Algebra 1 Linear inequalities Overview Solving linear inequalities Solving compound inequalities Solving absolute value equations and inequalities Linear inequalities in two variables.
Algebra 1 Systems of linear equations and inequalities Overview Graphing linear systems The substitution method for solving linear systems The elimination method for solving linear systems Systems of linear inequalities. Algebra 1 Exponents and exponential functions Overview Properties of exponents Scientific notation Exponential growth functions.
Algebra 1 Factoring and polynomials Overview Monomials and polynomials Special products of polynomials Polynomial equations in factored form. Algebra 1 Quadratic equations Overview Use graphing to solve quadratic equations Completing the square The quadratic formula.
Algebra 1 Radical expressions Overview The graph of a radical function Simplify radical expressions Radical equations The Pythagorean Theorem The distance and midpoint formulas. The imaginary number first appeared in print in the year An imaginary number possesses the unique property that when squared, the result is negative. Consider: The process of simplifying a radical containing a negative factor is the same as normal radical simplification.
The only difference is that the will be replaced with an " i ". As research with imaginary numbers continued, it was discovered that they actually filled a gap in mathematics and served a useful purpose. Imaginary numbers are essential to the study of sciences such as electricity, quantum mechanics, vibration analysis, and cartography. When the imaginary i was combined with the set of Real Numbers, the all encompassing set of Complex Numbers was formed. In Algebra 1, you will see that the "imaginary" number will be useful when solving quadratic equations.
We can no long impose the same kind of continuity conditions and get a straight answer - instead we have to form a sort of "barricade" in which the value of the square root jumps dramatically when we cross over this barricade. This is known as a branch cut.
The default choice, usually unspoken, is the standard branch. The idea of branch cuts leads into more advanced complex analysis topics of monodromy which pertains to "running around" a singularity, like crossing over the branch mentioned earlier and also Riemann surfaces , which can be thought of as what we get when we refuse to cut the plane into branches and instead consider a function multi-valued and look at its graph I am probably butchering that description though. Any number times itself is a positive number or zero , so you can't ever get to a negative number by squaring.
Since square roots undo squaring, negative numbers can't have square roots. First, we have to specify what set of numbers you are working in, because otherwise, the question is meaningless. Yes; they are called the "complex numbers". But because we're making a new number system, we can make up whatever we want. A side note: like the real numbers, there are two square roots for each number.
With the real numbers, we can choose the positive one as our "principal" square root. A positive times a positive is positive. Using this argument we have seen that the square root of a negative number cannot be positive or negative.
When we say that the square root of a negative number "doesn't exist", we mean that there is no real number solution. So firstly, the algorithm to compute square roots I'm guessing you're looking for is the one originating from the Newton Newton-Raphson Method. There's other methods but I'll post this one. Now the problem or so it seems with Newton's Method is it's stability.
There's also examples which I'd link but I'm restricted to only two links of functions for which Newton's Method cycles and for which it diverges for given initial conditions, which have real roots. Anyway if you wanted you could look up Newton's fractal it has some info for sequences converging to complex roots. A number multiplied by itself cannot be negative, so the square root of a negative number is not possible imaginary, I believe.
Those cool displays you see when music is playing? Yep, Complex Numbers are used to calculate them! Using something called "Fourier Transforms". In fact many clever things can be done with sound using Complex Numbers, like filtering out sounds, hearing whispers in a crowd and so on.
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