The quadratic formula. With the exception of the Pythagorean Theorem, it's probably the single most common mathematical formula people carry from high school. It's not a function as such, it's something that solves a function. Let me give an example:
Pick a number x, square it, add twice x to that, subtract 3 from that. You might want to figure out what x makes the answer equal to zero. It's the kind of thing we have to do in physics all the time. In this case, the answers (and you can find them by several methods) happen to be -3 and 1. Plug them in and you'll see that you end up with 0 as the answer.
In general if you have a quadratic equation like this:
The particular x that solves this equation (and there will be two of them - though they might be complex numbers, and they might be the same number, long story...) will be given by the quadratic formula:
Not so bad. But what if we're trying to solve a cubic equation, like this?
Well, just like the quadratic equation had two solutions, the cubic equation will have three. Here is the first:
Can you even read that? It's pretty tough for me. There's two more of course, for the other two solutions.
What about the quartic equation? It looks like this:
We can do it too. I'll spare you the horror, but it's about as much uglier than the cubic equation than the cubic equation is uglier than the quadratic equation. And there's four solutions, not just three.
What about the quintic equation? The one with an x^5 term?
Surprisingly - or perhaps mercifully - there isn't a solution. Some mathematicians much smarter than me proved it in 1824. At least there isn't a solution in anything with just fractions, powers, and roots like the ones we've seen here. But sometimes you really do need the actual numbers that solve quintic or even higher equations. Fortunately there's tricks we can do. These days computers automate most of them, but even before computers it was possible to get a decimal approximation to the solutions of these equations without needing an exact algebraic solution. And to be totally honest, that's pretty much all we ever do for any polynomial worse than the quadratic.
It's important to remember that there really are real live numbers that solve these polynomials of 5th and higher order, just not numbers we can express in terms of the standard arithmetic operations. But we're physicists. Experiments pretty much always give us decimal approximations of reality anyway, so we're used to it.
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"From each according to his ability, to each acording to his needs." Plus compost.
Dude, you really ought to put all these Sunday Function posts together in a book.
The cubic formula is quite unwieldy; it's something I can't carry in my head. I recall it's often easier to present if you declare 2 variables (often called p and q) and a determinant.
It's called "Handbook of Mathematical Functions", Edited by Abramowitz and Stegun", U.S. Department of Commerce (National Bureau of Standards).
To be a bit pedantic: there are algebraic solutions for some versions of the quintic equation, but no general algebraic solution that works on all quintics.
If people would like a book, I can recommend Mario Livio's book "The equation that couldn't be solved". It's light on equations, but covers the history of attempts to tackle the quintic equations, including the remarkable stories of Abel and Galois.
You may not know that solving that quadratic equation was one of the main problems that drove the creation of algebra itself by Arab mathematicians. Learned that from one of my math colleagues who teaches "math for the masses".
Since trig tables were being constructed at about the same time, I suspect they had to formulate the algorithm for completing the square so there was a means to compute the root needed to evaluate the half angle identities used to work your way from special angles to other ones.
"decimal approximations of reality".
What a great phrase.
There are schemes for computing exactly with the roots of any polynomial, one of them is implemented in Mathematica. After a long exact calculation you can convert the answer to decimal form with arbitrary precision. Problem is, it may take a lot of speed and memory to push the calculation through to the end.
"It's important to remember that there really are real live numbers that solve these polynomials of 5th and higher order, just not numbers we can express in terms of the standard arithmetic operations."
Well the solutions don't have to be real. Some polynomials of even degree do not have any real roots.
You may not know that solving that quadratic equation was one of the main problems that drove the creation of algebra itself by Arab mathematicians. Learned that from one of my math colleagues who teaches "math for the masses"
Iâm sorry to have to tell you this, CCPhysicist but your professor for âmath for the massesâ is wrong! The general solution to the quadratic equation was know to the Babylonians in the so called Old Babylonian period possibly as early as 1700 BCE. It was also known to the ancient Greeks and can be found in its geometrical form in Euclidâs âElementsâ from about 400 BCE; both sources were known to the Islamic algebraists. It is Euclidâs geometric proof that can be found in the book al-KitÄb al-muḫtaá¹£ar fÄ« ḥisÄb al-Äabr wa-l-muqÄbala by Ä al-KhwÄrizmÄ«. The Islamic algebraists further developed algebra in that they developed methods for finding individual solutions to specific forms of the cubic, bi-quadratic and quintic equations and in some cases polynomials of higher degrees, those susceptible to substitution.
Another book that has a great take on the solving of polynomial equations is Marcus du Sautoy's Symmetry. He's does a wonderful job of relating the history of finding solutions and placing the general problem in the context of symmetry.