rancid wraith
I was watching https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:complex/x9e81a4f98389efdf:complex-div/v/complex-conjugates and it didn't explain its origin. Is there a reason it was made like this, or was it just a convention?
I had no idea which category this question goes into, any idea which?
charred surgeBOT
gloomy wasp
i guess it's hard to talk about its "origin" but it likely comes from the property that any complex number multiplied by its conjugate produces a real number
which is useful when it comes to stuff like rewriting 1 / (a + bi) in the form c + di
tall oyster
Conjugates of a number are numbers that always appear as roots of a polynomial (where coefficients are of certain type) when the first one appears as a root.
When we are talking about polynomials with real coefficients, complex roots always appear in pairs and the roots are conjugates of each other. Therefore we say a + bi and a - bi are conjugates over real numbers.
With the same idea you could call √2 and -√2 conjugates over rationals because if you have a polynomial with rational coefficients having one of them as a root, the other one is also a root and we would say that √2 and -√2 are conjugates over rational numbers.
Historically complex numbers were discovered when trying to solve cubic equations. It was noticed that solutions of a cubic equation given by the cubic formula sometimes gives real roots using an expression containing square roots of negative numbers. Treating √-1 as an imaginary number i would then allow one to simplify the real solution. Sometimes cubic equations have only one real solution which means that i doesn't get eliminated entirely. Then someone eventually noticed that these non-real solutions seem to always appear in pairs and later someone started calling the solutions conjugates.
tall oyster
i guess it's hard to talk about its "origin" but it likely comes from the proper...
And this is indeed how we can characterize conjugates in the quadratic case using Vieta's formulas which includes complex conjugates. We have that sum and product of conjugates are real.
(And in similar manner sum and product of √2 and -√2 are rational)
rancid wraith
noticed that these non-real solutions seem to always appear in pairs and later someone started calling the solutions conjugates
Can you show me this situation before complex number was invented?