ But this expression was immediately dismissed (for Cardano knew $x^2+2x+10=0$ has no real solution).

Therefore, $y=x+5 = 2+5=7$ is the only solution to the original cubic equation.

$x^2+2x+10=0$ has no solution

Cardano also solved $y^3+3y^2-12y-18=0$ in a similar fashion:

Obtaining first the depressed cubic (with $y=x-1$ )

$x^3-15x=4.$

Next is the ad hoc factoring again:

$x^3-15x-4$

$= x^3-4x^2+4x^2-15x-4$

$= x^3-4x^2+4x^2-16x+x-4$

$= x^2(x-4)+4x(x-4)+(x-4)$

$= (x-4)(x^2+4x+1).$

Surely,

$(x-4)(x^2+4x+1)=0 \implies x=4$ is a solution.

Furthermore, two additional solutions: $x = \frac{-4\pm\sqrt{4^2-4}}{2} = -2\pm\sqrt{3}$ were obtained by applying the quadratic formula to $x^2+4x+10=0.$

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$x^3-15x=4$ has three solutions: $4, -2+\sqrt{3}, -2-\sqrt{3}$

But Cardano did not like the ad hoc factoring. He wanted a formula that readily solves the depressed cubic $x^3+px=q$ , one that resembles the formula for the quadratics (see “Deriving the quadratic formula without completing the square“).

His relentless search for such a formula took many years (see William Dunham’s “Journey through genuis“) but, lo and behold, he found one:

$x = \sqrt[3]{\frac{q}{2}+\sqrt{\frac{q^2}{4} + \frac{p^3}{27}}} - \sqrt[3]{-\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}}\quad\quad\quad(*)$

To be clear, (*) is not Cardano’s own making. The formula bears the name ‘Cardano’s formula’ today only because Cardano was the one who published it in his 1545 book “Ars Magna” but without its derivation. However, in a chapter titled “On the Cube and First Power Equal to the Number”, Cardnao did give acknowledgement to Scipio del Ferro and Niccolo Fontana, who had independently derived (*) around 1515, but had kept the knowledge away from the public.

We derive (*) as follows:

Consider an algebraic identity that reminiscent of the depressed cubic $x^3+px=q.$ Namely,

$(u-v)^3+\underbrace{3uv}_{p}(u-v)=\underbrace{u^3-v^3}_{q}.$

It suggests that if we can determine the quantity $u$ and $v$ in terms of $p$ and $q$ from

$\begin{cases} 3uv = p \\u^3-v^3=q\end{cases}\quad\quad\quad(1-1, 1-2)$

then $u-v$ is a solution to $x^3+px=q.$

Asume $u \ne 0$ , (1-1) gives

$v = \frac{p}{3u}.$

Substituting this into (1-2) yields

$u^3-\frac{p^3}{27u^3}=q.$

Multply both sides by $u^3$ and rearrange terms, we have a sixth-degree equation:

$u^6-qu^3-\frac{p^3}{27} =0.$

But it is also quadratic in $u^3$ .

$(u^3)^2-qu^3-\frac{p^3}{27} = 0.$

Therefore, using the formula for quadratics,

$u^3 = \frac{q}{2}\pm\sqrt{\frac{q^2}{4} + \frac{p^3}{27}}\implies u =\sqrt[3]{\frac{q}{2}\pm\sqrt{\frac{q^2}{4} + \frac{p^3}{27}}}.$

There are two cases to consider.

For

$u =\sqrt[3]{\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}},\quad\quad\quad(1-3)$

we have $v^3 \overset{(1-2)}{=} u^3-q = \frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}-q,$ i.e.,

$v = \sqrt[3]{-\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}}.\quad\quad\quad(1-4)$

It follows that

$x \overset{(*)}{=} u-v \overset{(1-3), (1-4)}{=} \sqrt[3]{\frac{q}{2}+\sqrt{\frac{q^2}{4} + \frac{p^3}{27}}} - \sqrt[3]{-\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}}\quad\quad\quad(1-5)$

For $u =\sqrt[3]{\frac{q}{2} - \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}}, \;v=\sqrt[3]{-\frac{q}{2} -\sqrt{\frac{q^2}{4} + \frac{p^3}{27}}}.$

$x = u-v = \sqrt[3]{\frac{q}{2} - \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}}-\sqrt[3]{-\frac{q}{2} -\sqrt{\frac{q^2}{4} + \frac{p^3}{27}}}$

$= -\sqrt[3]{-\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}} + \sqrt[3]{\frac{q}{2} +\sqrt{\frac{q^2}{4} + \frac{p^3}{27}}}$

$= \sqrt[3]{\frac{q}{2}+\sqrt{\frac{q^2}{4} + \frac{p^3}{27}}} - \sqrt[3]{-\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}},$

the same as (1-5).

If $u=0,$ we see that on the one hand,

$\left(p \overset{(1-1)}{=} 0 \implies v \overset{(1-2)}{=} -\sqrt[3]{q}\right) \implies u-v = 0-(-\sqrt[3]{q}) =\sqrt[3]{q}.$

On the other hand, letting $p =0$ in (*) yields

$u-v = \sqrt[3]{\frac{q}{2}+\sqrt{\frac{q^2}{4} + 0}} -\sqrt[3]{-\frac{q}{2} + \sqrt{\frac{q^2}{4} + 0}}$

$= \sqrt[3]{\frac{q}{2}+\frac{|q|}{2}} -\sqrt[3]{-\frac{q}{2} + \frac{|q|}{2}}$

$= \begin{cases} \sqrt[3]{\frac{q}{2}+\frac{q}{2}} - \sqrt[3]{-\frac{q}{2}+\frac{q}{2}} = \sqrt[3]{q}, \;q \ge 0\\ \sqrt[3]{\frac{q}{2}-\frac{q}{2}} - \sqrt[3]{-\frac{q}{2}-\frac{q}{2}} = \sqrt[3]{q},\;q<0\end{cases}$

Cardano first tested (*) on cubic $x^3+6x=20$ by letting $p=6, q=20.$ . The formula yields

$x = \sqrt[3]{\frac{20}{2}+\sqrt{\frac{20^2}{4} + \frac{6^3}{27}}} - \sqrt[3]{-\frac{20}{2} + \sqrt{\frac{20^2}{4} + \frac{6^3}{27}}}= \sqrt[3]{10+\sqrt{108}} - \sqrt[3]{-10+\sqrt{108}}.$

It came as a surprise to Cardano initially. But he quickly realized that this sophisticated looking expression is nothing more than “2”, the unique solution of $x^3+6x=20$ , in disguise.

Today, this is easily checked by a CAS:

For a mathematical proof, see “A Delightful Piece of Mathematics“.

Cardano then tested the formula on cubic $x^3-15x=4.$ Substituting $p = -15, q=4$ into it gave

$x = \sqrt[3]{2+\sqrt{-121}} - \sqrt[3]{-2+\sqrt{-121}}\quad\quad\quad(**)$

He was startled by the result!

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The presence of $\sqrt{-121}$ alone did not surprise him for he had seen negative number under the square root before (while solving $x^2+2x+10=0,$ a quadratic clearly has no solution). What really perplexed Cardano this time was the fact that square root of negative number appearing in the result for a cubic that has three real solutions!

Cardano thus sought the value of $\sqrt[3]{2+\sqrt{-121}} - \sqrt[3]{-2+\sqrt{-121}}$ to see which solution, amongst $4, -2+\sqrt{3}$ and $-2+\sqrt{3}$ it represents.

He started with $\sqrt[3]{2+\sqrt{-121}}$ . At once, Cardano noticed that $2+\sqrt{-121}$ is a number in the form of

$a+\sqrt{-b}.$

And he speculated that the result of calculating $\sqrt[3]{2+\sqrt{-121}}$ has the same manner.

So Cardano wanted to find $a$ and $b$ such that

$\sqrt[3]{2+\sqrt{-121}} = a+\sqrt{-b}.$

He proceeded as follows:

Cubing both sides gives

$2+\sqrt{-121} = a^3 +3a^2\sqrt{-b}-3ab-b\sqrt{-b} = a^3-3ab+3a^2\sqrt{-b}-b\sqrt{-b}.$

Equating the similar parts on both sides yields a system of nonlinear algebraic equations

$\begin{cases} a^3-3ab=2 \\ 3a^2\sqrt{-b}-b\sqrt{-b} = \sqrt{-121} \end{cases}\quad\quad\quad(1-6, 1-7)$

Squaring both (1-6) and (1-7) gives:

$\begin{cases} a^6-6a^4b+9a^2b^2=4 \\ -9a^4b+6a^2b^2-b^3=-121\end{cases}\quad\quad\quad(1-8, 1-9)$

and subtracting (1-9) from (1-8) results in

$a^6+3a^4b+3a^2b^2+b^3 =125 \implies (a^2+b)^3 = 125\implies a^2+b=5$

or,

$b = 5-a^2.$

Substituting it back into (1-6) yields

$4a^3-15a=2 \implies a^3 = \frac{15}{4}a + \frac{1}{2}.$

And so,

$a^3-\frac{15}{4}a = \frac{1}{2}$

This is a depressed cubic with $p=-\frac{15}{4}, q=\frac{1}{2}.$ By Cardano’s formula,

$a=\sqrt[3]{\frac{1}{4} + \sqrt{\frac{1}{4}(\frac{1}{2})^2 + \frac{1}{27}(\frac{-15}{4})^3}}-\sqrt[3]{-\frac{1}{4} + \sqrt{\frac{1}{4}(\frac{1}{2})^2 + \frac{1}{27}(\frac{-15}{4})^3}}$

$=\sqrt[3]{\frac{1}{4} + \sqrt{\frac{1}{16} - \frac{3375}{27\cdot64}}}-\sqrt[3]{-\frac{1}{4} + \sqrt{\frac{1}{16} - \frac{3375}{27\cdot64}}}$

$=\sqrt[3]{\frac{1}{4} + \sqrt{\frac{1}{16}-\frac{3375}{1728}}}-\sqrt[3]{-\frac{1}{4} + \sqrt{\frac{1}{16}-\frac{3375}{1728}}}$

That is,

$a =\sqrt[3]{\frac{1}{4} + \sqrt{\underline{-\frac{121}{64}}}}-\sqrt[3]{-\frac{1}{4} + \sqrt{\underline{-\frac{121}{64}}}}$

So solving $4a^3-15a=2$ for $a$ by Cardano’s formula resulting in having to calculate another square root of a negative number. Cardano was put right back to where he had started. With the frustration he called the cubic “irreducible” and pursued the matter no further.

It would be another generation before Rafael Bombelli (1576-72) took upon the challenge of calculating $\sqrt[3]{2+\sqrt{-121}}$ again.

Bombelli’s was an engineer who knew how to drain the swampy marshes, and only between his engineering projects was he actively engaged in mathematics. Being practical and sound minded, he read the near-mystical $\sqrt{-121}$ not as the square root of a negative number but a symbolic representation for a new type of number that extends the real number. He imagined a set for a new type of number that

[1] Has every real number as its member.

[2] The arithmetic operations ( $+, \cdot$ ) are so defined that the commutative, associative and distributive law are obeyed.

[3] There is a member $\bold{i}$ such that $\bold{i}\cdot\bold{i} = -1.$ i.e.,

$\bold{i}^2=-1.\quad\quad\quad(2-1)$

Bombelli sanity checked his idea by consider any quadratic equation

$ax^2+bx+c=0.$

That is

$x^2+2hx+g=0$

where $g=\frac{c}{a}, h=\frac{b}{2a}$ which can be written as

$(x+h)^2+g-h^2=0\quad\quad\quad(2-2)$

or,

$(x+h)^2=h^2-g.$

if $h^2-g$ is positive, then it has a square root, and $-h + \sqrt{h^2-g}$ is a solution of the equation (so is the number $-h-\sqrt{h^2-g}).$ If $h^2-g$ is not positive, then $g-h^2$ is, and therefore has a square root $\sqrt{g-h^2}.$

Let

$x = -h + \sqrt{g-h^2}\cdot \bold{i}.\quad\quad\quad(2-3)$

The left side of (2-2) becomes

$(\underline{-h+\sqrt{g-h^2}\cdot \bold{i}}+h)^2+g-h^2$

$= (g-h^2)\cdot \bold{i}^2 +g-h^2$

$=(g-h^2)\cdot(\bold{i}^2+1)$

$\overset{[3]}{=}(g-h^2)\cdot (-1+1)$

$=(g-h^2)\cdot 0$

$= 0$

That is, (2-3) is a solution of (2-2).

Bombelli was elated as it suggested that by considering his set which contains the real numbers and $\bold{i},$ all quadratic equations have solutions!

It also gave him much needed confidence in showing what $\sqrt[3]{2+\sqrt{-121}} - \sqrt[3]{-2+\sqrt{-121}}$ really is.

Right away, Bombelli saw

$(\sqrt{121}\cdot\bold{i})^2\overset{[1],[2]}{=}(\sqrt{121})^2\cdot\bold{i}^2 \overset{[3]}{=} 121\cdot (-1) = -121$

so he replaced $\sqrt{-121}$ in $\sqrt[3]{2+\underline{\sqrt{-121}}}$ with $\sqrt{121}\cdot\bold{i}:$

$\sqrt[3]{2+\sqrt{-121}}=\sqrt[3]{2+\sqrt{121}\cdot\bold{i}} = \sqrt[3]{2+11\cdot\bold{i}}.$

He then anticipated that the value of $\sqrt[3]{2+11\cdot\bold{i}}$ is a new type of number $a+b\cdot\bold{i}$ where $a$ and $b$ are real numbers. i.e.,

$\sqrt[3]{2+11\cdot\bold{i}} = a + b\cdot \bold{i},\quad a,b \in R.\quad\quad\quad(3-1)$

And finally, he proceeded to find $a$ and $b$ from (3-1).

As an illustration, we solve (3-1) for $a, b$ as follows:

Cubing it gives

$2+11\cdot\bold{i} = (a+b\cdot\bold{i})^3.$

Since

$(a+b\cdot \bold{i})^3 = a^3+3a^2(b\bold{i})+3a(b\bold{i})^2+(b\cdot\bold{i})^3$

$=a^3+3a^2b\bold{i}+3ab^2\bold{i}^2+b^3\bold{i}^2\bold{i}$

$\overset{[3]}{=}a^3+3a^2b\bold{i}-3ab^2-b^3\bold{i}$

$= a^3-3ab^2 + (3a^2b-b^3)\cdot\bold{i},$

this is

$2+11\cdot\bold{i} = a^3-3ab^2 + (3a^2b-b^3)\cdot\bold{i}.$

Equating similar terms on both sides yields a system of nonlinear equations:

$\begin{cases} a^3-3ab^2=2 \\ 3a^2b-b^3=11 \end{cases}$

After factoring, it becomes

$\begin{cases} a(a^2-3b^2)=2 \\ b(3a^2-b^2)=11 \end{cases}\quad\quad\quad(3-2, 3-3)$

Assuming $a$ and $b$ are both integers, then $a$ and $a^2-3b^2$ on the left side of (3-2) are two integer factors of $2.$ Since $2$ has only two factors, namely, $1$ and $2.$ If $a=1$ then from (3-1), $1\cdot(1-3b^2) =2 \implies b^2 < 0,$ a contradiction. However, $(a=2, 2(4-b^2)=2)$ yields $b=-1$ or $+1.$ While $(a=2, b=-1) \implies b(3a^2-b^2)=(-1)\cdot(3\cdot 2^2-(-1)^2)=-11$ contradicts (3-3), $(a=2,b=1)$ gives $1\cdot(3\cdot 2^2-1^2)=11.$ Therefore, $a=2, b=1$ is the solution to (3-2, 3-3). i.e.,

$\sqrt[3]{2+\sqrt{-121}} = 2+\bold{i}.\quad\quad\quad(3-4)$

It is also easy to see that (3-4) is true as follows:

$(2+\bold{i})^3=2^3+3\cdot 2^2\cdot \bold{i}+ 3\cdot 2\cdot \bold{i}^2+\bold{i}^3$

$= 8 + 12\cdot \bold{i} +6\cdot \bold{i}^2 + \bold{i}^2\cdot \bold{i}$

$\overset{[3]}{=} 8 + 12\cdot \bold{i}-6-\bold{i}$

$= 2 + 11\cdot \bold{i}$

$= 2 + \sqrt{121}\cdot \bold{i}$

$= 2+\sqrt{-121}.$

And so

$2 + \sqrt{-121} = (2+\bold{i})^3 \implies \sqrt[3]{2+\sqrt{-121}} = 2 + \bold{i}.$

Similarly, Bombelli obtained (see Exercise-1)

$\sqrt[3]{-2+\sqrt{-121}} = -2+\bold{i}.\quad\quad\quad(3-5)$

By (3-4) and (3-5) Bombelli was able to reproduce the solution to cubic $x^3-15x=4$ :

$x = \sqrt[3]{2+\sqrt{-121}}-\sqrt[3]{-2+\sqrt{-121}}=2+\bold{i}-(-2+\bold{i})=4.$

Thus, with $\bold{i}$ and the ordinary rules of real numbers’ arithmetic, Bombelli broke the mental logjam concerning negative number under the square root.

Satisfied with his work that unlocked what seemed to be an impassable barrier, Bombelli moved on without constructing his set for the new type of number in a logically unobjectionable way. The world had to wait another two hundred years for that (see “Mr. Hamilton does complex numbers”). Still, Bombelli deserves the credit for not only recognizing numbers of a new type have a role to play in algebra, but also giving $\bold{i}$ its initial impetus and now undisputed legitimacy.