Yeah, this is one of the craziest stories in the history of math, and deserves to be better known!

In the mid-1500s, Cardano (building on del Ferro and Tartaglia, not entirely amicably...) solved the cubic equation. The solution goes something like this. https://twitter.com/3blue1brown/status/1253747012245221376

2/ You want to solve ax^3+bx^2+cx+d=0. First make it "monic" (leading coefficient 1) by dividing by "a". Renaming coefficients, we have:

x^3+Bx^2+Cx+D=0.

Now "depress" it (kill off x^2 term) by replacing x=y-B/3. After some algebra, we have:

y^3+my+n=0.

[Aside: when you do...

3/ the same with a quadratic, ax^2+bx+c=0, divide by "a", and set y=x-B/2, you get y^2+n=0. This recently "rediscovered" method of solving quadratics is what @3blue1brown explained in his last live session. Shows how poorly we teach quad equations that NYT thinks it's new...]

4/ Back to y^3+my+n=0. By a beautiful method (which I can explain elsewhere), Cardano derives the formula:

y=u-v,

where (crt=cube root)

u=crt[-n/2+sqrt R],
v=crt[n/2+sqrt R],

and

R=n^2/4+m^3/27.

Now, I made two claims in my message to Grant:

5/

(1) That negative numbers did *not* exist at this time,

but shockingly that

(2) Complex numbers were needed as an "abstract tool" to solve "real" problems!

Let me first argue (1). Cardano first solves y^3+my=n, then y^3=my+n, then y^3+n=my, etc. Aren't these all the same??

6/ No! Because to Cardano (like Euclid!), y^3 literally means a y×y×y cube!

That is, they are not overloading the "-" symbol (as we do today, and confuse poor pre-algebra students!) to mean both subtraction and negation. Negative numbers don't exist!

So how could complexes???

7/ Look back at Cardano's formula. Let's solve, e.g.,

y^3-15y-4=0.

One can easily stumble on to y=4 as a solution. But what does Cardano's formula give? y=u-v where:

R=n^2/4+m^3/27 = -121

and

u=crt[-n/2+sqrt R]=crt[2+sqrt(-121)],
v=crt[-2+sqrt(-121)].

Huh? Here's Bombelli's

8/ workaround: Ok, "imagine" that all of this makes sense, which it doesn't, and notice (writing i=sqrt(-1)) that:

(2+i)^3=2^3+3*2^2*i+3*2*i^2+i^3=2+sqrt(-121).

So we must have that

u=crt[2+sqrt(-121)]=2+i.

And similarly, v=-2+i. So:

y=u-v=(2+i)-(-2+i) = 4!!!

This nonsense

9/ isn't "real", but it's a *useful tool* to get at the real answer. Complex numbers existed in this limbo long before they became "the complex plane" in work of Euler/Argand/Wessel. But there you have it, complex numbers were needed well before negatives!

#mbtos #iteachmath

PS A great exposition of this story, as well as the Cardano/Tartaglia feud, is in Dunham's fantastic book, "A Journey Through Genius" (which is where I first read about it).