Shoutbox

Excellent and interesting question though. But, you will learn that in the futur. It's the same as you didn't learn about integrals and deviations yet. Everything will come in time, and you first need to understand other things before you can understand this

Anyways, here is a shot (and some interesting history of maths)...

No, taking a square root of a negative number is, as you pointed out, impossible in the real world. However, that doesn't mean you can't do maths with the "impossible". It's something like the number "infinity". Infinity doesn't exist in real life, everything is limited. But you still need something to do maths with and to represent "something" beyond our scope of thinking.

This is the same for taking a square root of a negative number. In some calculations like quadratic and cubic equations (like used ealier in this thread) you will see that you can have a result which actually can't exist, but still it _is_ a result.

In the past, people stopped there, they couldn't go further. But one day, somebody said, what if we just invent a new number, and lets call it i (from "imaginary number"), i would be equal to the square root to -1. From that time on people could do maths and work out solutions further where they were stuck in the past.

note: such things happened in the past also (and still are happening now). It is only since the 13th century that negative numbers were slowly considered real numbers in the Western world. Before that, anything lower then 0 simpy didn't exist.

So, a whole new group of numbers emerged, the "complex numbers" written as "(x, y)" where x is a real number and y the imaginary number. So the complex number (5, 4) means 5+4i or 5+4*SQR(-1)...

So you can do maths with them too of course:
(5, 4) + (5, 4) = 5+4i + 5+4i = 10 + 8i

And you can even solve this strange question: divide the number 10 into two parts so that their product is 40! Impossible you would think, well not with complex numbers, the solution: 5+SQR(–15) and 5–SQR(–15). The proof:

[5+SQR(–15)]  +  [5–SQR(–15)]
=>  5 + 5 + SQR(–15) - SQR(–15)
=>  5 + 5
=>  10, hey presto!

[5+SQR(–15)]  *  [5–SQR(–15)]
=>  5²  -  5*SQR(–15)  +  5*SQR(–15)  -  SQR(–15)²
=>  25  -  -15
=>  25  +  15
=>  40, hey presto!

Note: this is actually a "famous" calculation/question from Cardano (16th century). He didn't actualy invented the imaginary number i and also didn't do anything further with this calculation or understood what he just found. This was actually the result of a competition to solve the cubic equation. Which was thought to be impossible. Cardona solved it by using negative numbers (a great controversy in that time) and this "complex" number thing was only a small byproduct of it.

Cardano did not go further into this, and it is only later that they were called complex numbers. A few years later Bombelli gave several examples involving these new type of numbers.

eg:

One of Cardano's cubic formulas gives the solution to the equation:
x³ = cx + d
as
x = SQR³[d/2 + SQR(e)] + SQR³[d/2 – SQR(e)]    where e = (d/2)² – (c/3)³
I've used "SQR³(x)" to indicate the third root of x

The 'problem' is that e could be negative, but there isn't any real number which you can multipy by itself and get a negative number, so Cardano was stuck there. Nevertheless Bombelli used this formula to solve the equation:
x³ = 15x + 4

according to Cardano's formula:
c = 15
d = 4
thus e = (4/2)² – (15/3)³  =>  2² - 5³  =>  -121  (a negative number no less!!!)
and thus x = SQR³[2 + SQR(–121)] + SQR³[2 – SQR(–121)]

Now, the square root of –121 is not a real number; it's neither positive, negative, nor zero. Nevertheless, Bombelli continued to work with this expression until he found equations that lead him to the solution 4, a real number!
So there _is_ a real solution eventhough SQR(–121) can't exist.

This shows nicely how complex numbers do have their use to solve things, even though they can't exist. Prior to the discovery of this, people said there was no solution. Although 4³ is equal to 15 * 4 + 4....


There is more detail to this story though:

When cubic formulas were a big hush-hush in the 16th century, Cardano noted that the sum of the three solutions to a cubic equation (x³ + bx² + cx + d = 0) is –b, the negation of the coefficient of x².

By the 17th century the theory of equations had developed so far so that someone called Girard developped a principle of algebra out of this. He also said that an nth degree equation always had n solutions. This is what we now call "the fundamental theorem of algebra". But He never was able to proof this principle though, just because some solutions couldn't "exist".

Another guy also studied this relation between solutions and coefficients, and was able to show that Cardano and Girard were right if you disregarded negative solutions as "false" and those other impossible solutions as "imaginary".

And slowly negative numbers became a common good and raised in status (by the results and efforts of other people), but it was not until the 18th century that complex numbers became in real use, roughly 200 years after their first "discovery" . They weren't considered to be real numbers though, but they were useful in the theories and formulas back in that time.

But still nobody was able to proof the fundamental theorem of algebra which stated that there are n solutions of an nth degree equation. So, although the complex numbers were not fully understood, the square root of -1 was being used more and more.

By the end of the 18th century numbers of the form x + yi were used pretty often by research mathematicians. For example Euler, a very influencial mathematician used it often in his trigonometric functions. And because of this and his influences, it was Gauss, and others like Wessel and Argand, who began to study complex numbers in function of xy-planes (as coordinates of planes) and that complex numbers were fully understood. And thus in 1799, Gauss published the first proof to the Girard's principle of algebra.

(You might know Gauss from the "Gauss-curve", it's the same one. And Euler is also a name which you certainly will hear in the next years in school, if you already didn't. And this whole deal with complex numbers will certainly be on the agenda and you'll notice that it will all start with talk about coördinates of planes and the polar coördinates system )